BusyBeaverWiki - User contributions [en]

SKI Calculus

2026-05-11T01:42:18Z

Sligocki: link LC

A '''SKI calculus''' program is a binary tree where the leaves are combinators, the three symbols <code>S</code>, <code>K</code>, <code>I</code>. Using parentheses to notate the tree, a simple example of a SKI program is <code>(((SK)S)((KI)S))</code>. We can omit parentheses by assuming they are left-binding by default, so we simplify our program to <code>SKS(KIS)</code>.

Like [[Lambda Calculus|lambda calculus]], SKI calculus has a process called beta-reduction. We change the tree according to any reducible redex.

* <code>Ix -> I</code>
* <code>Kxy -> Kx</code>
* <code>Sxyz -> Sxz(yz)</code>

Note that <code>xyz</code> represent any valid trees, not just single combinators. We repeat this process and we say it terminates if the combinator cannot be beta-reduced.

Busy Beaver for SKI calculus ('''BB_SKI''') is a variation of the [[Busy Beaver for lambda calculus|Busy Beaver problem for lambda calculus]]. BB_SKI(n) is defined as the size of the largest output of a terminating program of size n.

== Champions ==
{| class="wikitable"
! n !! Value !! Champion !! Discoverered by
|-
| 1 || = 1 || S || ?
|-
| 2 || = 2 || SS || ?
|-
| 3 || = 3 || SSS || ?
|-
| 4 || = 4 || SSSS || ?
|-
| 5 || = 6 || SSS(SS) || ?
|-
| 6 || ≥ 17 || SSS(SI)S || ?
|-
|7
|≥ 40
|S(SS)S(SS)S
|?
|-
|8
|≥ 41
|SII(S(S(SS)))S
|?
|-
|9
|≥ 79
|SII(SS(SSS))S
|?
|-
|10
|≥ 164
|SII(SS(SS(SS)))S
|?
|-
|11
|≥ 681
|SII(SS(SS(SSS)))S
|?
|-
|12
|≥ 1530
|SII(SS(SS(SS(SS))))S
|?
|-
|13
|≥ 65537
|S(S(SI))I(S(S(KS)K)I)KK
|?
|-
|14
|≥ 2^256+1
|S(S(S(SI)))I(S(S(KS)K)I)KK
|?
|-
|15
|> 2^2^2^2^21
|S(S(SSS)I)I(S(S(KS)K)I)KK
|?
|-
|16
|> 2^^19
|S(S(S(SSS))I)I(S(S(KS)K)I)KK
|?
|-
|17
|> 2^^2^128
|SSK(S(S(KS)K)I)(S(SI(SI))I)KK
|?
|-
|18
|> 2^^2^2^2^2^21
|SSK(S(S(KS)K)I)(S(S(SSS)I)I)KK
|?
|-
|19
|> 2^^^2^128
|S(SSK(S(SI(SI))I))I(S(S(KS)K)I)KK
|?
|-
|20
|> 2^^^2^2^2^2^21
|S(SSK(S(S(SSS)I)I))I(S(S(KS)K)I)KK
|?
|-
|21
|> 2^^^2^^19
|S(SSK(S(S(S(SSS))I)I))I(S(S(KS)K)I)KK
|?
|-
|25
|> Graham's Number
|SII(SI(SI(K(S(K(S(K(SS(K(K(S(S(KS)K)I)))))(SI)))K))))
| 2014MELO03
|}

== SK calculus ==
We can remove the <code>I</code> combinator and replace it by <code>(SKS)</code>, <code>(SKK)</code> or any <code>(SKx)</code>. These terms have a straightforward binary encoding where (prefix) application is 1, K=00, and S=01. Since n combinators take n-1 applications to combine, their code length is 2n + n-1 = 3n-1 bits.

