BusyBeaverWiki - User contributions [en]

CounterScript

2026-04-22T19:03:24Z

Azerty: /* Champions */ Set BBCS(8) and BBCS(9) status to proven.

'''CounterScript''' is a minimal counter-based programming language designed for '''Busy Beaver–style program enumeration and analysis'''.

It features a very small instruction set operating over an unbounded set of nonnegative integer variables (''counters'').

'''BBCS'''(n) is the Busy Beaver function for '''CounterScript''' programs.

= Definition =
A CounterScript program consists of a finite sequence of instructions drawn from three primitives:

* Increment a counter
* Conditional decrement
* While loop

All counters are initialized to <code>0</code> and can grow without bound.

The model is closely related to '''Minsky register machines,''' '''Fractran''' and '''Brainfuck''', and is therefore [[Turing-complete]].

== Instructions ==
Instructions are executed one by one from left to right.
{| class="wikitable"
|+
!Name
!Command
!Definition
|-
|Increment
|<code>A++</code>
|Increment counter A by 1
|-
|Decrement
|<code>A--</code>
|Decrement counter A by 1 if A > 0
|-
|While loop
|<code>while A {...}</code>
|Executes the loop body while <code>x > 0</code>
|}

= Program size =
The '''length''' of a CounterScript program is defined as the total number of instructions.

By example, <code>A++; while A {A--; B++;}</code> has length 4.

== Champions ==
<div class="toccolours mw-collapsible">
{| class="wikitable"
!n
!BBCS(n)
!Champion
|-
|1
|1
|<code>A++;</code>
|-
|2
|2
|<code>A++; A++;</code>
|-
|3
|3
|<code>A++; A++; A++;</code>
|-
|4
|4
|<code>A++; A++; A++; A++;</code>
|-
|5
|5
|<code>A++; A++; A++; A++; A++;</code>
|-
|6
|6
|<code>A++; A++; A++; A++; A++; A++;</code>
|-
|7
|7
|<code>A++; A++; A++; A++; A++; A++; A++;</code>
|-
|8
|9
|<code>A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|9
|12
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|10
|≥16
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|-
|11
|≥20
|<code>A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|-
|12
|≥25
|<code>A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++; B++;}</code>
|}
</div>

[[Category:Functions]]

Busy Beaver for lambda calculus

2026-04-11T16:35:57Z

Azerty: /* De Bruijn */

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>\ge 10^{10^4}</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits]. 28 Jan 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

Busy Beaver for lambda calculus

2026-04-11T16:35:15Z

Azerty: /* De Bruijn */

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>\ge 10^{10^4}</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable" mw-collapsible
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits]. 28 Jan 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

Busy Beaver for lambda calculus

2026-04-11T16:34:16Z

Azerty: /* De Bruijn */

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>\ge 10^{10^4}</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable" mw="collapsible"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits]. 28 Jan 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

CounterScript

2026-04-11T16:32:22Z

Azerty: /* Champions */

'''CounterScript''' is a minimal counter-based programming language designed for '''Busy Beaver–style program enumeration and analysis'''.

It features a very small instruction set operating over an unbounded set of nonnegative integer variables (''counters'').

'''BBCS'''(n) is the Busy Beaver function for '''CounterScript''' programs.

= Definition =
A CounterScript program consists of a finite sequence of instructions drawn from three primitives:

* Increment a counter
* Conditional decrement
* While loop

All counters are initialized to <code>0</code> and can grow without bound.

The model is closely related to '''Minsky register machines,''' '''Fractran''' and '''Brainfuck''', and is therefore [[Turing-complete]].

== Instructions ==
Instructions are executed one by one from left to right.
{| class="wikitable"
|+
!Name
!Command
!Definition
|-
|Increment
|<code>A++</code>
|Increment counter A by 1
|-
|Decrement
|<code>A--</code>
|Decrement counter A by 1 if A > 0
|-
|While loop
|<code>while A {...}</code>
|Executes the loop body while <code>x > 0</code>
|}

= Program size =
The '''length''' of a CounterScript program is defined as the total number of instructions.

By example, <code>A++; while A {A--; B++;}</code> has length 4.

== Champions ==
<div class="toccolours mw-collapsible">
{| class="wikitable"
!n
!BBCS(n)
!Champion
|-
|1
|1
|<code>A++;</code>
|-
|2
|2
|<code>A++; A++;</code>
|-
|3
|3
|<code>A++; A++; A++;</code>
|-
|4
|4
|<code>A++; A++; A++; A++;</code>
|-
|5
|5
|<code>A++; A++; A++; A++; A++;</code>
|-
|6
|6
|<code>A++; A++; A++; A++; A++; A++;</code>
|-
|7
|7
|<code>A++; A++; A++; A++; A++; A++; A++;</code>
|-
|8
|≥9
|<code>A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|9
|≥12
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|10
|≥16
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|}
</div>

CounterScript

2026-04-11T16:32:00Z

Azerty: Created page

'''CounterScript''' is a minimal counter-based programming language designed for '''Busy Beaver–style program enumeration and analysis'''.

It features a very small instruction set operating over an unbounded set of nonnegative integer variables (''counters'').

'''BBCS'''(n) is the Busy Beaver function for '''CounterScript''' programs.

= Definition =
A CounterScript program consists of a finite sequence of instructions drawn from three primitives:

* Increment a counter
* Conditional decrement
* While loop

All counters are initialized to <code>0</code> and can grow without bound.

The model is closely related to '''Minsky register machines,''' '''Fractran''' and '''Brainfuck''', and is therefore [[Turing-complete]].

== Instructions ==
Instructions are executed one by one from left to right.
{| class="wikitable"
|+
!Name
!Command
!Definition
|-
|Increment
|<code>A++</code>
|Increment counter A by 1
|-
|Decrement
|<code>A--</code>
|Decrement counter A by 1 if A > 0
|-
|While loop
|<code>while A {...}</code>
|Executes the loop body while <code>x > 0</code>
|}

= Program size =
The '''length''' of a CounterScript program is defined as the total number of instructions.

By example, <code>A++; while A {A--; B++;}</code> has length 4.

== Champions ==
<div class="toccolours mw-collapsible mw-collapsed">
{| class="wikitable"
!n
!BBCS(n)
!Champion
|-
|1
|1
|<code>A++;</code>
|-
|2
|2
|<code>A++; A++;</code>
|-
|3
|3
|<code>A++; A++; A++;</code>
|-
|4
|4
|<code>A++; A++; A++; A++;</code>
|-
|5
|5
|<code>A++; A++; A++; A++; A++;</code>
|-
|6
|6
|<code>A++; A++; A++; A++; A++; A++;</code>
|-
|7
|7
|<code>A++; A++; A++; A++; A++; A++; A++;</code>
|-
|8
|≥9
|<code>A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|9
|≥12
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|10
|≥16
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|}
</div>

TMBR: March 2026

2026-04-10T16:20:18Z

Azerty: /* BB Adjacent */ Added CounterScript

{{TMBRnav|February 2026|April 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for March 2026. We celebrated bbchallenge's fourth birthday on 8 March. This month was quite a substantial month in Beaver research, as after more than 200 days, the last 3 informally solved [[BB(3,3)]] holdouts were [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised by mxdys in Rocq.] Following the now-tradition, 3 '''new''' [[BB(2,5)]] machines were proven to not halt! This leaves us with 60 informal holdouts. There has been a 4.37% reduction in [[BB(6)]], and quite a lot of results in [[Fractran]]: With the help of AI Agent Claude Opus 4.6, it was proven tentatively that BBf(21) = 31,957,632 and that a [[Cryptid]] exists in BBf(22), which means that [[Fractran#Fenrir|BBf(22) Is Hard]].

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]] which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Misc ==
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}} was shown to be a probviously non-halting [[Cryptid]] similar to [[Space Needle]] by DrDisentangle.
* mxdys made a new Turing Machine Visualizer using longitudinal acceleration (for shift-overflow mixed-digit non-unary counters)[https://discord.com/channels/960643023006490684/1239205785913790465/1488164940303958136 <nowiki>[1]</nowiki>] which is available at https://ccz181078.github.io/TM/LongAcc/index.html.

== Meta ==

* Famous math Youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== BB Adjacent ==
* [[Fractran]]: In BBf(21), Claude Opus 4.6 gave a proof that all 140 holdouts do not halt. This tentatively proves that BBf(21) = 31,957,632.
* [[Fractran]]: A Cryptid was discovered in BBf(22) with the help of Claude Opus 4.6, which was named [[wikipedia:Fenrir|Fenrir]] from [[wikipedia:Norse_mythology|Nordic mythology.]]
* [[Fractran]]: Katelyn Doucette started working on a program to visualize fractran spacetime diagrams just like for TMs:
** [https://discord.com/channels/960643023006490684/1438019511155691521/1484649168638185689 Frankenstein's Monster]
** [https://discord.com/channels/960643023006490684/1438019511155691521/1484648223133012149 BBf(20) champion]
** [https://discord.com/channels/960643023006490684/1438019511155691521/1484692739961655356 Hydra]
** [https://discord.com/channels/960643023006490684/1438019511155691521/1484692800917344366 Space Needle]
*Discord user azertyordinal introduced an esolang called CounterScript and its Busy Beaver function BBCS.
**Discord user realsheepthe made a cryptid of length 23 that computes a variation of [[Hydra function]] and a length 92 program that may surpass Graham's number.

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1214
|1161
|53
|4.37%
|-
|[[BB(7)]]
|18,195,192
|18,036,852
|158,340
|0.87%
|-
|[[BB(2,5)]]
|72
|69
|3
|4.17%
|-
|[[BB(2,6)]]
|548,993
|545,005
|3,988
|0.73%
|}
* [[BB(6)]]: 1e14 machines: '''170'''. 1e15 machines: '''237'''. '''53 solved machines'''.
** [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Holdouts lists] for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines[https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775 <nowiki>[3]</nowiki>] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more[https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600 <nowiki>[5]</nowiki>] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], and then [https://discord.com/channels/960643023006490684/1477591686514212894/1483655806917279816 another 9 days later], therefore reducing that holdout count by 2.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 released] a new holdout list of '''1161''' machines. The new informal holdout count is '''1159''', and the Rocq-verified holdout count is '''1187.'''
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to '''18,036,852''' (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR] and mxdys confirmed two[https://discord.com/channels/960643023006490684/1259770421046411285/1483043448855461989 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1483043657778069564 <nowiki>[7]</nowiki>] machines to be non-halting. Thus, the new holdout count is '''69''', or '''60''' considering informal proofs.
*[[BB(2,6)]]:
**A new filtering run by Andrew Ducharme has reduced to number of holdouts from 548,993 to '''545,005'''.[https://discord.com/channels/960643023006490684/1084047886494470185/1487675435473961132 <nowiki>[8]</nowiki>]
*[[BB(3,3)]]:
**mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] three remaining informal results (650, 412, 279) into Rocq.

[[Category:This Month in Beaver Research|2026-03]]

1RB1RC 1LC1LE 1RA1RD 0RF0RE 1LA0LB ---1RA

2026-03-31T19:56:52Z

Azerty: /* Analysis by @mxdys */

{{machine|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA}}{{unsolved|Does this TM run forever?}}
{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA}} is a [[BB(6)]] [[Cryptid]] found by @mxdys and shared on Discord on 27 Jul 2024. It's closely related to [[Hydra]] and [[Antihydra]].

== Analysis by @mxdys ==
<pre>
antihydra variant:
1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA

(n,m) := 0^inf C> 01011 01^n 1 01^m 0^inf

start from (3,1)
(2n+1,m) --> (3n+3,m+2)
(2n,m+1) --> (3n+3,m)
(4n,0) --> (9n+6,1)
(4n+2,0) --> halt
</pre>
The rules are proven in [https://github.com/ccz181078/busycoq/blob/BB6/verify/AntiHydra2.v Rocq].

== Analysis by @dyuan01 ==
Using B(a, b) = (3a-6, b), I've come up with these rules:
<pre>
Start from B(3, 1)
B(2n, m+1) → B(3n, m)
B(2n+1, m) → B(3n+1, m+2)
B(4n+2, 0) → B(9n+4, 1)
B(4n, 0) → halt
</pre>
This is actually just Hydra except you start at B(3, 1) instead of B(3, 0) and you don't necessarily halt if you reach -1. In fact, this can only halt if Hydra halts.

[[Category:BB(6)]][[Category:Cryptids]]

Busy Beaver for lambda calculus

2026-03-23T21:22:11Z

Azerty: /* De Bruijn */ Updated length 14 champion.

Brainfuck

2026-02-11T07:58:05Z

Azerty: Added some informations about how bf works.

Brainfuck is a Turing complete language invented by Urban Muller.

Brainfuck operates on an array of memory cells, each initially set to zero. In the original implementation, the array was 30,000 cells long and each cell values would be limited to numbers from 0 to 255, but this may not be part of the language specification. A braincuk program can work with different, and sometimes unbounded, array length and cells size.

Like in a [[Turing machine]], there is a pointer, initially pointing to the first memory cell.

Programs are composed of commands, represented with symbols. The commands are:

TODO

[page under construction]

User:Azerty/Lambda Calculus Busy Beaver

2026-02-07T17:34:48Z

Azerty: Created page with "We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers. For n ≤ 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20. {| class="wikitable" !BBλ_db(n)..."

We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n ≤ 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}
Most champions are currently taken from binary BBλ.

Busy Beaver for lambda calculus

2026-02-06T07:40:49Z

Azerty: /* De Bruijn */

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n. Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) is not defined for those values. The smallest closed lambda term is <code>\1</code> which has size 4.

For the rest of n ≤ 20: BBλ(n) = n is trivial and can be achieved via picking any n bit term already in normal form. For example <code>\\...\1</code> and <code>\\...\2</code> with k lambdas has size 2k+2 and 2k+3 respectively (for k ≥ 1 and k ≥ 2 respectively).

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|21 || = 22 || <code>\(\1 1) (1 (\2))</code>
| <code>\(1 (\2)) (1 (\2))</code> || John Tromp & Bertram Felgenhauer
|-
|22 || = 24 || <code>\(\1 1 1) (1 1)</code>
| <code>\(1 1) (1 1) (1 1)</code> || John Tromp & Bertram Felgenhauer
|-
|23 || = 26 || <code>\(\1 1) (1 (\\2))</code>
| <code>\(1 (\\2)) (1 (\\2))</code> || John Tromp & Bertram Felgenhauer
|-
|24 || = 30 || <code>\(\1 1 1) (1 (\1))</code>
| <code>\(1 (\1)) (1 (\1)) (1 (\1))</code> || John Tromp & Bertram Felgenhauer
|-
|25 || = 42 || <code>\(\1 1) (\1 (2 1))</code>
| <code>\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|26 || = 52 || <code>(\1 1) (\\2 (1 2))</code>
| <code>\\2 (\\2 (1 2)) (1 (2 (\\2 (1 2))))</code> || John Tromp & Bertram Felgenhauer
|-
|27 || = 44 || <code>\\(\1 1) (\1 (2 1))</code>
| <code>\\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|28 || = 58 || <code>\(\1 1) (\1 (2 (\2))))</code>
| <code>\1 (\\1 (3 (\2))) (1 (\2 (\\1 (4 (\2)))))</code>|| John Tromp & Bertram Felgenhauer
|-
| 29 || = 223|| <code>\(\1 1) (\1 (1 (2 1)))</code>
| <pre>
\B (B (1 B))
where:
B = (A (A (1 A)))
A = (1 (\1 (1 (2 1))))
</pre>||John Tromp & Bertram Felgenhauer
|-
|30
|= 160
|<code>(\1 1 1) (\\2 (1 2))</code>
| <pre>
\\2 B A (1 (2 B A))
where:
B = (\\2 A (1 (2 A)))
A = (\\2 (1 2))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|31
|= 267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
| <pre>
\\2 A (2 A (C (2 A)))
where:
C = (2 A (2 A (1 B (2 A))))
B = (\3 A (3 A (1 (3 A))))
A = (\\2 (2 (1 2)))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|32
|= 298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|
|John Tromp & Bertram Felgenhauer
|-
|33
|= 1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
| <pre>
\C (C (C (1 C)))
where:
C = (B (B (B (1 B)))
B = (A (A (A (1 A)))
A = (1 (\1 (1 (1 (2 1)))))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|34 || <math>= \ 5 \left(2^{2^{2^2}}\right) + 6 = 327\,686</math>
| <code>(\1 1 1 1) (\\2 (2 1))</code>
| <math>C(2^{2^{2^2}})</math>|| John Tromp & Bertram Felgenhauer
|-
|35 || <math>= 5 \left(3^{3^3}\right) + 6 > 3.8 \times 10^{13}</math>
| <code>(\1 1 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^3})</math>|| John Tromp & Bertram Felgenhauer
|-
|36 || <math>= 5 \left(2^{2^{2^3}}\right) + 6 > 5.7 \times 10^{77}</math>
| <code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
| <math>C(2^{2^{2^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|37 || <math> = BB\lambda(35)+2</math>
| <code>\(\1 1 1) (\\2 (2 (2 1)))</code>
| <math>\lambda x. C(3^{3^3})</math>||mxdys, tromp, dyuan, sligocki
|-
|38 || <math>\ge 5 \left(2^{2^{2^{2^2}}}\right) + 6 > 10^{10^4}</math>
| <code>(\1 1 1 1 1) (\\2 (2 1))</code>
| <math>C(2^{2^{2^{2^2}}})</math>|| John Tromp & Bertram Felgenhauer
|-
|39 || <math>\ge 5 \left(3^{3^{3^3}}\right) + 6 > 10^{10^{12}}</math>
| <code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^{3^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|40 || <math> > (2\uparrow\uparrow)^{15} 33 > 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{15} 33</math>|| mxdys and racheline
|-
|41 || <math>\ge 5 \left(3^{3^{85}}\right) + 6 > 10^{10^{40}}</math>
| <code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge BB\lambda(40)+2</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math> and <math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge BB\lambda(43)+2</math>
| <code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| ||
|-
|46 || <math> \ge BB\lambda(44)+2</math>
| <code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| ||
|-
|47 ||
|
| ||
|-
|48 || <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>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
<math>> \text{Graham's number}</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|<math>C(N) \text{ for } N \approx f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|<math>C(N) \text{ for } N \approx f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|
{| class="wikitable"
|-
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|}
|
|Patcail
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|Patcail
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
<math>> \text{TREE}(G64)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|331
| lim(BMS)
|<code>too large to show</code>
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & John Tromp & 50_ft_lock]
|
|-
|1850
|> Loader's number
|<code>too large to show</code>
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 John Tromp]
|}

Where <math>C(n)</math> represents the Church numeral ''n'' (<math>\lambda f x. f^n(x)</math>) written as <code>\\2 (2 ... (2 1)...)</code> with ''n'' 2s in this text representation.

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<code>1</code>
|1
|-
|3
|
|0
|-
|4
|<code>\1</code>
|4
|-
|5
|<code>\2</code>
|5
|-
|6
|<code>\\1</code>
|6
|-
|7
|<code>\\2</code>
|7
|-
|8
|<code>1 (\1)</code>
|<math>f(4) = 26</math>
|-
|9
|<code>\\2</code>
|9
|-
|10
|<code>1 (\\1)</code>
|<math>f(6) = 36</math>
|-
|11
|<code>1 (\\2)</code>
|<math>f(7) = 41</math>
|-
|12
|<code>1 (1 (\1))</code>
|<math>f^{2}(4) = 266</math>
|-
|13
|<code>1 (\\2)</code>
|<math>f(9) = 51</math>
|-
|14
|<code>1 (1 (\\1))</code>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<code>1 (1 (\\2))</code>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<code>1 (1 (1 (\1)))</code>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<code>1 (1 (\\\2))</code>
|<math>f^2(9) = f(51)</math>
|-
|18
|<code>1 (\1) 1 (\1)</code>
|<math>f^4(4) </math>
|-
|19
|<code>1 (1 (1 (\\2)))</code>
|<math>f^3(7)</math>
|-
|20
|<code>1 (\\1) 1 (\1)</code>
|<math>f^6(4)</math>
|-
|21
|<code>1 (\\2) 1 (\1)</code>
|<math>f^7(4)</math>
|-
|22
|<code>1 (1 (\1)) 1 (\1)</code>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2023/11/24/largest-number The largest number representable in 64 bits]. 24 Nov 2023. John Tromp.
* [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

Busy Beaver for lambda calculus

2026-02-05T19:57:34Z

Azerty: /* De Bruijn */ Added values for n < 7

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n. Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) is not defined for those values. The smallest closed lambda term is <code>\1</code> which has size 4.

For the rest of n ≤ 20: BBλ(n) = n is trivial and can be achieved via picking any n bit term already in normal form. For example <code>\\...\1</code> and <code>\\...\2</code> with k lambdas has size 2k+2 and 2k+3 respectively (for k ≥ 1 and k ≥ 2 respectively).

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|21 || = 22 || <code>\(\1 1) (1 (\2))</code>
| <code>\(1 (\2)) (1 (\2))</code> || John Tromp & Bertram Felgenhauer
|-
|22 || = 24 || <code>\(\1 1 1) (1 1)</code>
| <code>\(1 1) (1 1) (1 1)</code> || John Tromp & Bertram Felgenhauer
|-
|23 || = 26 || <code>\(\1 1) (1 (\\2))</code>
| <code>\(1 (\\2)) (1 (\\2))</code> || John Tromp & Bertram Felgenhauer
|-
|24 || = 30 || <code>\(\1 1 1) (1 (\1))</code>
| <code>\(1 (\1)) (1 (\1)) (1 (\1))</code> || John Tromp & Bertram Felgenhauer
|-
|25 || = 42 || <code>\(\1 1) (\1 (2 1))</code>
| <code>\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|26 || = 52 || <code>(\1 1) (\\2 (1 2))</code>
| <code>\\2 (\\2 (1 2)) (1 (2 (\\2 (1 2))))</code> || John Tromp & Bertram Felgenhauer
|-
|27 || = 44 || <code>\\(\1 1) (\1 (2 1))</code>
| <code>\\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|28 || = 58 || <code>\(\1 1) (\1 (2 (\2))))</code>
| <code>\1 (\\1 (3 (\2))) (1 (\2 (\\1 (4 (\2)))))</code>|| John Tromp & Bertram Felgenhauer
|-
| 29 || = 223|| <code>\(\1 1) (\1 (1 (2 1)))</code>
| <pre>
\B (B (1 B))
where:
B = (A (A (1 A)))
A = (1 (\1 (1 (2 1))))
</pre>||John Tromp & Bertram Felgenhauer
|-
|30
|= 160
|<code>(\1 1 1) (\\2 (1 2))</code>
| <pre>
\\2 B A (1 (2 B A))
where:
B = (\\2 A (1 (2 A)))
A = (\\2 (1 2))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|31
|= 267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
| <pre>
\\2 A (2 A (C (2 A)))
where:
C = (2 A (2 A (1 B (2 A))))
B = (\3 A (3 A (1 (3 A))))
A = (\\2 (2 (1 2)))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|32
|= 298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|
|John Tromp & Bertram Felgenhauer
|-
|33
|= 1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
| <pre>
\C (C (C (1 C)))
where:
C = (B (B (B (1 B)))
B = (A (A (A (1 A)))
A = (1 (\1 (1 (1 (2 1)))))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|34 || <math>= \ 5 \left(2^{2^{2^2}}\right) + 6 = 327\,686</math>
| <code>(\1 1 1 1) (\\2 (2 1))</code>
| <math>C(2^{2^{2^2}})</math>|| John Tromp & Bertram Felgenhauer
|-
|35 || <math>= 5 \left(3^{3^3}\right) + 6 > 3.8 \times 10^{13}</math>
| <code>(\1 1 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^3})</math>|| John Tromp & Bertram Felgenhauer
|-
|36 || <math>= 5 \left(2^{2^{2^3}}\right) + 6 > 5.7 \times 10^{77}</math>
| <code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
| <math>C(2^{2^{2^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|37 || <math> = BB\lambda(35)+2</math>
| <code>\(\1 1 1) (\\2 (2 (2 1)))</code>
| <math>\lambda x. C(3^{3^3})</math>||mxdys, tromp, dyuan, sligocki
|-
|38 || <math>\ge 5 \left(2^{2^{2^{2^2}}}\right) + 6 > 10^{10^4}</math>
| <code>(\1 1 1 1 1) (\\2 (2 1))</code>
| <math>C(2^{2^{2^{2^2}}})</math>|| John Tromp & Bertram Felgenhauer
|-
|39 || <math>\ge 5 \left(3^{3^{3^3}}\right) + 6 > 10^{10^{12}}</math>
| <code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^{3^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|40 || <math> > (2\uparrow\uparrow)^{15} 33 > 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{15} 33</math>|| mxdys and racheline
|-
|41 || <math>\ge 5 \left(3^{3^{85}}\right) + 6 > 10^{10^{40}}</math>
| <code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge BB\lambda(40)+2</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math> and <math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge BB\lambda(43)+2</math>
| <code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| ||
|-
|46 || <math> \ge BB\lambda(44)+2</math>
| <code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| ||
|-
|47 ||
|
| ||
|-
|48 || <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>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
<math>> \text{Graham's number}</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|<math>C(N) \text{ for } N \approx f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|<math>C(N) \text{ for } N \approx f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|
{| class="wikitable"
|-
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|}
|
|Patcail
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|Patcail
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
<math>> \text{TREE}(G64)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|Patcail
|-
|
|
|
|
|
|-
|331
| lim(BMS)
|<code>too large to show</code>
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & John Tromp & 50_ft_lock]
|
|-
|1850
|> Loader's number
|<code>too large to show</code>
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 John Tromp]
|}

Where <math>C(n)</math> represents the Church numeral ''n'' (<math>\lambda f x. f^n(x)</math>) written as <code>\\2 (2 ... (2 1)...)</code> with ''n'' 2s in this text representation.