=== Champions ===
{| class="wikitable"
! n !! bits !! Value !! Champion !! Discoverered by
|-
| 1 || 2 || = 1 || S || ?
|-
| 2 || 5 || = 2 || SS || ?
|-
| 3 || 8 || = 3 || SSS || ?
|-
| 4 || 11 || = 4 || SSSS || ?
|-
| 5 || 14 || = 6 || SSS(SS) || ?
|-
| 6 || 17 || ≥ 10 || SSS(SS)S || ?
|-
|7
| 20
|≥ 40
|S(SS)S(SS)S
|?
|-
|8
| 23
|≥ 41
|S(S(SS)S(SS)S)
|?
|-
|9
| 26
|≥ 42
|S(S(S(SS)S(SS)S))
|?
|-
|10
| 29
|≥ 66
|SS(SSS)(SS(SS))S
|?
|-
|11
| 32
|≥ 79
|SS(SSS)(SS(SSS))S
|?
|-
|12
| 35
|≥ 164
|SS(SKK)(SS)(SS(SS))S
|?
|-
|13
| 38
|≥ 681
|SS(SKK)(SS)(SS(SSS))S
|?
|-
|14
| 41
|≥ 1530
|SS(SKK)(SS)(SS(SS(SS)))S
|?
|-
|15
| 44
|≥ 7811
|SS(SKK)(SS)(SS(SS(SSS)))S
|?
|}

== See Also ==
[https://web.archive.org/web/20250217071017/https://komiamiko.me/math/ordinals/2020/06/21/ski-numerals.html Lower bounds of this function] (archived)

[https://dallaylaen.github.io/ski-interpreter/ SKI interpreter]

[[Category:Functions]]

General Recursive Function

2026-05-08T16:54:20Z

Sligocki: /* Champions */ BBµ(17) ≥ 2090

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C^k(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C^k(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R^{k+1}(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M^k(f) \in GRF_k</math> is the µ-operator which allows unlimited search.
The arity superscripts for C, R and M are redundant (except for <math>C^k(h)</math>) and so are sometimes omitted below.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

''R'' can be recursively evaluated following its definition directly or it can be simulated as a bounded for loop like this python code:<pre>
# f = R(g,h)
def f(n, *xs):
acc = g(*xs)
for k in range(n):
acc = h(k, acc, *xs)
return acc
</pre>The iterative function ''h'' is passed two synthetic args in the front: (''k'') iteration number (0-indexed) and (''acc'') "accumulator" from previous iterations.
Both the base and iterative cases can be parameterized by values <math>x_2, x_3, ..., x_k</math>.

If you restrict functions to only those constructible with Z, S, P, C, R (no M) then this defines the Primitive Recursive Functions, a subset of the General Recursive Functions that always halt.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return (this is the only way for a GRF to not halt).

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|≥ 47
|<math>M(C(R(S, R(P^2_1, R(P^3_2, C(R(S, P^3_1), P^5_3, P^5_2)))), P^1_1, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1501347538287067267 5 May 2026]
|
|-
|17
|≥ 2090
|<math>M^0(C^1(R^2(P^1_1, R^3(P^2_1, R^4(R^3(R^2(S, C^3(S, P^3_2)), P^4_1), P^5_2))), P^1_1, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1502314241766588468 8 May 2026]
|
|-
|18
|
|
|
|
|-
|19
|
|
|
|
|-
|20
|
|
|
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

Main Page

2026-05-08T14:39:54Z

Sligocki: /* This Month in Beaver Research (TMBR) */ Update TMBR links

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 fewer 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: April 2026]]
* [[TMBR: March 2026]]
* [[TMBR: February 2026]]

[[TMBR: May 2026]] is currently work in progress, This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

TMBR: April 2026

2026-05-08T02:43:50Z

Sligocki: Cleanup lede and remove hard-coded link naming (so that automatic works correctly).

{{TMBRnav|March 2026|May 2026}}

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed below 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids. The first BBµ champion was found that takes advantage of the minimization (M) operator. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
**BBf(n) was added to OEIS: [[oeis:A395424|A395424]]
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** A couple of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482] and size 81, by Shawn Ligocki on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>).
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits].
**Tromp's BB Lambda paper got published in the journal [https://www.mdpi.com/1099-4300/28/5/494 Entropy].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction. He also found several new high-scoring halters, current places 4 through 8 in the 4x3 Busy Beaver game. 4th place is {{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}} with approximate sigma score ~10↑↑1023.47221. [https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-08T02:36:23Z

Sligocki: /* BB Adjacent */ OEIS

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
**BBf(n) was added to OEIS: [[oeis:A395424|A395424]]
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A couple of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] and size 81, by Shawn Ligocki on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>).
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits].
**Tromp's BB Lambda paper got published in the journal [https://www.mdpi.com/1099-4300/28/5/494 Entropy].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction. He also found several new high-scoring halters, current places 4 through 8 in the 4x3 Busy Beaver game. 4th place is {{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}} with approximate sigma score ~10↑↑1023.47221. [https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

General Recursive Function

2026-05-07T18:35:17Z

Sligocki: /* Champions */ BBµ(17) ≥ 1244

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C^k(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C^k(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R^{k+1}(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M^k(f) \in GRF_k</math> is the µ-operator which allows unlimited search.
The arity superscripts for C, R and M are redundant (except for <math>C^k(h)</math>) and so are sometimes omitted below.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

''R'' can be recursively evaluated following its definition directly or it can be simulated as a bounded for loop like this python code:<pre>
# f = R(g,h)
def f(n, *xs):
acc = g(*xs)
for k in range(n):
acc = h(k, acc, *xs)
return acc
</pre>The iterative function ''h'' is passed two synthetic args in the front: (''k'') iteration number (0-indexed) and (''acc'') "accumulator" from previous iterations.
Both the base and iterative cases can be parameterized by values <math>x_2, x_3, ..., x_k</math>.

If you restrict functions to only those constructible with Z, S, P, C, R (no M) then this defines the Primitive Recursive Functions, a subset of the General Recursive Functions that always halt.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return (this is the only way for a GRF to not halt).

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|≥ 47
|<math>M(C(R(S, R(P^2_1, R(P^3_2, C(R(S, P^3_1), P^5_3, P^5_2)))), P^1_1, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1501347538287067267 5 May 2026]
|
|-
|17
|≥ 1244
|<math>M^{0}(R^{1}(C^{0}(S, Z^{0}), R^{2}(S, R^{3}(P^{2}_{1}, C^{4}(R^{2}(S, R^{3}(P^{2}_{1}, P^{4}_{1})), P^{4}_{4}, P^{4}_{2})))))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1502007318382903438 7 May 2026]
|
|-
|18
|
|
|
|
|-
|19
|
|
|
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T18:28:25Z

Sligocki: /* Primitive Recursion */ Inline some math

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C^k(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C^k(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R^{k+1}(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M^k(f) \in GRF_k</math> is the µ-operator which allows unlimited search.
The arity superscripts for C, R and M are redundant (except for <math>C^k(h)</math>) and so are sometimes omitted below.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly or it can be simulated as a bounded for loop like this python code:<pre>
# f = R(g,h)
def f(n, *xs):
acc = g(*xs)
for k in range(n):
acc = h(k, acc, *xs)
return acc
</pre>The iterative function ''h'' is passed two synthetic args in the front: (''k)'' iteration number (0-indexed) and (''acc)'' "accumulator" from previous iterations. Both the base and iterative cases can be parameterized by values <math>x_2, x_3, \dots, x_k</math>.

If you restrict functions to only those constructible with Z, S, P, C, R (no M) then this defines the Primitive Recursive Functions, a subset of the General Recursive Functions that always halt.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return (this is the only way for a GRF to not halt).

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T18:27:38Z

Sligocki: Add example of how to evaluate R using iteration (instead of recursion) and mention PRF

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C^k(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C^k(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R^{k+1}(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M^k(f) \in GRF_k</math> is the µ-operator which allows unlimited search.
The arity superscripts for C, R and M are redundant (except for <math>C^k(h)</math>) and so are sometimes omitted below.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly or it can be simulated as a bounded for loop like this python code:<pre>
# f = R(g,h)
def f(n, *xs):
acc = g(*xs)
for k in range(n):
acc = h(k, acc, *xs)
return acc
</pre>The iterative function ''h'' is passed two synthetic args in the front: (''k)'' iteration number (0-indexed) and (''acc)'' "accumulator" from previous iterations. Both the base and iterative cases can be parameterized by values <math>x_2, x_3, \dots, x_k</math>.