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<code>1</code>
|1
|-
|3
|
|0
|-
|4
|<code>\1</code>
|4
|-
|5
|<code>\2</code>
|5
|-
|6
|<code>\\1</code>
|6
|-
|7
|<code>\\2</code>
|7
|-
|8
|<code>1 (\1)</code>
|<math>f(4) = 26</math>
|-
|9
|<code>\\2</code>
|9
|-
|10
|<code>1 (\\1)</code>
|<math>f(6) = 36</math>
|-
|11
|<code>1 (\\2)</code>
|<math>f(7) = 41</math>
|-
|12
|<code>1 (1 (\1))</code>
|<math>f^{2}(4) = 266</math>
|-
|13
|<code>1 (\\2)</code>
|<math>f(9) = 51</math>
|-
|14
|<code>1 (1 (\\1))</code>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<code>1 (1 (\\2))</code>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<code>1 (1 (1 (\1)))</code>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<code>1 (1 (\\\2))</code>
|<math>f^2(9) = f(51)</math>
|-
|18
|<code>1 (\1) 1 (\1)</code>
|<math>f^4(4) </math>
|-
|19
|<code>1 (1 (1 (\\2)))</code>
|<math>f^3(7)</math>
|-
|20
|<code>1 (\\1) 1 (\1)</code>
|<math>f^6(4)</math>
|-
|21
|<code>1 (\\2) 1 (\1)</code>
|<math>f^7(4)</math>
|-
|22
|<code>1 (1 (\1)) 1 (\1)</code>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n ≤ 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2023/11/24/largest-number The largest number representable in 64 bits]. 24 Nov 2023. John Tromp.
* [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

Busy Beaver for lambda calculus

2026-02-05T10:00:31Z

Azerty: /* De Bruijn */ More precise lower bounds.

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n. Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) is not defined for those values. The smallest closed lambda term is <code>\1</code> which has size 4.

For the rest of n ≤ 20: BBλ(n) = n is trivial and can be achieved via picking any n bit term already in normal form. For example <code>\\...\1</code> and <code>\\...\2</code> with k lambdas has size 2k+2 and 2k+3 respectively (for k ≥ 1 and k ≥ 2 respectively).

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!# Beta reductions
!Normal form
!Discovered By
|-
|21 || = 22 || <code>\(\1 1) (1 (\2))</code>
|1|| <code>\(1 (\2)) (1 (\2))</code> || John Tromp & Bertram Felgenhauer
|-
|22 || = 24 || <code>\(\1 1) (1 (\\1))</code><code>\(\1 1 1) (1 1)</code>
|1
1
| <code>\(1 (\\1)) (1 (\\1))</code> <code>\(1 1) (1 1) (1 1)</code>
|John Tromp & Bertram Felgenhauer
|-
|23 || = 26 || <code>\(\1 1) (1 (\\2))</code>
|1|| <code>\(1 (\\2)) (1 (\\2))</code> || John Tromp & Bertram Felgenhauer
|-
|24 || = 30 || <code>\(\1 1 1) (1 (\1))</code>
|1|| <code>\(1 (\1)) (1 (\1)) (1 (\1))</code> || John Tromp & Bertram Felgenhauer
|-
|25 || = 42 || <code>\(\1 1) (\1 (2 1))</code>
|3|| <code>\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|26 || = 52 || <code>(\1 1) (\\2 (1 2))</code>
|3|| <code>\\2 (\\2 (1 2)) (1 (2 (\\2 (1 2))))</code> || John Tromp & Bertram Felgenhauer
|-
|27 || = 44 || <code>\\(\1 1) (\1 (2 1))</code>
|3|| <code>\\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|28 || = 58 || <code>\(\1 1) (\1 (2 (\2))))</code>
|3|| <code>\1 (\\1 (3 (\2))) (1 (\2 (\\1 (4 (\2)))))</code>|| John Tromp & Bertram Felgenhauer
|-
| 29 || = 223|| <code>\(\1 1) (\1 (1 (2 1)))</code>
|4|| <pre>
\B (B (1 B))
where:
B = (A (A (1 A)))
A = (1 (\1 (1 (2 1))))
</pre>||John Tromp & Bertram Felgenhauer
|-
|30
|= 160
|<code>(\1 1 1) (\\2 (1 2))</code> and
<code>(\1 (1 1)) (\\2 (1 2))</code>
|7
5
|<pre>
\\2 B A (1 (2 B A))
where:
B = (\\2 A (1 (2 A)))
A = (\\2 (1 2))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|31
|= 267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|6
|<pre>
\\2 A (2 A (C (2 A)))
where:
C = (2 A (2 A (1 B (2 A))))
B = (\3 A (3 A (1 (3 A))))
A = (\\2 (2 (1 2)))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|32
|= 298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|
|
|John Tromp & Bertram Felgenhauer
|-
|33
|= 1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|5
|<pre>
\C (C (C (1 C)))
where:
C = (B (B (B (1 B)))
B = (A (A (A (1 A)))
A = (1 (\1 (1 (1 (2 1)))))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|34 || <math>= \ 5 \left(2^{2^{2^2}}\right) + 6 = 327\,686</math>
| <code>(\1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^2}})</math>|| John Tromp & Bertram Felgenhauer
|-
|35 || <math>= 5 \left(3^{3^3}\right) + 6 > 3.8 \times 10^{13}</math>
| <code>(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^3})</math>|| John Tromp & Bertram Felgenhauer
|-
|36 || <math>= 5 \left(2^{2^{2^3}}\right) + 6 > 5.7 \times 10^{77}</math>
| <code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
| || <math>C(2^{2^{2^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|37 || <math> = BB\lambda(35)+2</math>
| <code>\(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>\lambda x. C(3^{3^3})</math>||mxdys, tromp, dyuan, sligocki
|-
|38 || <math>\ge 5 \left(2^{2^{2^{2^2}}}\right) + 6 > 10^{10^4}</math>
| <code>(\1 1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^{2^2}}})</math>|| John Tromp & Bertram Felgenhauer
|-
|39 || <math>\ge 5 \left(3^{3^{3^3}}\right) + 6 > 10^{10^{12}}</math>
| <code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{3^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|40 || <math> > (2\uparrow\uparrow)^{15} 33 > 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{15} 33</math>|| mxdys and racheline
|-
|41 || <math>\ge 5 \left(3^{3^{85}}\right) + 6 > 10^{10^{40}}</math>
| <code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge BB\lambda(40)+2</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math> and <math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge BB\lambda(43)+2</math>
| <code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || ||
|-
|46 || <math> \ge BB\lambda(44)+2</math>
| <code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|47 ||
|
| || ||
|-
|48 || <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>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
<math>> \text{Graham's number}</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|
{| class="wikitable"
|-
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|}
|
|
|Patcail
|-
|...
|
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|...
|
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
<math>> \text{TREE}(G64)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|331
| lim(BMS)
|<code>too large to show</code>
|
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & John Tromp & 50_ft_lock]
|
|-
|1850
|> Loader's number
|<code>too large to show</code>
|
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 John Tromp]
|}

Where <math>C(n)</math> represents the Church numeral ''n'' (<math>\lambda f x. f^n(x)</math>) written as <code>\\2 (2 ... (2 1)...)</code> with ''n'' 2s in this text representation.

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<code>1</code>
|1
|-
|3
|
|0
|-
|4
|<code>\1</code>
|4
|-
|5
|<code>\2</code>
|5
|-
|6
|<code>\\1</code>
|6
|-
|7
|<code>\\2</code>
|7
|-
|8
|<code>1 (\1)</code>
|<math>f(4) = 26</math>
|-
|9
|<code>\\2</code>
|9
|-
|10
|<code>1 (\\1)</code>
|<math>f(6) = 36</math>
|-
|11
|<code>1 (\\2)</code>
|<math>f(7) = 41</math>
|-
|12
|<code>1 (1 (\1))</code>
|<math>f^{2}(4) = 266</math>
|-
|13
|<code>1 (\\2)</code>
|<math>f(9) = 51</math>
|-
|14
|<code>1 (1 (\\1))</code>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<code>1 (1 (\\2))</code>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<code>1 (1 (1 (\1)))</code>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<code>1 (1 (\\\2))</code>
|<math>f^2(9) = f(51)</math>
|-
|18
|<code>1 (\1) 1 (\1)</code>
|<math>f^4(4) </math>
|-
|19
|<code>1 (1 (1 (\\2)))</code>
|<math>f^3(7)</math>
|-
|20
|<code>1 (\\1) 1 (\1)</code>
|<math>f^6(4)</math>
|-
|21
|<code>1 (\\2) 1 (\1)</code>
|<math>f^7(4)</math>
|-
|22
|<code>1 (1 (\1)) 1 (\1)</code>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size.

To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers.

By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

It is believed that d
{| class="wikitable"
!n
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2023/11/24/largest-number The largest number representable in 64 bits]. 24 Nov 2023. John Tromp.
* [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

Busy Beaver for lambda calculus

2026-02-05T09:54:56Z

Azerty: /* De Bruijn */ Added new lower bounds and added expression size explanation.

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n. Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) is not defined for those values. The smallest closed lambda term is <code>\1</code> which has size 4.

For the rest of n ≤ 20: BBλ(n) = n is trivial and can be achieved via picking any n bit term already in normal form. For example <code>\\...\1</code> and <code>\\...\2</code> with k lambdas has size 2k+2 and 2k+3 respectively (for k ≥ 1 and k ≥ 2 respectively).