If you restrict functions to only those constructible with Z, S, P, C, R (no M) then this defines the Primitive Recursive Functions, a subset of the General Recursive Functions that always halt.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return (this is the only way for a GRF to not halt).

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T18:14:26Z

Sligocki: /* Combinators */ Add arity superscript to combinators and note they are usually redundant.

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C^k(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C^k(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R^{k+1}(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M^k(f) \in GRF_k</math> is the µ-operator which allows unlimited search.
The arity superscripts for C, R and M are redundant (except for <math>C^k(h)</math>) and so are sometimes omitted below.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

TMBR: May 2026

2026-05-04T18:08:19Z

Sligocki: GRF

{{TMBRnav|April 2026|June 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).''

== BB Adjacent ==
[[General Recursive Function|General Recursive Functions]]:

* Some new cryptids were hand-built:
** Size 56, by Shawn on 2 May (simulating 5x+1 problem starting at 7).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Size 49, by aparker, star and Shawn on 3 May (simulating [[wikipedia:Brocard's_problem|Brocard's problem]]).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf]
* The first non-trivial divergent GRF was found (size 15). It halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n.

== Holdouts ==

*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''533,764''' via Enumerate.py, a '''0.44%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 24K subtasks out of the 1 million subtasks (or '''24%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[2]</nowiki>]

[[Category:This Month in Beaver Research|2026-05]]

TMBR: April 2026

2026-05-04T18:03:47Z

Sligocki: /* BB Adjacent */ Remove May BBµ cryptid, we just found another, so probably makes sense to just move these to May.

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A couple of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] and size 81, by Shawn Ligocki on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>).
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits].
**Tromp's BB Lambda paper got published in the journal [https://www.mdpi.com/1099-4300/28/5/494 Entropy].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction. He also found several new high-scoring halters, current places 4 through 8 in the 4x3 Busy Beaver game. 4th place is {{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}} with approximate sigma score ~10↑↑1023.47221. [https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

General Recursive Function

2026-05-04T18:00:45Z

Sligocki: /* Champions */ Remove other size 2 champions, we haven't included ties in any other size, let's keep it simple for now.

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T17:58:18Z

Sligocki: /* Tetrahedral Divisibility */ link

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T17:57:34Z

Sligocki: /* Cryptids */ Add Tetr Div GRF

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Notable Divergent GRF ==

=== Tetrahedral Divisibility ===
The GRF <math>M^{0}(C^{1}(R^{2}(S, R^{3}(P^{2}_{1}, R^{4}(P^{3}_{1}, R^{5}(R^{4}(P^{3}_{3}, P^{5}_{1}), P^{6}_{2})))), S, S))</math> (Size 15) halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>. aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n, therefore this GRF diverges.

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-05-04T17:47:26Z

Sligocki: /* Cryptids */ Build table of known results

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
So far all [[Cryptids]] have been constructed by hand. None found "in the wild" yet.
{| class="wikitable"
|+Hand Constructed Cryptids
!Size
!Problem
!Authors
!Ref
|-
|49
|[[wikipedia:Brocard's_problem|Brocard's problem]]
|aparker, star and Shawn 3 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf brocard.mgrf] [https://discord.com/channels/960643023006490684/1447627603698647303/1500605707748245524 Discord]
|-
|56
|5x+1 trajectory of 7
|Shawn Ligocki 2 May 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf collatz.mgrf]
|-
|81
|Erdos Ternary Conjecture
|Shawn Ligocki 28 Apr 2026
|[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf erdos.mgrf]
|-
|139
|Antihydra-like problem
|Jacob Mandelson 8 Apr 2026
|[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 Orig (size 141)] [https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983 Size 139]
|}

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
Most champions are primitive recursive functions. In other words they do not use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

TMBR: April 2026

2026-05-03T20:33:27Z

Sligocki: typo

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in [[Fractran]], [[GRF]] and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the Min operator. GRF Cryptids down to size 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74 bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: A size 29 program that is tetrational,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-03T16:03:30Z

Sligocki: Add Hydra Sums info

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in [[Fractran]], [[GRF]] and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the Min operator. GRF Cryptids down to size 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74 bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: A size 29 program that is tetrational,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-03T15:59:20Z

Sligocki: /* BB Adjacent */ Remove extra bit about champions w/o Min since now that was all supersceded by Min champions!