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!# Beta reductions
!Normal form
!Discovered By
|-
|21 || = 22 || <code>\(\1 1) (1 (\2))</code>
|1|| <code>\(1 (\2)) (1 (\2))</code> || John Tromp & Bertram Felgenhauer
|-
|22 || = 24 || <code>\(\1 1) (1 (\\1))</code><code>\(\1 1 1) (1 1)</code>
|1
1
| <code>\(1 (\\1)) (1 (\\1))</code> <code>\(1 1) (1 1) (1 1)</code>
|John Tromp & Bertram Felgenhauer
|-
|23 || = 26 || <code>\(\1 1) (1 (\\2))</code>
|1|| <code>\(1 (\\2)) (1 (\\2))</code> || John Tromp & Bertram Felgenhauer
|-
|24 || = 30 || <code>\(\1 1 1) (1 (\1))</code>
|1|| <code>\(1 (\1)) (1 (\1)) (1 (\1))</code> || John Tromp & Bertram Felgenhauer
|-
|25 || = 42 || <code>\(\1 1) (\1 (2 1))</code>
|3|| <code>\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|26 || = 52 || <code>(\1 1) (\\2 (1 2))</code>
|3|| <code>\\2 (\\2 (1 2)) (1 (2 (\\2 (1 2))))</code> || John Tromp & Bertram Felgenhauer
|-
|27 || = 44 || <code>\\(\1 1) (\1 (2 1))</code>
|3|| <code>\\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|28 || = 58 || <code>\(\1 1) (\1 (2 (\2))))</code>
|3|| <code>\1 (\\1 (3 (\2))) (1 (\2 (\\1 (4 (\2)))))</code>|| John Tromp & Bertram Felgenhauer
|-
| 29 || = 223|| <code>\(\1 1) (\1 (1 (2 1)))</code>
|4|| <pre>
\B (B (1 B))
where:
B = (A (A (1 A)))
A = (1 (\1 (1 (2 1))))
</pre>||John Tromp & Bertram Felgenhauer
|-
|30
|= 160
|<code>(\1 1 1) (\\2 (1 2))</code> and
<code>(\1 (1 1)) (\\2 (1 2))</code>
|7
5
|<pre>
\\2 B A (1 (2 B A))
where:
B = (\\2 A (1 (2 A)))
A = (\\2 (1 2))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|31
|= 267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|6
|<pre>
\\2 A (2 A (C (2 A)))
where:
C = (2 A (2 A (1 B (2 A))))
B = (\3 A (3 A (1 (3 A))))
A = (\\2 (2 (1 2)))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|32
|= 298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|
|
|John Tromp & Bertram Felgenhauer
|-
|33
|= 1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|5
|<pre>
\C (C (C (1 C)))
where:
C = (B (B (B (1 B)))
B = (A (A (A (1 A)))
A = (1 (\1 (1 (1 (2 1)))))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|34 || <math>= \ 5 \left(2^{2^{2^2}}\right) + 6 = 327\,686</math>
| <code>(\1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^2}})</math>|| John Tromp & Bertram Felgenhauer
|-
|35 || <math>= 5 \left(3^{3^3}\right) + 6 > 3.8 \times 10^{13}</math>
| <code>(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^3})</math>|| John Tromp & Bertram Felgenhauer
|-
|36 || <math>= 5 \left(2^{2^{2^3}}\right) + 6 > 5.7 \times 10^{77}</math>
| <code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
| || <math>C(2^{2^{2^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|37 || <math> = BB\lambda(35)+2</math>
| <code>\(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>\lambda x. C(3^{3^3})</math>||mxdys, tromp, dyuan, sligocki
|-
|38 || <math>\ge 5 \left(2^{2^{2^{2^2}}}\right) + 6 > 10^{10^4}</math>
| <code>(\1 1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^{2^2}}})</math>|| John Tromp & Bertram Felgenhauer
|-
|39 || <math>\ge 5 \left(3^{3^{3^3}}\right) + 6 > 10^{10^{12}}</math>
| <code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{3^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|40 || <math> > (2\uparrow\uparrow)^{15} 33 > 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{15} 33</math>|| mxdys and racheline
|-
|41 || <math>\ge 5 \left(3^{3^{85}}\right) + 6 > 10^{10^{40}}</math>
| <code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge BB\lambda(40)+2</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math> and <math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge BB\lambda(43)+2</math>
| <code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || ||
|-
|46 || <math> \ge BB\lambda(44)+2</math>
| <code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|47 ||
|
| || ||
|-
|48 || <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>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
<math>> \text{Graham's number}</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|
{| class="wikitable"
|-
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|}
|
|
|Patcail
|-
|...
|
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|...
|
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
<math>> \text{TREE}(G64)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|
|
|Patcail
|-
|
|
|
|
|
|
|-
|331
| lim(BMS)
|<code>too large to show</code>
|
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & John Tromp & 50_ft_lock]
|
|-
|1850
|> Loader's number
|<code>too large to show</code>
|
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 John Tromp]
|}

Where <math>C(n)</math> represents the Church numeral ''n'' (<math>\lambda f x. f^n(x)</math>) written as <code>\\2 (2 ... (2 1)...)</code> with ''n'' 2s in this text representation.

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<code>1</code>
|1
|-
|3
|
|0
|-
|4
|<code>\1</code>
|4
|-
|5
|<code>\2</code>
|5
|-
|6
|<code>\\1</code>
|6
|-
|7
|<code>\\2</code>
|7
|-
|8
|<code>1 (\1)</code>
|<math>f(4) = 26</math>
|-
|9
|<code>\\2</code>
|9
|-
|10
|<code>1 (\\1)</code>
|<math>f(6) = 36</math>
|-
|11
|<code>1 (\\2)</code>
|<math>f(7) = 41</math>
|-
|12
|<code>1 (1 (\1))</code>
|<math>f^{2}(4) = 266</math>
|-
|13
|<code>1 (\\2)</code>
|<math>f(9) = 51</math>
|-
|14
|<code>1 (1 (\\1))</code>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<code>1 (1 (\\2))</code>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<code>1 (1 (1 (\1)))</code>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<code>1 (1 (\\\2))</code>
|<math>f^2(9) = f(51)</math>
|-
|18
|<code>1 (\1) 1 (\1)</code>
|<math>f^4(4) </math>
|-
|19
|<code>1 (1 (1 (\\2)))</code>
|<math>f^3(7)</math>
|-
|20
|<code>1 (\\1) 1 (\1)</code>
|<math>f^6(4)</math>
|-
|21
|<code>1 (\\2) 1 (\1)</code>
|<math>f^7(4)</math>
|-
|22
|<code>1 (1 (\1)) 1 (\1)</code>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size.

To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers.

By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.
{| class="wikitable"
!n
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(10 \uparrow\uparrow 16)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2023/11/24/largest-number The largest number representable in 64 bits]. 24 Nov 2023. John Tromp.
* [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

User:Azerty/Busy Beaver Functions

2026-01-26T09:22:48Z

Azerty: /* De Bruijn */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^{\omega^\omega}}(10^{19,727})</math>
|<code>(\1 1 1 1 1) (\\\1 (\\2 (2 1)) 3 2 1)</code>
|-
|BBL(23)
|<math>f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(25)
|<math>f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2026-01-23T09:58:42Z

Azerty: /* De Bruijn */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^{\omega^\omega}}(10^{19,727})</math>
|<code>(\1 1 1 1 1) (\\\1 (\\2 (2 1)) 3 2 1)</code>
|-
|BBL(23)
|<math>f_{\omega^{\omega^\omega}}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(26)
|<math>f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 (2 1)))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

TMBR: January 2026

2026-01-18T09:48:30Z

Azerty: /* Champions */ I didn't find these BLB and TT champions.

{{TMBRnav|December 2025|February 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

This is the first edition of TMBR in 2026.

== Champions ==

* nickdrozd discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* creeperman7002 discovered <code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>, a [[TT|TT(4,2)]] TM which runs for 48,186 steps and <code>1TA2PB3TB---_3TA1PB1TA1PA</code>, a [[TT|TT(2,4)]] TM which runs for more than 3.467*1015 steps.
* A new lower bound of <math>f_{\varepsilon_0 \omega^{\omega^{3}}}\left(4\right)</math> was computed for the [[Busy Beaver for lambda calculus#Champions|BBλ(91)]] champion.

== Blog Posts ==

* 16 Jan 2026. Nick Drozd. [https://nickdrozd.github.io/2026/01/16/ai-lin-busy-beaver.html AI Reproduction of Lin's Busy Beaver Proof].

== BB Adjacent ==

* [https://discord.com/channels/960643023006490684/1458010522967609425/1458010522967609425 Uniform-Action Busy Beavers] were introduced, and lower bounds have been given up to BBu(6) (with BBu(2)=6, BBu(3)=17 and BBu(4)=29)

== Theory ==
@ConePine [https://discord.com/channels/960643023006490684/960643023530762341/1458876275749032232 shared an idea] on how the value of BB(5) can be proved without enumerating Turing machines.

== Holdouts ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 machines[https://discord.com/channels/960643023006490684/1456755170527543420/1456755170527543420 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1456757189787127971/1456757189787127971 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1456757980203585678/1456757980203585678 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1456758622762696815/1456758622762696815 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1456759118818971709/1456759118818971709 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1456761765688901724/1456761765688901724 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1456762231491657830/1456762231491657830 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1456310285462802587/1456760768472289321 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456932659740545217 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933097009320089 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933714629099593 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456934113943621653 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457147385309565021 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457409447084429594 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457497643478683770 <nowiki>[4 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457644806984437760 <nowiki>[10 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457992404597739672 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1458413343088840734 <nowiki>[17 more]</nowiki>] simulated out, plus 2 solved machines).
** Alistaire found a halting machine in the list mentioned above, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456317703156531211 Discord]. Approximate score: 4e12. Later, he found another halting machine in the same list, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456381492019069192 Discord] - approximate score: 1.5e18.
** @mxdys shared a list of machines that seem to be provable. See [https://discord.com/channels/960643023006490684/1460495597386731643/1460495597386731643 Discord].
*[[BB(7)]]:
**Andrew Ducharme further continued reducing the number of holdouts, from 20,387,509 to '''20,197,978''' TMs, a 0.93% reduction. This puts the total compute time on the current BB(7) pipeline just over '''50,000''' hours.[https://discord.com/channels/960643023006490684/1369339127652159509/1459719965396566078]
**The holdout count was further reduced to '''19,879,953''' (a 1.57% reduction), which breaks the 20 million barrier.[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365]

[[Category:This Month in Beaver Research|2025-12]]

User:Azerty/Busy Beaver Functions

2026-01-16T08:11:34Z

Azerty: /* De Bruijn */ Added BO champions

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^2+2}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(19)
|<math>f_{\omega^\omega}(26)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(20)
|<math>f_{\omega^\omega}(f_{\omega+1}(f_{\omega}(8)))</math>
|<code>(\1 (\1 (\\\1 3 2 1) 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(21)
|<math>f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(22)
|<math>f_{\omega^{\omega^\omega}}(26)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(23)
|<math>f_{\omega^{\omega^\omega}}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(26)
|<math>f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 (2 1)))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2026-01-15T08:54:31Z

Azerty: /* De Bruijn */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^2+2}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(19)
|<math>f_{\omega^\omega}(26)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(20)
|<math>f_{\omega^\omega}(f_{\omega+1}(f_{\omega}(8)))</math>
|<code>(\1 (\1 (\\\1 3 2 1) 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(21)
|<math>f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(22)
|<math>f_{\omega^{\omega^\omega}}(26)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(23)
|<math>f_{\omega^{\omega^\omega}}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^{\omega^{\omega+2}}}(2)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(25)
|<math>f_{\omega^{\omega^{\omega^\omega}}}(26)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(26)
|<math>f_{\epsilon_0 \times \omega^{\omega^3}}(4)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|-
|BBL(28)
|<math>f_{\epsilon_{\omega^{\omega^3}}}(3)</math>
|<code>(\1 1 (\\1 2 (\\1 2 1) 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2026-01-15T08:53:49Z

Azerty: /* De Bruijn */ Added new champion.

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^2+2}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(19)
|<math>f_{\omega^\omega}(26)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(20)
|<math>f_{\omega^\omega}(f_{\omega+1}(f_{\omega}(8)))</math>
|<code>(\1 (\1 (\\\1 3 2 1) 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(21)
|<math>f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(22)
|<math>f_{\omega^{\omega^\omega}}(26)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(23)
|<math>f_{\omega^{\omega^\omega}}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^{\omega^{\omega+2}}}(2)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(25)
|<math>f_{\omega^{\omega^{\omega^\omega}}}(26)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(26)
|<math>f_{\epsilon_0 \times \omega^{\omega^3}}(4)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|-
|BBL(27)
|<math>f_{\epsilon_{\omega^{\omega^3}}}(3)</math>
|<code>(\1 1 (\\1 2 (\\1 2 1) 1) (\1 (\2 1 2))) (\\2 3 (\3 1 2))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2026-01-11T14:19:41Z

Azerty: /* De Bruijn */ Added more lower bounds.

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^2+2}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(19)
|<math>f_{\omega^\omega}(26)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(20)
|<math>f_{\omega^\omega}(f_{\omega+1}(f_{\omega}(8)))</math>
|<code>(\1 (\1 (\\\1 3 2 1) 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(21)
|<math>f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(22)
|<math>f_{\omega^{\omega^\omega}}(26)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(23)
|<math>f_{\omega^{\omega^\omega}}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^{\omega^{\omega+2}}}(2)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 1))</code>
|-
|BBL(25)
|<math>f_{\omega^{\omega^{\omega^\omega}}}(26)</math>
|<code>(\1 1 (\\\\\1 5 4 3 2 1) 1 1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(26)
|<math>f_{\epsilon_0 \times \omega^{\omega^3}}(4)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2026-01-11T14:04:46Z

Azerty: /* De Bruijn */ Updated lower bound

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\epsilon_0 \times \omega^{\omega^3}}(4)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Champions Zoology

2026-01-05T14:41:09Z

Azerty:

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Examples
!Scale
|-
|Trivial
|
|<math>O(n)</math>
|-
|Multiplier
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BLB(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(2^n)</math>
|-
|Chaotic
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(2^n)</math>
|-
|Exponentiater
|
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} BB(2,4)
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}} BB(5,2)
* {{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}} BB(3,3)
|<math>O(2^n)</math>
|}

Blanking Busy Beaver Function

2026-01-04T21:20:54Z

Azerty: /* Champions */ I found the BLBi(7) champion!

The '''Blanking Busy Beaver Function''' (BLB(n,m)) is a [[Busy Beaver Functions|Busy Beaver Function]] which measures the largest amount of steps done by any [[Turing machine]] with n states and m symbols before blanking the tape. Blanking Busy Beaver machines are allowed to be halting or non-halting. As machines with just a single state cannot blank the tape, BLB(1,m) is nonexistent for any amount of symbols m. Similarly, BLB(n,1) is nonexistent for any amount of states n, as the tape of [[BB(n,1)|1-symbol machines]] is always blank.