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in [[Fractran]], [[GRF]] and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the Min operator. GRF Cryptids down to size 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74 bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: A size 29 program that is tetrational,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-03T15:58:24Z

Sligocki: Expand lede and group Racheline's discoveries together.

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in [[Fractran]], [[GRF]] and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the Min operator. GRF Cryptids down to size 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74 bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: A size 29 program that is tetrational,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o M) up to size 20, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

BBµ

2026-05-03T15:53:03Z

Sligocki: Redirected page to General Recursive Function

#REDIRECT [[General Recursive Function]]

GRF

2026-05-03T15:52:49Z

Sligocki: Redirected page to General Recursive Function

#REDIRECT [[General Recursive Function]]

TMBR: April 2026

2026-05-03T15:43:11Z

Sligocki: /* BB Adjacent */ get rid of duplicate =

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o M) up to size 20, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-03T15:39:54Z

Sligocki: /* BB Adjacent */ GRF updates

{{TMBRnav|March 2026|May 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 April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: Size 141, by Jacob on 8 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] Size 81, by Shawn on 28 Apr.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] Size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o M) up to size 20, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. Therefore the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-05-01T21:13:19Z

Sligocki: 2T rule steps

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a chaotic [[probviously]] halting [[BB(6)]] TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407

Low level rules:
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

High level rules:<pre>if a ≥ floor(c/3):
A(a, b, 3k) --> A(4k+2b+1, 1, a-k+1) X -> X+4
A(a, b, 3k+1) --> A(4k+2b+3, 1, a-k+1) X -> X+5
A(a, b, 3k+2) --> A(4k+2b+5, 1, a-k+1) X -> X+6
if a < floor(c/3):
A(k, b, 3k+3) --> Halt
A(k, b, 3k+4) --> Halt
A(k, b, 3k+5+c) --> A(4k+2b+4, 2, c) X -> X+3

with X=a+2b+c</pre>where <math>X = a+2b+c</math> is a sort of norm on these configs that we can see grows only linearly.

Simulated out to 2T high level rules:
<pre>
1_000_000_000 A( 1_449_166_375, 1, 3_050_820_388)
10_000_000_000 A( 39_348_725_977, 1, 5_651_355_297)
100_000_000_000 A( 150_379_323_247, 1, 299_620_772_649)
500_000_000_000 A(2_123_188_460_901, 1, 126_811_502_474)
1_000_000_000_000 A(2_018_953_178_979, 1, 2_481_046_137_608)
1_500_000_000_000 A( 415_020_273_351, 1, 6_334_978_834_030)
2_000_000_000_000 A(1_111_898_336_656, 2, 7_888_101_240_482)
</pre>
=== Probabilistic Model ===
[[File:Tent Distr 5B.png|thumb|Distribution of ''r'' values over the first 5B steps (1000 buckets).]]
[[File:Skewed tent map.png|thumb|Map governing update of ''r'' stat in the limit. It is a Skewed or [[wikipedia:Tent_map#Asymmetric_tent_map|Asymmetric Tent Map]].]]
Consider <math>r = \frac{a}{a+c}</math>. This value seems empirically to be be extremely uniform on the range [0,1]. And for large values of a,c there is a clear reason why: The update function is a [[wikipedia:Tent_map#Asymmetric_tent_map|Skewed Tent Map]], a map which is known to have long-run time average distribution completely uniform.