== Champions ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|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(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|≥ 329
|{{TM|1RB2LC2LA_1LC---2RA_2RC2LB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== [[Instruction-Limited Busy Beaver|Instruction-limited]] Blanking Busy Beaver (BLBi(n)) ==
BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol on the tape.

=== Champions ===
Note that BLBi(1) and BLBi(2) are nonexistent as Turing machines with one or two instructions cannot blank the tape.
{| class="wikitable"
|+
!Instructions
!Steps
!Champions
|-
|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}}
|}

==References==
[[Category:Functions]]

Blanking Busy Beaver Function

2026-01-04T17:46:38Z

Azerty: /* Champions */

The '''Blanking Busy Beaver Function''' (BLB(n,m)) is a [[Busy Beaver Functions|Busy Beaver Function]] which measures the largest amount of steps done by any [[Turing machine]] with n states and m symbols before blanking the tape. Blanking Busy Beaver machines are allowed to be halting or non-halting. As machines with just a single state cannot blank the tape, BLB(1,m) is nonexistent for any amount of symbols m. Similarly, BLB(n,1) is nonexistent for any amount of states n, as the tape of [[BB(n,1)|1-symbol machines]] is always blank.

== Champions ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|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(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|≥ 329
|{{TM|1RB2LC2LA_1LC---2RA_2RC2LB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== [[Instruction-Limited Busy Beaver|Instruction-limited]] Blanking Busy Beaver (BLBi(n)) ==
BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol on the tape.

=== Champions ===
Note that BLBi(1) and BLBi(2) are nonexistent as Turing machines with one or two instructions cannot blank the tape.
{| class="wikitable"
|+
!Instructions
!Steps
!Champions
|-
|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
|?
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

==References==
[[Category:Functions]]

Blanking Busy Beaver Function

2026-01-04T17:46:14Z

Azerty: /* Champions */ BLB(2) is proven

The '''Blanking Busy Beaver Function''' (BLB(n,m)) is a [[Busy Beaver Functions|Busy Beaver Function]] which measures the largest amount of steps done by any [[Turing machine]] with n states and m symbols before blanking the tape. Blanking Busy Beaver machines are allowed to be halting or non-halting. As machines with just a single state cannot blank the tape, BLB(1,m) is nonexistent for any amount of symbols m. Similarly, BLB(n,1) is nonexistent for any amount of states n, as the tape of [[BB(n,1)|1-symbol machines]] is always blank.

== Champions ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|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(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|≥ 329
|{{TM|1RB2LC2LA_1LC---2RA_2RC2LB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== [[Instruction-Limited Busy Beaver|Instruction-limited]] Blanking Busy Beaver (BLBi(n)) ==
BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol on the tape.

=== Champions ===
Note that BLBi(1) and BLBi(2) are nonexistent as Turing machines with one or two instructions cannot blank the tape.
{| class="wikitable"
|+
!Instructions
!Steps
!Champions
|-
|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
|?
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

==References==
[[Category:Functions]]

User:Azerty

2026-01-02T21:55:29Z

Azerty:

I am an amateur interested in computability theory, large numbers and BB functions.

My web pages:

* [https://azertywastaken.github.io/TuringMachinesSimulator/tm_simulator Turing machines simulator]
* [https://azertywastaken.github.io/TuringMachinesSimulator/bb_champions List of lower bounds and champions]

My GitHub repositories:

* [https://github.com/AzertyWasTaken/OrdinalNotations Ordinal notations]
* [https://github.com/AzertyWasTaken/SearchBB Search BB champions script]

My wiki user pages:

* [[User:Azerty/Busy Beaver Functions|List of lower bounds and champions]]
* [[User:Azerty/Turing Completeness|List of Turing-complete systems]]
* [[User:Azerty/Champions Zoology|Champions Zoology]]

User:Azerty

2026-01-02T16:19:51Z

Azerty: Added a link to my champions web page.

I am an amateur interested in computability theory, large numbers and BB functions.

My web pages:

* [https://azertywastaken.github.io/TuringMachinesSimulator/tm_simulator Turing machines simulator]
* [https://azertywastaken.github.io/TuringMachinesSimulator/bb_champions List of lower bounds and champions]

My GitHub repositories:

* [https://github.com/AzertyWasTaken/OrdinalNotations Ordinal notations]
* [https://github.com/AzertyWasTaken/SearchBB Search BB champions script]

My user pages:

* [[User:Azerty/Busy Beaver Functions|List of lower bounds and champions]]
* [[User:Azerty/Turing Completeness|List of Turing-complete systems]]
* [[User:Azerty/Champions Zoology|Champions Zoology]]

Main Page

2026-01-02T16:01:10Z

Azerty: Added BB(n,1) = n

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with less states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: November 2025|TMBR November 2025]]
* [[TMBR: October 2025|TMBR October 2025]]
* [[TMBR: September 2025|TMBR September 2025]]

[[TMBR: December 2025|TMBR December 2025]] is currently work in progress.

== Contribute to this wiki ==
This wiki is collaborative, feel free to contribute by editing existing pages or creating new ones:

<inputbox>
type=create
width=100
break=no
buttonlabel=Create new article
default=(Article title)
</inputbox>

==Notes==
<references />

User:Azerty/Turing Completeness

2026-01-01T15:15:05Z

Azerty: /* Turing-complete systems */

This is a list of systems that are Turing-complete.

To be Turing-complete, a system must have unbounded memory, access to that memory, and loops that can repeat forever.

== Turing-complete systems ==

* 2-symbol Turing machine on a blank tape
* Uniform-action Turing machines with 2 states and a blank tape
* 15-state 2-symbol Turing machine
* 9-state 3-symbol Turing machine
* 6-state 4-symbol Turing machine
* 5-state 5-symbol Turing machine
* 4-state 6-symbol Turing machine
* 3-state 9-symbol Turing machine
* 2-state 18-symbol Turing machine
* 24-instruction Turing machines

== Open problems about Turing-completenesss ==

* Turing machines with minimum states and symbols count
* Turing machines with minimum instructions count
* Turing machines with minimum move left transitions count
* Turing machines with minimum states with 2 defined transitions and 2 symbols count
* Turing machines with minimum symbols with 2 defined transitions and 2 states count
* Turing machines with 2 symbols and a blank tape but each state must write the same symbol
* Turing machines with 2 symbols and a blank tape but each state must move in the same direction
* Turing machines with 2 states and a blank tape but each symbol must write the same symbol
* Turing machines with 2 states and a blank tape but each symbol must go to the same state

Busy Beaver for lambda calculus

2026-01-01T14:58:29Z

Azerty: /* De Bruijn */ Updated De Bruijn Busy Beaver champions and discoverers.

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n. Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) is not defined for those values. The smallest closed lambda term is <code>\1</code> which has size 4.

For the rest of n ≤ 20: BBλ(n) = n is trivial and can be achieved via picking any n bit term already in normal form. For example <code>\\...\1</code> and <code>\\...\2</code> with k lambdas has size 2k+2 and 2k+3 respectively (for k ≥ 1 and k ≥ 2 respectively).

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!# Beta reductions
!Normal form
!Discovered By
|-
|21 || = 22 || <code>\(\1 1) (1 (\2))</code>
|1|| <code>\(1 (\2)) (1 (\2))</code> || John Tromp & Bertram Felgenhauer
|-
|22 || = 24 || <code>\(\1 1) (1 (\\1))</code><code>\(\1 1 1) (1 1)</code>
|1
1
| <code>\(1 (\\1)) (1 (\\1))</code> <code>\(1 1) (1 1) (1 1)</code>
|John Tromp & Bertram Felgenhauer
|-
|23 || = 26 || <code>\(\1 1) (1 (\\2))</code>
|1|| <code>\(1 (\\2)) (1 (\\2))</code> || John Tromp & Bertram Felgenhauer
|-
|24 || = 30 || <code>\(\1 1 1) (1 (\1))</code>
|1|| <code>\(1 (\1)) (1 (\1)) (1 (\1))</code> || John Tromp & Bertram Felgenhauer
|-
|25 || = 42 || <code>\(\1 1) (\1 (2 1))</code>
|3|| <code>\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|26 || = 52 || <code>(\1 1) (\\2 (1 2))</code>
|3|| <code>\\2 (\\2 (1 2)) (1 (2 (\\2 (1 2))))</code> || John Tromp & Bertram Felgenhauer
|-
|27 || = 44 || <code>\\(\1 1) (\1 (2 1))</code>
|3|| <code>\\1 (\1 (2 1)) (1 (1 (\1 (2 1))))</code> || John Tromp & Bertram Felgenhauer
|-
|28 || = 58 || <code>\(\1 1) (\1 (2 (\2))))</code>
|3|| <code>\1 (\\1 (3 (\2))) (1 (\2 (\\1 (4 (\2)))))</code>|| John Tromp & Bertram Felgenhauer
|-
| 29 || = 223|| <code>\(\1 1) (\1 (1 (2 1)))</code>
|4|| <pre>
\B (B (1 B))
where:
B = (A (A (1 A)))
A = (1 (\1 (1 (2 1))))
</pre>||John Tromp & Bertram Felgenhauer
|-
|30
|= 160
|<code>(\1 1 1) (\\2 (1 2))</code> and
<code>(\1 (1 1)) (\\2 (1 2))</code>
|7
5
|<pre>
\\2 B A (1 (2 B A))
where:
B = (\\2 A (1 (2 A)))
A = (\\2 (1 2))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|31
|= 267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|6
|<pre>
\\2 A (2 A (C (2 A)))
where:
C = (2 A (2 A (1 B (2 A))))
B = (\3 A (3 A (1 (3 A))))
A = (\\2 (2 (1 2)))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|32
|= 298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|
|
|John Tromp & Bertram Felgenhauer
|-
|33
|= 1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|5
|<pre>
\C (C (C (1 C)))
where:
C = (B (B (B (1 B)))
B = (A (A (A (1 A)))
A = (1 (\1 (1 (1 (2 1)))))
</pre>
|John Tromp & Bertram Felgenhauer
|-
|34 || <math>= \ 5 \left(2^{2^{2^2}}\right) + 6 = 327\,686</math>
| <code>(\1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^2}})</math>|| John Tromp & Bertram Felgenhauer
|-
|35 || <math>= 5 \left(3^{3^3}\right) + 6 > 3.8 \times 10^{13}</math>
| <code>(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^3})</math>|| John Tromp & Bertram Felgenhauer
|-
|36 || <math>= 5 \left(2^{2^{2^3}}\right) + 6 > 5.7 \times 10^{77}</math>
| <code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
| || <math>C(2^{2^{2^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|37 || <math> = BB\lambda(35)+2</math>
| <code>\(\1 1 1) (\\2 (2 (2 1)))</code>
| || <math>\lambda x. C(3^{3^3})</math>||mxdys, tromp, dyuan, sligocki
|-
|38 || <math>\ge 5 \left(2^{2^{2^{2^2}}}\right) + 6 > 10^{10^4}</math>
| <code>(\1 1 1 1 1) (\\2 (2 1))</code> and
<code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
| || <math>C(2^{2^{2^{2^2}}})</math>|| John Tromp & Bertram Felgenhauer
|-
|39 || <math>\ge 5 \left(3^{3^{3^3}}\right) + 6 > 10^{10^{12}}</math>
| <code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{3^3}})</math>|| John Tromp & Bertram Felgenhauer
|-
|40 || <math> > (2\uparrow\uparrow)^{15} 33 > 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{15} 33</math>|| mxdys and racheline
|-
|41 || <math>\ge 5 \left(3^{3^{85}}\right) + 6 > 10^{10^{40}}</math>
| <code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
| || <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge BB\lambda(40)+2</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math> and <math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge BB\lambda(43)+2</math>
| <code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
| || ||
|-
|46 || <math> \ge BB\lambda(44)+2</math>
| <code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
| || ||
|-
|47 ||
|
| || ||
|-
|48 || <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>
| || <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) > \text{Graham's number}</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|
|-
|63
|<math>> f_{\omega^3}\left(2\right)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|
|<math>C(N) \text{ for } N \approx f_{\omega^3}\left(2\right)</math>
|Patcail & [https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|
|-
|91
|<math>> f_{\varepsilon_0 + 1}\left(3\right)</math>
|
{| class="wikitable"
|-
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|}
|
|
|50_ft_lock
|-
|...
|
|
|
|
|
|-
|1850
|> Loader's number
|<code>too large to show</code>
|
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 John Tromp]
|}