If we assume that this sequence has (pseudo-)random uniform distribution, then we can calculate an exact probability of halting. Let <math>S = a+c</math> and consider at each step that the model has a fixed value of S and b, but the value of ''a'' (and consequently also ''c'') are chosen uniformly at random. If S = 4k+3, there is exactly 1/S choices that will lead to halt: A(k, b, 3k+3). The same is true for S = 4k+4. But for S = 4k+{0,1}, all choices must reset (hit a non-halt rule). Assuming that S is equally likely to have each remainder, we can say that there is a <math>\frac{1}{2S}</math> chance of halting at each high level rule. We can see above that X grows linearly with each rule application, the same is true for S since <math>b \in \{1,2\}</math>. Thus <math>\sum_t \frac{1}{2S(t)} \to \infty</math> and thus <math>P(\text{never halt}) = \prod_t \left( 1 - \frac{1}{2S(t)} \right) = 0</math>. In other words, this random model is guaranteed to eventually halt.

This model supports a survival function <math>V(T) = \prod_{t=0}^T \left( 1 - \frac{1}{2(S_0 + \mu t)} \right) </math> for <math>\mu \approx 4.5 </math> the average increase in S per step. With a bit of approximation you can get <math>V(t) \approx \left( 1 + \frac{\mu t}{S_0} \right)^{-\frac{1}{2\mu}} </math> and solving for ''t'' (to find the time when only P = V(t) have survived) you get <math>t(P) = \frac{S_0}{\mu} (P^{2\mu} - 1) </math>, if you further assume that <math>S_0 \approx N\mu </math> (the sim started from S=0, and after N steps was at <math>N\mu </math>), then you see that the median halting time (given that you have already simulated for N steps without halt) is 512N.

I (Shawn) simulated starting from all a,c such that S=a+c=2048 (and all <math>b \in \{1,2\} </math> and there were the empirical survival rates:<pre>S=2048: 4098 configs, 2264 halted within 20_000_000 steps

Survival at multiples of S:
step 2_048: 3491/4098 (85.2%)
step 4_096: 3407/4098 (83.1%)
step 8_192: 3205/4098 (78.2%)
step 16_384: 3205/4098 (78.2%)
step 32_768: 2704/4098 (66.0%)
step 65_536: 2704/4098 (66.0%)
step 131_072: 2350/4098 (57.3%)
step 262_144: 2235/4098 (54.5%)
step 524_288: 2070/4098 (50.5%)
step 1_048_576: 2070/4098 (50.5%)
step 2_097_152: 2070/4098 (50.5%)
step 4_194_304: 1834/4098 (44.8%)
step 8_388_608: 1834/4098 (44.8%)
step 16_777_216: 1834/4098 (44.8%)</pre>These seem to roughly match the model which expects that the median halting time would have been 1,048,576.
[[Category:BB(6)]]

General Recursive Function

2026-04-30T15:10:42Z

Sligocki: Add definition of size back

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (the number of atoms and combinators in the definition, [[General Recursive Function#Size|see details below]]). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Size ===
The size of a GRF is the number of atoms and combinators in the definition:

* <math>|Z^k| = |P^k_i| = |S| = 1</math>
* <math>|C(h, g_1, \cdots, g_m)| = 1 + |h| + |g_1| + \cdots + |g_m|</math>
* <math>|R(g, h)| = 1 + |g| + |h|</math>
* <math>|M(f)| = 1 + |f|</math>

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-04-30T02:58:01Z

Sligocki: /* Champions */ BBµ(14) ≥ 32!

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (defined below). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 32
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(P^{1}_{1}, P^{3}_{1}), P^{4}_{2}, P^{4}_{1}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-04-29T23:40:46Z

Sligocki: /* Champions */ New BBµ(14) champ again

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (defined below). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 13
|<math>M^{0}(C^{1}(R^{2}(P^{1}_{1}, R^{3}(P^{2}_{1}, C^{4}(R^{2}(S, P^{3}_{1}), P^{4}_{2}, P^{4}_{4}))), P^{1}_{1}, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|
|
|
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-04-29T21:16:01Z

Sligocki: /* Champions */ New BBµ(14) ≥ 9 champion

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (defined below). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 9
|<math>M(C(R(P^1_1, R(P^2_1, C(R(S, P^3_1), P^4_2, P^4_1))), P^1_1, S))</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 29 Apr 2026]
|
|-
|15
|
|
|
|
|-
|16
|≥ 10
|<math>C(AckDiag[0, Plus[2]], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492394919728517160 11 Apr 2026]
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-29T16:33:37Z