Where <math>C(n)</math> represents the Church numeral ''n'' (<math>\lambda f x. f^n(x)</math>) written as <code>\\2 (2 ... (2 1)...)</code> with ''n'' 2s in this text representation.
== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BB(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<code>1</code>
|1
|-
|3
|
|0
|-
|4
|<code>\1</code>
|4
|-
|5
|<code>\2</code>
|5
|-
|6
|<code>\\1</code>
|6
|-
|7
|<code>\\2</code>
|7
|-
|8
|<code>1 (\1)</code>
|<math>f(4) = 26</math>
|-
|9
|<code>\\2</code>
|9
|-
|10
|<code>1 (\\1)</code>
|<math>f(6) = 36</math>
|-
|11
|<code>1 (\\2)</code>
|<math>f(7) = 41</math>
|-
|12
|<code>1 (1 (\1))</code>
|<math>f^{2}(4) = 266</math>
|-
|13
|<code>1 (\\2)</code>
|<math>f(9) = 51</math>
|-
|14
|<code>1 (1 (\\1))</code>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<code>1 (1 (\\2))</code>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<code>1 (1 (1 (\1)))</code>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<code>1 (1 (\\\2))</code>
|<math>f^2(9) = f(51)</math>
|-
|18
|<code>1 (\1) 1 (\1)</code>
|<math>f^4(4) </math>
|-
|19
|<code>1 (1 (1 (\\2)))</code>
|<math>f^3(7)</math>
|-
|20
|<code>1 (\\1) 1 (\1)</code>
|<math>f^6(4)</math>
|-
|21
|<code>1 (\\2) 1 (\1)</code>
|<math>f^7(4)</math>
|-
|22
|<code>1 (1 (\1)) 1 (\1)</code>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size.
{| class="wikitable"
!n
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|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>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|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>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|Patcail & [https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|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>
|50_ft_lock
|}

== See Also ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2023/11/24/largest-number The largest number representable in 64 bits]. 24 Nov 2023. John Tromp.
* [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
[[category:Functions]]

1RB0LB2LA 1LA0RC0LB 2RC2RB0LC

2026-01-01T13:04:05Z

Azerty: Created a page so it's easier to find the article (by nickdrozd) about BBB(3,3).

{{TM|1RB1LF_1LB1LC_1RD0LE_---0RB_0RC0LA_1RC0RF}} is the current BBB(3,3) champion. It runs for <math>10 \uparrow \uparrow 6</math> steps before quasihalting.

== Source ==
https://nickdrozd.github.io/2025/03/24/bbb-3-3.html

User:Azerty/Busy Beaver Functions

2025-12-31T12:54:38Z

Azerty: /* Restricted move and write per state */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(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>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2025-12-31T12:54:26Z

Azerty: /* Restricted write per state */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBws(5)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3,2)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4,2)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(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>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2025-12-31T12:31:19Z

Azerty: /* Turmite semi-infinite tape */

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2,2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3,2)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4,2)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3,2)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4,2)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3,1)
|82
|<code>1PB---1TC_2PC1TA0PB_2TB2TC2PB</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(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>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Champions Zoology

2025-12-31T09:15:35Z

Azerty:

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Examples
!Scale
|-
|Chaotic
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(n^2)</math>
|-
|Multiplier
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BLB(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(n^3)</math>
|-
|Exponentiater
|
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} BB(2,4)
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}} BB(5,2)
* {{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}} BB(3,3)
|<math>O(2^n)</math>
|}

Blanking Busy Beaver Function

2025-12-31T08:51:40Z

Azerty: /* Champions */ New BLB(3,3) champion!

The '''Blanking Busy Beaver Function''' (BLB(n,m)) is a [[Busy Beaver Functions|Busy Beaver Function]] which measures the largest amount of steps done by any [[Turing machine]] with n states and m symbols before blanking the tape. Blanking Busy Beaver machines are allowed to be halting or non-halting. As machines with just a single state cannot blank the tape, BLB(1,m) is nonexistent for any amount of symbols m. Similarly, BLB(n,1) is nonexistent for any amount of states n, as the tape of [[BB(n,1)|1-symbol machines]] is always blank.

== Champions ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|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(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|≥ 329
|{{TM|1RB2LC2LA_1LC---2RA_2RC2LB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== [[Instruction-Limited Busy Beaver|Instruction-limited]] Blanking Busy Beaver (BLBi(n)) ==
BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol on the tape.

=== Champions ===
Note that BLBi(1) and BLBi(2) are nonexistent as Turing machines with one or two instructions cannot blank the tape.
{| class="wikitable"
|+
!Instructions
!Steps
!Champions
|-
|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
|?
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

==References==
[[Category:Functions]]

User:Azerty/Champions Zoology

2025-12-31T08:40:59Z

Azerty: Added a multiplier example.

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Examples
!Scale
|-
|Chaotic
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(n^2)</math>
|-
|Multiplier
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BLB(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(n^3)</math>
|-
|Exponentier
|
* {{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}} BBt(3,3)
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} BB(2,4)
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}} BB(5,2)
* {{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}} BB(3,3)
|<math>O(2^n)</math>
|}

User:Azerty/Busy Beaver Functions

2025-12-30T22:14:07Z

Azerty:

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

I only add functions that meet the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2,2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3,2)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4,2)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3,2)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4,2)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(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>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2025-12-30T13:58:25Z

Azerty: /* De Bruijn */ Added new champions.

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

I only add functions that follow the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2,2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3,2)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4,2)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3,2)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4,2)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(18)
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBL(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>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Champions Zoology

2025-12-30T09:44:02Z

Azerty: Added more Collatz-like examples.

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Examples
!Scale
|-
|Chaotic
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(n^2)</math>
|-
|Multiplier
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BBt(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(n^3)</math>
|-
|Counter
|
* {{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}} BBt(3,3,2)
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} BB(2,4)
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}} BB(5,2)
* {{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}} BB(3,3)
|<math>O(2^n)</math>
|}

User:Azerty/Champions Zoology

2025-12-30T09:08:14Z

Azerty:

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Examples
!Scale
|-
|Chaotic
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(n^2)</math>
|-
|Multiplier
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BBt(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(n^3)</math>
|-
|Counter
|
* {{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}} BBt(3,3,2)
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|
* {{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB|halt}} BBwsms(5,2)
|<math>O(2^n)</math>
|}

User:Azerty/Champions Zoology

2025-12-29T14:28:54Z

Azerty: Added Collatz-like function.

This is a zoology of champions and what they compute.
{| class="wikitable"
!Classification
!Description
!Examples
!Scale
|-
|Chaotic
|Have a chaotic behavior and repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}} BB(2,2)
* {{TM|1RB---_1LB0RC_1LC1LA|halt}} BB(3,2)
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}} BB(4,2)
|<math>O(n^2)</math>
|-
|Multiplier
|Create a number then multiply it.
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}} BB(2,3)
* {{TM|1RB1LB_1LA1LC_1RC0LC|halt}} BBt(3,2)
|<math>O(n^2)</math>
|-
|[[Piecewise Affine Function]]
|Iterate a function built by gluing together straight-line formulas, each one used on a different part of the input.
|
* <code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code> TT(3,3)
|<math>O(n^3)</math>
|-
|Counter
|Count in base ≥ 2 until the number reaches a certain length.
|
* {{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}} BBt(3,3,2)
* {{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}} BBti(8)
|<math>O(2^n)</math>
|-
|[[Collatz-like]]
|Iterate a function that return a larger number depending of the input modulo.
|
* {{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB|halt}} BBwsms(5,2)
|<math>O(2^n)</math>
|}

User:Azerty/Busy Beaver Functions

2025-12-29T14:19:45Z

Azerty: Added restricted move and write per state.

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

I only add functions that follow the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBws(2,2)
|6
|{{TM|1RB1LB_1LA---}}
|-
|BBws(3,2)
|20
|{{TM|1RB---_0LC0RC_1LC1LA}}
|-
|BBws(4,2)
|53
|{{TM|1RB1LB_1LC1RD_1RA1LA_---0RC}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== Restricted move and write per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must write the same symbol and go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBwsms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBwsms(3,2)
|17
|{{TM|1RB---_0RC0RC_1LC1LA}}
|-
|BBwsms(4,2)
|29
|{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---}}
|-
|BBwsms(5,2)
|422
|{{TM|1RB1RA_1RC1RE_1LD---_1LA1LD_---0RB}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(23)
|<math>f_{\omega^2}(4)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^2}(27)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\omega^\omega}(4)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\omega^\omega}(27)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\omega^\omega}(10^{12})</math>
|<code>(\1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(28)
|<math>f_{\omega^\omega}(10^{10^{12}})</math>
|<code>(\1 1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(34)
|<math>f_{\omega^{\omega^\omega}}(4)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Turing Completeness

2025-12-29T12:40:55Z

Azerty: /* Turing-complete systems */ Added more problems.

This is a list of systems that are Turing-complete.

To be Turing-complete, a system must have unbounded memory, access to that memory, and loops that can repeat forever.

== Turing-complete systems ==

* Turing machines with 2 symbols and a blank tape
* Turing machines with 2 states and a blank tape
* Turing machines with 15 states and 2 symbols
* Turing machines with 9 states and 3 symbols
* Turing machines with 6 states and 4 symbols
* Turing machines with 5 states and 5 symbols
* Turing machines with 4 states and 6 symbols
* Turing machines with 3 states and 9 symbols
* Turing machines with 2 states and 18 symbols
* Turing machines with 24 instructions

== Open problems about Turing-completenesss ==

* Turing machines with minimum states and symbols count
* Turing machines with minimum instructions count
* Turing machines with minimum move left transitions count
* Turing machines with minimum states with 2 defined transitions and 2 symbols count
* Turing machines with minimum symbols with 2 defined transitions and 2 states count
* Turing machines with 2 symbols and a blank tape but each state must write the same symbol
* Turing machines with 2 symbols and a blank tape but each state must move in the same direction
* Turing machines with 2 states and a blank tape but each symbol must write the same symbol
* Turing machines with 2 states and a blank tape but each symbol must move in the same direction
* Turing machines with 2 states and a blank tape but each symbol must go to the same state

User:Azerty/Busy Beaver Functions

2025-12-29T12:22:05Z

Azerty: /* Classic */ New BB(5,2,1) champion!

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

I only add functions that follow the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|242
|{{TM|1RB0LA_1RC0LA_1LD0RE_---1LA_---0RB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(23)
|<math>f_{\omega^2}(4)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^2}(27)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\omega^\omega}(4)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\omega^\omega}(27)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\omega^\omega}(10^{12})</math>
|<code>(\1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(28)
|<math>f_{\omega^\omega}(10^{10^{12}})</math>
|<code>(\1 1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(34)
|<math>f_{\omega^{\omega^\omega}}(4)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2025-12-29T09:27:33Z

Azerty: Added Turmite restricted move per state function.

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

I only add functions that follow the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|197
|{{TM|1RB---_1LC0RB_0LD1LD_0LE---_0RA1LB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite restricted move per state ===
Maximum number of steps done by a n-state, m-symbol turmite machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTms(2,2,1)
|4
|<code>1TA0TB</code>
|-
|TTms(2,2)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTms(3,2,1)
|12
|<code>1TA1TB_---0PC_0PA---</code>
|-
|TTms(3,2)
|25
|<code>1TA1TB_0PC0PA_---0PA</code>
|-
|TTms(4,2,1)
|30
|<code>1TA1TB_1PD0PC_0PA---_0PB---</code>
|-
|TTms(4,2)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

=== Turmite restricted move per state instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTmsi(2)
|4
|<code>1TA0TB</code>
|-
|TTmsi(3)
|10
|<code>1TA1TB_---0PA</code>
|-
|TTmsi(4)
|16
|<code>1TA1TB_2PB0PA</code>
|-
|TTmsi(5)
|26
|<code>1TA1TB0TA_2PB0PA---</code>
|-
|TTmsi(6)
|36
|<code>1TA1TB0TC_2PB0PA---_0TB------</code>
|-
|TTmsi(7)
|247
|<code>1TA1TB_0PD0PC_0PA0PB_0PC---</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(23)
|<math>f_{\omega^2}(4)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^2}(27)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\omega^\omega}(4)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\omega^\omega}(27)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\omega^\omega}(10^{12})</math>
|<code>(\1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(28)
|<math>f_{\omega^\omega}(10^{10^{12}})</math>
|<code>(\1 1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(34)
|<math>f_{\omega^{\omega^\omega}}(4)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty/Busy Beaver Functions

2025-12-29T09:06:08Z

Azerty: Added restricted move per state BB function.