Sligocki: Run -> 1.5T

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a chaotic [[probviously]] halting [[BB(6)]] TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407

Low level rules:
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

High level rules:<pre>if a ≥ floor(c/3):
A(a, b, 3k) --> A(4k+2b+1, 1, a-k+1) X -> X+4
A(a, b, 3k+1) --> A(4k+2b+3, 1, a-k+1) X -> X+5
A(a, b, 3k+2) --> A(4k+2b+5, 1, a-k+1) X -> X+6
if a < floor(c/3):
A(k, b, 3k+3) --> Halt
A(k, b, 3k+4) --> Halt
A(k, b, 3k+5+c) --> A(4k+2b+4, 2, c) X -> X+3

with X=a+2b+c</pre>where <math>X = a+2b+c</math> is a sort of norm on these configs that we can see grows only linearly.

Simulated out to 1.5T high level rules:
<pre>
1_000_000_000 A( 1_449_166_375, 1, 3_050_820_388)
10_000_000_000 A( 39_348_725_977, 1, 5_651_355_297)
100_000_000_000 A( 150_379_323_247, 1, 299_620_772_649)
500_000_000_000 A(2_123_188_460_901, 1, 126_811_502_474)
1_000_000_000_000 A(2_018_953_178_979, 1, 2_481_046_137_608)
1_500_000_000_000 A( 415_020_273_351, 1, 6_334_978_834_030)
</pre>
=== Probabilistic Model ===
[[File:Tent Distr 5B.png|thumb|Distribution of ''r'' values over the first 5B steps (1000 buckets).]]
[[File:Skewed tent map.png|thumb|Map governing update of ''r'' stat in the limit. It is a Skewed or [[wikipedia:Tent_map#Asymmetric_tent_map|Asymmetric Tent Map]].]]
Consider <math>r = \frac{a}{a+c}</math>. This value seems empirically to be be extremely uniform on the range [0,1]. And for large values of a,c there is a clear reason why: The update function is a [[wikipedia:Tent_map#Asymmetric_tent_map|Skewed Tent Map]], a map which is known to have long-run time average distribution completely uniform.

If we assume that this sequence has (pseudo-)random uniform distribution, then we can calculate an exact probability of halting. Let <math>S = a+c</math> and consider at each step that the model has a fixed value of S and b, but the value of ''a'' (and consequently also ''c'') are chosen uniformly at random. If S = 4k+3, there is exactly 1/S choices that will lead to halt: A(k, b, 3k+3). The same is true for S = 4k+4. But for S = 4k+{0,1}, all choices must reset (hit a non-halt rule). Assuming that S is equally likely to have each remainder, we can say that there is a <math>\frac{1}{2S}</math> chance of halting at each high level rule. We can see above that X grows linearly with each rule application, the same is true for S since <math>b \in \{1,2\}</math>. Thus <math>\sum_t \frac{1}{2S(t)} \to \infty</math> and thus <math>P(\text{never halt}) = \prod_t \left( 1 - \frac{1}{2S(t)} \right) = 0</math>. In other words, this random model is guaranteed to eventually halt.

This model supports a survival function <math>V(T) = \prod_{t=0}^T \left( 1 - \frac{1}{2(S_0 + \mu t)} \right) </math> for <math>\mu \approx 4.5 </math> the average increase in S per step. With a bit of approximation you can get <math>V(t) \approx \left( 1 + \frac{\mu t}{S_0} \right)^{-\frac{1}{2\mu}} </math> and solving for ''t'' (to find the time when only P = V(t) have survived) you get <math>t(P) = \frac{S_0}{\mu} (P^{2\mu} - 1) </math>, if you further assume that <math>S_0 \approx N\mu </math> (the sim started from S=0, and after N steps was at <math>N\mu </math>), then you see that the median halting time (given that you have already simulated for N steps without halt) is 512N.