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

I only add functions that follow the requirements below:

* The function must grow uncomputably fast, like BB(n).
* The function must not have similar champions to other function, like Σ(n).
* The function must not be arbitrary complex.

== Turing machines ==

=== [[Busy Beaver Functions|Classic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BB(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BB(2,3,1)
|14
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3,2)]]
|21
|{{TM|1RB---_1LB0RC_1LC1LA|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BB(4,2,1)
|38
|{{TM|1RB---_1LB0LC_1RD1LD_0RB---|halt}}
|-
|BB(3,3,2)
|77
|{{TM|1RB---0RC_2LC------_2RC0LC1RA|halt}}
|-
|[[BB(4)|BB(4,2)]]
|107
|{{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|-
|BB(2,4,1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BB(5,2,1)
|197
|{{TM|1RB---_1LC0RB_0LD1LD_0LE---_0RA1LB|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BB(3,3,1)
|3,932,964
|{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|[[BB(5)|BB(5,2)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|-
|BB(6,2,1)
|17,397,627,083
|{{TM|1RB0LA_1RC1RD_1LB1RA_---0RE_1LF0RC_1LA---|halt}}
|-
|[[BB(3,3)]]
|<math>1.191 \times 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC|halt}}
|-
|BB(3,4,2)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|[[BB(2,5)]]
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|[[BB(2,6)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB---3LB2LA|halt}}
|-
|[[BB(6)|BB(6,2)]]
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(4,3)]]
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10^{28}</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|-
|[[BB(7)|BB(7,2)]]
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_---0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(3,4)]]
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(9,2)
|<math>f_\omega(10 \uparrow^{7} 10 \uparrow^{7} 2)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(3,5)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BB(10,2)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11,2)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BB(12,2)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BB(14,2)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BB(15,2)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16,2)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|-
|[[BB(2,7)]]
|
|
|}

=== [[Instruction-Limited Busy Beaver|Instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBi(3)
|6
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|38
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
|-
|BBi(8)
|<math>6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 3,314,360</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|-
|BBi(11)
|<math>10 \uparrow^{15} 4</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(14)
|<math>f_{\omega}(10 \uparrow^{15} 4)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|-
|BBi(19)
|<math>f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_---0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BBi(21)
|<math>f_\omega^2(10 {\uparrow}^{2} 10)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_---0LI_0LD1LE|halt}}
|-
|BBi(23)
|<math>f_{\omega}^4(10 {\uparrow}^{2} 65,534)</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_---1LA_1RF1LL|halt}}
|-
|BBi(27)
|<math>f_{\omega + 1}(65,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM---_0LN1LF_0LJ---|halt}}
|-
|BBi(29)
|<math>f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_---0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BBi(31)
|<math>f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_---0LJ_1LH1LO_1RB1RP|halt}}
|}

=== Semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turing machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBt(2,2,1)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,2)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBt(2,3,1)
|9
|{{TM|1RB---2RB_2LB2RA0RB|halt}}
|-
|BBt(3,2,1)
|11
|{{TM|1RB---_0RC0RB_1LB1LC|halt}}
|-
|BBt(3,2)
|12
|{{TM|0RB0LB_0RC1RC_1LA0LC|halt}}
|-
|BBt(2,3)
|17
|{{TM|1RB0RA0RB_2LA2RB1LB|halt}}
|-
|BBt(2,4,1)
|36
|{{TM|1RB3RA0RB0RA_2LA2RB1LB---|halt}}
|-
|BBt(2,4)
|49
|{{TM|1RB2RA2RB2LA_2LB0RA3RB0LA|halt}}
|-
|BBt(4,2,1)
|55
|{{TM|0RB---_1RC0LD_1RD1RB_1LB0RC|halt}}
|-
|BBt(4,2)
|63
|{{TM|0RB0LD_0RC1LC_0RD1LA_1LB1RD|halt}}
|-
|BBt(5,2,1)
|70
|{{TM|0RB---_0RC1LC_1RD0RE_1RE1RD_1LE1LB|halt}}
|-
|BBt(3,3,2)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBt(3,3)
|202
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB|halt}}
|-
|BBt(4,3,2)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== Semi-infinite tape instructions ===
Maximum number of right steps done by a Turing machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBti(3)
|3
|{{TM|1RB---_1LB0RB|halt}}
|-
|BBti(4)
|8
|{{TM|1RB---_1RC---_1LC0RC|halt}}
|-
|BBti(5)
|18
|{{TM|1RB------_1RC------_2LC2RC0RC|halt}}
|-
|BBti(6)
|36
|{{TM|1RB------_1RC------_1RD------_2LD2RD0RD|halt}}
|-
|BBti(7)
|93
|{{TM|1RB------_0RC0RB1LC_1LB2RC0LB|halt}}
|-
|BBti(8)
|425
|{{TM|1RB------_1RC------_0RD0RC1LD_1LC2RD0LC|halt}}
|}

=== [[Blanking Busy Beaver Function|Blanking]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLB(2,2,1)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLB(2,2)
|8
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3,2,1)
|16
|{{TM|1RB---_1LC0LC_1RC0LB}}
|-
|BLB(2,3,1)
|22
|{{TM|1RB2LB---_2LA0RB1LB}}
|-
|BLB(3,2)
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(2,3)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(4,2,1)
|169
|{{TM|1RB---_0RC0LA_1LC1LD_0RB0RD}}
|-
|BLB(3,3,2)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLB(3,3,1)
|308
|{{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}
|-
|BLB(4,2)
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5,2)
|32,810,047
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6,2)
|65,538,549
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(2,4)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|Blanking instructions]] ===
Maximum number of steps done by a Turing machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|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)
|173
|{{TM|1RB------_2RC2RB1LB_2LC2RB0RC}}
|-
|BLBi(8)
|<math>1.367 \times 10^{12}</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

=== Restricted move per state ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k+1 undefined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBms(2,2,1)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBms(2,2)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBms(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBms(3,2)
|19
|{{TM|1RB1RA_0RC---_1LC0LA}}
|-
|BBms(4,2,1)
|32
|{{TM|1RB---_1LC---_0LD0LC_1RD0RA}}
|-
|BBms(4,2)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBms(5,2,1)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBms(5,2)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== Restricted move per state instructions ===
Maximum number of steps done by a Turing machine with n defined instructions before halting. TMs must go to the same direction per state. TMs halt when they are on an undefined transititon.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBmsi(2)
|4
|{{TM|0RB_1LA|halt}}
|-
|BBmsi(3)
|6
|{{TM|1RB---_1LB1LA}}
|-
|BBmsi(4)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBmsi(5)
|23
|{{TM|1RB------_0RC------_1LC2LB0LA}}
|-
|BBmsi(6)
|66
|{{TM|1RB------_1LB0LC---_1RC2RA1RB}}
|-
|BBmsi(7)
|75
|{{TM|1RB0RD_0RC0RA_1LC0LA_0RB---}}
|-
|BBmsi(8)
|87
|{{TM|1RB1RD_1RC---_1RD0RA_1LE0LD_---0LA}}
|-
|BBmsi(9)
|219
|{{TM|1RB1RE_0LC1LC_1RA1RD_1LD0LB_0RC---}}
|}

=== [[Reversible Turing Machine|Reversible]] ===
Maximum number of steps done by a n-state, m-symbol reversible Turing machine with k+1 instructions transitions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBr(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|BBr(2,2)
|6
|{{TM|0RB---_1LA1RB|halt}}
|-
|BBr(3,2)
|17
|{{TM|0RB---_0LC1RA_1RB1LC|halt}}
|-
|BBr(4,2)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC---|halt}}
|-
|BBr(5,2)
|388
|{{TM|1RB0RD_1RC0RB_1RD---_1LE1LA_0LE0LA|halt}}
|-
|BBr(6,2)
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA---_1RF1RA|halt}}
|-
|BBr(7,2)
|<math>10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB---|halt}}
|}

=== [[Maximum Consecutive Ones Function|Consecutive ones]] ===
Maximum number of consecutive 1s left on the tape by a n-state, m-symbol Turing machine with k+1 undefined instructions after halting. TMs must leave all their 1s consecutively.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBn(2,2,1)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|BBn(2,2)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBn(3,2)
|6
|{{TM|1RB1LC_1RC---_1LA0LB|halt}}
|-
|BBn(2,3)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBn(4,2)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_---0RA|halt}}
|-
|BBn(3,3)
|12
|{{TM|1RB1RA---_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|BBn(5,2)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_---0LB|halt}}
|-
|BBn(4,3)
|<math>10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_---1LA1LD_0RB2RA2RD|halt}}
|}

=== [[Non-halting Turing machine#Translated cycler preperiod|Preperiodic]] ===
Maximum number of steps done by a n-state, m-symbol Turing machine with k undefined instructions before turning into a translated cycler.
{| class="wikitable"
!Domain
!Lower Bound
!
|-
|BBS(2,2,1)
|4
|{{TM|1RB---_1LB0RB}}
|-
|BBS(2,2)
|9
|{{TM|1RB0LB_1LA0RB}}
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(2,4)
|<math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|-
|BBP(5,2,1)
|<math>5.418 \times 10^{51}</math>
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|-
|BBP(5,2)
|<math>10^{14,006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BBP(3,3)
|<math>10 {\uparrow}^{2} 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}

=== [[Non-halting Turing machine#Translated cycler period|Periodic]] ===
Maximum translated cycler period done by a n-state, m-symbol Turing Machine with k undefined instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBP(2,2,1)
|3
|{{TM|1RB---_0LB1RA}}
|-
|BBP(2,2)
|9
|{{TM|1RB0RB_1LB1RA}}
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(3,3)
|1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|-
|BBP(2,4)
|33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|-
|BBP(4,2)
|212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|-
|BBP(5,2,1)
|8,468,569,863
|{{TM|1RB1RD_1LC0RC_1RA1LD_0RE0LB_---1RC}}
|}

== Turing machine variations ==

=== [[Terminating Turmite|Turmite]] ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k+1 undefined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TT(2,2,1)
|6
|<code>0PB_1TA</code>
|-
|TT(2,2)
|13
|<code>1TB---_1PA0PB</code>
|-
|TT(2,3,1)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TT(3,2,1)
|20
|<code>1TB---_1PC0PB_1PA---</code>
|-
|TT(3,2)
|82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4,2,1)
|139
|<code>1PB---_0PD0TC_---0TA_1TA1PA</code>
|-
|TT(2,3)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3,1)
|683
|<code>1TB0PA---_1PC1PB0TC_2PA1PA---</code>
|-
|TT(4,2)
|758
|<code>1TB---_0PD1PB_1PA1TA_0PC0PD</code>
|-
|TT(2,4)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TT(3,3)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite instructions ===
Maximum number of steps done by a Turmite machine with n defined instructions before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTi(2)
|6
|<code>0PB_1TA</code>
|-
|TTi(3)
|13
|<code>1TB---_1PA0PB</code>
|-
|TTi(4)
|20
|<code>1TB------_1PA2PB0PB</code>
|-
|TTi(5)
|223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TTi(7)
|1,068
|<code>1TB3TB2PB---_2TB1PA0PA2TB</code>
|-
|TTi(8)
|45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|}

=== Turmite semi-infinite tape ===
Maximum number of right steps done by a n-state, m-symbol Turmite machine with k undefined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTt(2,2,1)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTt(2,2)
|4
|<code>1PB0TB_1TA0PB</code>
|-
|TTt(2,3,1)
|9
|<code>1PB2TB---_1TA0PB0TB</code>
|-
|TTt(3,2,1)
|9
|<code>0PB---_1PC1TC_1TB0PC</code>
|-
|TTt(3,2)
|23
|<code>0PB0PC_1PC0TA_1TB1TC</code>
|-
|TTt(2,3)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTt(2,4,1)
|44
|<code>1PB2PB0PB---_2TB2TA3TA1TB</code>
|-
|TTt(4,2,1)
|48
|<code>0PB---_1TB1PC_0TB1TD_0PC0PD</code>
|-
|TTt(3,3,2)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTt(3,3)
|134
|<code>1PB2PC2TB_1TC0TA0PB_2TC1TA2TA</code>
|-
|TTt(4,2)
|166
|<code>1PB0TB_0PC0TA_0PD0PB_1TB1TA</code>
|-
|TTt(2,4)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite semi-infinite tape instructions ===
Maximum number of right steps done by a Turmite machine with n defined instructions on a semi-infinite tape before its head leave the tape. The head starts in the first cell. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTti(3)
|3
|<code>0PB---_1TB1PB</code>
|-
|TTti(4)
|8
|<code>0PB---_0PC---_1TC1PC</code>
|-
|TTti(5)
|21
|<code>0PB---_0PC---_0PD---_1TC0TD</code>
|-
|TTti(6)
|39
|<code>2PB1TA1TB_2TA0TA0PB</code>
|-
|TTti(7)
|70
|<code>1PB---0PB_0PC---1PC_2TC0TA2PC</code>
|-
|TTti(8)
|194
|<code>0PB2PB3TB0PA_1TB2TB3PB3TA</code>
|}

=== Turmite blanking ===
Maximum number of steps done by a n-state, m-symbol Turmite machine with k undefined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLT(2,2,1)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLT(2,2)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLT(2,3,1)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLT(3,2,1)
|24
|<code>1PB---_0PC0TC_1TA0TA</code>
|-
|TLT(3,2)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(4,2,1)
|84
|<code>1TB1PC_1TA0PA_1PA0TB</code>
|-
|TLT(2,3)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLT(4,2)
|267
|<code>1TB0TD_1PC0PA_0PA0TD_1PD0PB</code>
|-
|TLT(2,4)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLT(3,3)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

=== Turmite blanking instructions ===
Maximum number of steps done by a turmite machine with n defined instructions before blanking the tape again. TMs must write a non-zero symbol on the first step. TMs are disqualified if they halt before blanking the tape.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TLTi(3)
|7
|<code>1TB0TB_0PA---</code>
|-
|TLTi(4)
|12
|<code>1PB0PA_1TA0TA</code>
|-
|TLTi(5)
|24
|<code>1TB2PB0PA_2TA---0TA</code>
|-
|TLTi(6)
|108
|<code>1TB0TA0PB_2PA2PB0PA</code>
|-
|TLTi(8)
|357
|<code>1TB2TB0TB2PA_3TA2PA0PA3PB</code>
|-
|TLTi(9)
|2,122
|<code>1TB2PA0TA_1TA0TB2PC_2PB0PA0TC</code>
|}

== Lambda calculus ==

=== [[Busy Beaver for SKI calculus#SK calculus|SK calculus]] ===
Maximum normal form size of any closed SK term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|SKBB(4)
|4
|<code>SSSS</code>
|-
|SKBB(5)
|6
|<code>SSS(SS)</code>
|-
|SKBB(6)
|8
|<code>SSS(SSS)</code>
|-
|SKBB(7)
|10
|<code>SSS(SSSS)</code>
|-
|SKBB(8)
|23
|<code>SSS(S(SKS))S</code>
|}

=== BCKW calculus ===
Maximum normal form size of any closed BCKW term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCKWBB(3)
|3
|<code>BBB</code>
|-
|BCKWBB(4)
|5
|<code>WB(BB)</code>
|-
|BCKWBB(5)
|7
|<code>WB(BBB)</code>
|-
|BCKWBB(6)
|11
|<code>WB(WB(BB))</code>
|-
|BCKWBB(7)
|15
|<code>WB(WB(BBB))</code>
|-
|BCKWBB(8)
|23
|<code>WB(WB(WB(BB)))</code>
|}

=== [[Busy Beaver for lambda calculus#De Bruijn|De Bruijn]] ===
Maximum normal form size of any closed lambda term of size n.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBL(7)
|7
|<code>\1 1 1 1 1 1</code>
|-
|BBL(8)
|16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBL(9)
|68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBL(10)
|<math>7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(11)
|<math>10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBL(12)
|<math>10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBL(13)
|<math>10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBL(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>
|-
|BBL(15)
|<math>f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBL(23)
|<math>f_{\omega^2}(4)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 1))</code>
|-
|BBL(24)
|<math>f_{\omega^2}(27)</math>
|<code>(\1 1 (\\1 (\\1 2 1) 2 1) (\1 1) 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBL(25)
|<math>f_{\omega^\omega}(4)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|-
|BBL(26)
|<math>f_{\omega^\omega}(27)</math>
|<code>(\1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(27)
|<math>f_{\omega^\omega}(10^{12})</math>
|<code>(\1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(28)
|<math>f_{\omega^\omega}(10^{10^{12}})</math>
|<code>(\1 1 1 1 (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 (2 1)))</code>
|-
|BBL(34)
|<math>f_{\omega^{\omega^\omega}}(4)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) (\\\1 3 2 1) (\\1 2 1) (\1 1) 1) (\\\2 (2 1))</code>
|}

== Tag system ==

=== 2-tag system ===
Maximum number of transformations performed with a 2-tag system of size n. Its size if the sum of the number of letters and the number of different letters.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TTBB(4)
|1
|<code>Δ: aa P: a</code>
|-
|TTBB(5)
|2
|<code>Δ: aaa P: a</code>
|-
|TTBB(6)
|3
|<code>Δ: aaaa P: a</code>
|-
|TTBB(7)
|4
|<code>Δ: aaaaa P: a</code>
|-
|TTBB(8)
|5
|<code>Δ: aaaaaa P: a</code>
|-
|TTBB(9)
|8
|<code>Δ: aaa P: bbb, a</code>
|-
|TTBB(10)
|13
|<code>Δ: aaaa P: bbb, a</code>
|-
|TTBB(11)
|20
|<code>Δ: aaaaa P: bbb, a</code>
|-
|TTBB(12)
|24
|<code>Δ: aaa P: bc, a, aaa</code>
|}

=== Cyclic tag ===
Maximum number of transformations that can be done with a cyclic tag system of size n. Its size if the sum of the number of bits and the number of instructions.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|CTBB(2)
|1
|<code>Δ: 1 P: _</code>
|-
|CTBB(3)
|2
|<code>Δ: 11 P: _</code>
|-
|CTBB(4)
|4
|<code>Δ: 11 P: 0</code>
|-
|CTBB(5)
|6
|<code>Δ: 111 P: 0</code>
|-
|CTBB(6)
|9
|<code>Δ: 111 P: 00</code>
|-
|CTBB(7)
|12
|<code>Δ: 1111 P: 00</code>
|-
|CTBB(8)
|16
|<code>Δ: 1111 P: 000</code>
|-
|CTBB(9)
|20
|<code>Δ: 11111 P: 000</code>
|-
|CTBB(10)
|25
|<code>Δ: 11111 P: 0000</code>
|}

=== Tree rewriting ===
Maximum number of transformations that can be done with a tree rewriting system of size n. Its size if the number of brackets.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|TRBB(2)
|1
|<code>Δ: () P: () > _</code>
|-
|TRBB(3)
|2
|<code>Δ: ()() P: () > _</code>
|-
|TRBB(4)
|3
|<code>Δ: ()()() P: () > _</code>
|-
|TRBB(5)
|4
|<code>Δ: ()()()() P: () > _</code>
|-
|TRBB(6)
|6
|<code>Δ: ()()() P: () > (), () > _</code>
|-
|TRBB(7)
|8
|<code>Δ: ()()()() P: () > (), () > _</code>
|-
|TRBB(8)
|10
|<code>Δ: ()()()()() P: () > (), () > _</code>
|-
|TRBB(14)
|38
|<code>Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()</code>
|-
|TRBB(23)
|672
|<code>Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _</code>
|-
|TRBB(38)
|<math>10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 54</math>
|<code>Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _</code>
|-
|TRBB(45)
|<math>10 {\uparrow}^{3} 130</math>
|<code>Δ: ((()))((())) P: ((())) > ((()))(()()), (()()) > (()())(()), (()) > (())(()()()), (()()()) > (()()())()(), ()()() > _, () > _</code>
|-
|TRBB(69)
|<math>10 {\uparrow}^{4} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 4</math>
|<code>Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _</code>
|}

== Fractran ==

=== [[Fractran]] ===
Maximum number of instructions applied by a fractran program of size n before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BBf(10)
|7
|<code>[729/2, 1/3]</code>
|-
|BBf(11)
|10
|<code>[27/2, 25/3, 1/5]</code>
|-
|BBf(12)
|13
|<code>[81/2, 25/3, 1/5]</code>
|-
|BBf(13)
|17
|<code>[81/2, 125/3, 1/5]</code>
|-
|BBf(14)
|21
|<code>[243/2, 125/3, 1/5]</code>
|-
|BBf(15)
|28
|<code>[1/45, 4/5, 3/2, 25/3]</code>
|-
|BBf(16)
|53
|<code>[1/45, 4/5, 3/2, 125/3]</code>
|-
|BBf(17)
|107
|<code>[5/6, 49/2, 3/5, 40/7]</code>
|-
|BBf(18)
|211
|<code>[5/6, 49/2, 3/5, 80/7]</code>
|-
|BBf(19)
|370
|<code>[5/6, 49/2, 3/5, 160/7]</code>
|-
|BBf(20)
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|-
|BBf(21)
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|-
|BBf(22)
|<math>1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|}

== Brainfuck ==

=== Brainfuck ===
Maximum number of instructions read by a brainfuck program with n instructions (uses <code>+-<>[]</code> symbols). Cells size are unbounded and can have negative values.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BFBB(6)
|6
|<code>++++++</code>
|-
|BFBB(7)
|8
|<code>++++[-]</code>
|-
|BFBB(8)
|12
|<code>+++[--+]</code>
|-
|BFBB(9)
|16
|<code>++++[--+]</code>
|-
|BFBB(10)
|20
|<code>+++++[--+]</code>
|-
|BFBB(11)
|38
|<code>+[[>]-<+<+]</code>
|}

=== Boolfuck ===
Maximum number of instructions read by a boolfuck program with n instructions (uses <code>*<>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BLBB(7)
|7
|<code>>>>>>>></code>
|-
|BLBB(8)
|9
|<code>*>*[<<>]</code>
|-
|BLBB(9)
|11
|<code>*>>>*[<*]</code>
|-
|BLBB(10)
|14
|<code>*>*>*[<<>]</code>
|-
|BLBB(11)
|17
|<code>*>>>*[<<>*]</code>
|-
|BLBB(12)
|21
|<code>*>>>>*[<<>*]</code>
|-
|BLBB(13)
|25
|<code>*>>>>>*[<<>*]</code>
|-
|BLBB(14)
|58
|<code>+>+[<+<+[>]+<]</code>
|-
|BLBB(17)
|109
|<code>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(19)
|112
|<code>+>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(20)
|180
|<code>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(22)
|183
|<code>+>+>>+>>+>>+[>>+[<]+<]</code>
|-
|BLBB(23)
|269
|<code>+>>+>>+>>+>>+[>>+[<]+<]</code>
|}

=== BitChanger ===
Maximum number of instructions read by a bitchanger program with n instructions (uses <code><}[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BCBB(7)
|7
|<code><nowiki>}}}}}}}</nowiki></code>
|-
|BCBB(8)
|9
|<code><nowiki>}}}}<[<]</nowiki></code>
|-
|BCBB(9)
|13
|<code><nowiki>}}}<[}<<]</nowiki></code>
|-
|BCBB(10)
|17
|<code><nowiki>}}}}<[}<<]</nowiki></code>
|-
|BCBB(11)
|21
|<code><nowiki>}}}}}<[}<<]</nowiki></code>
|-
|BCBB(12)
|25
|<code><nowiki>}}}}}}<[}<<]</nowiki></code>
|}

=== Bitter ===
Maximum number of instructions read by a bitter program with n instructions (uses <code><>[]</code> symbols).
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|BTBB(7)
|7
|<code>>>>>>>></code>
|-
|BTBB(8)
|9
|<code>>>>><[<]</code>
|-
|BTBB(9)
|11
|<code>>>>>><[<]</code>
|-
|BTBB(10)
|13
|<code>>>>>>><[<]</code>
|-
|BTBB(11)
|15
|<code>>>>>>>><[<]</code>
|-
|BTBB(12)
|17
|<code>>>>>>>>><[<]</code>
|}

== Minsky machines ==

=== [[Register machine|Minsky machines]] ===
Maximum number of steps done by a Minsky machine before halting.
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|MBB(1)
|1
|<code>0+*</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>
|}

User:Azerty

2025-12-28T21:09:21Z

Azerty: Added champions zoology link.

I am an amateur interested in computability theory, large numbers and BB functions.

My GitHub repositories:

* [https://azertywastaken.github.io/TuringMachinesSimulator/tm_simulator Turing machines simulator]
* [https://github.com/AzertyWasTaken/OrdinalNotations Ordinal notations]
* [https://github.com/AzertyWasTaken/SearchBB Search BB champions script]

My user pages:

* [[User:Azerty/Busy Beaver Functions|List of lower bounds and champions]]
* [[User:Azerty/Turing Completeness|List of Turing-complete systems]]
* [[User:Azerty/Champions Zoology|Champions Zoology]]