I (Shawn) simulated starting from all a,c such that S=a+c=2048 (and all <math>b \in \{1,2\} </math> and there were the empirical survival rates:<pre>S=2048: 4098 configs, 2264 halted within 20_000_000 steps

Survival at multiples of S:
step 2_048: 3491/4098 (85.2%)
step 4_096: 3407/4098 (83.1%)
step 8_192: 3205/4098 (78.2%)
step 16_384: 3205/4098 (78.2%)
step 32_768: 2704/4098 (66.0%)
step 65_536: 2704/4098 (66.0%)
step 131_072: 2350/4098 (57.3%)
step 262_144: 2235/4098 (54.5%)
step 524_288: 2070/4098 (50.5%)
step 1_048_576: 2070/4098 (50.5%)
step 2_097_152: 2070/4098 (50.5%)
step 4_194_304: 1834/4098 (44.8%)
step 8_388_608: 1834/4098 (44.8%)
step 16_777_216: 1834/4098 (44.8%)</pre>These seem to roughly match the model which expects that the median halting time would have been 1,048,576.
[[Category:BB(6)]]

Beaver Math Olympiad

2026-04-28T17:11:58Z

Sligocki: /* 8. 1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch) */ Improve floor bracket sizes

'''Beaver Mathematical Olympiad''' (BMO) is an attempt to re-formulate the halting problem for some particular Turing machines as a mathematical problem in a style suitable for a hypothetical math olympiad.

The purpose of the BMO is twofold. First, statements where non-essential details (related to tape encoding, number of steps, etc.) are discarded are more suitable to be shared with mathematicians who perhaps are able to help. Second, it's a way to jokingly highlight how a hard question could appear deceptively simple.

BMO problems have been formalized in Lean and added to the DeepMind formal-conjectures database ([https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Other/BeaverMathOlympiad.lean Github]).

== Unsolved problems ==

=== 1. {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (1, 2)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n-b_n, 4b_n+2) & \text{if }a_n \ge b_n \\
(2a_n+1, b_n-a_n) & \text{if }a_n < b_n
\end{cases}</math>

for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_i = b_i</math>?

The first 10 values of <math>(a_n, b_n)</math> are <math>(1, 2), (3, 1), (2, 6), (5, 4), (1, 18), (3, 17), (7, 14), (15, 7), (8, 30), (17, 22)</math>.

=== 2. [[Hydra]] and [[Antihydra]] ===

Let <math>(a_n)_{n \ge 0}</math> be a sequence such that <math>a_{n+1} = a_n+\left\lfloor\frac{a_n}{2}\right\rfloor</math> for all non-negative integers <math>n</math>.

# If <math>a_0=3</math>, does there exist a non-negative integer <math>k</math> such that the list of numbers <math>a_0, a_1, a_2, \dots, a_k</math> have more than twice as many even numbers as odd numbers? ([[Hydra]])
# If <math>a_0=8</math>, does there exist a non-negative integer <math>k</math> such that the list of numbers <math>a_0, a_1, a_2, \dots, a_k</math> have more than twice as many odd numbers as even numbers? ([[Antihydra]])

=== 5. {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (0, 5)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n+1, b_n-f(a_n)) & \text{if } b_n \ge f(a_n) \\
(a_n, 3b_n+a_n+5) & \text{if } b_n < f(a_n)
\end{cases}</math>

where <math>f(x)=10\cdot 2^x-1</math> for all non-negative integers <math>x</math>.

Does there exist a positive integer <math>i</math> such that <math>b_i = f(a_i)-1</math>?

=== 6. {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}} ===
Let <math>f(b) = b + k + 3a</math>, where <math>k</math> and <math>a</math> are non-negative integers satisfying <math>b = (2a+1)\cdot 2^k</math>.

Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?

=== 8. {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n-\left\lfloor\frac{b_n}{2}\right\rfloor-3, 3\left\lfloor\frac{b_n+1}{2}\right\rfloor+6) & \text{if }a_n > \left\lfloor\frac{b_n}{2}\right\rfloor \\
(3a_n+5, b_n-2a_n) & \text{if }a_n \le \left\lfloor\frac{b_n}{2}\right\rfloor
\end{cases}</math>

for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_n = \left\lfloor \frac{b_n}{2} \right\rfloor + 1</math>?

== Solved problems ==

=== 3. {{TM|1RB0RB3LA4LA2RA_2LB3RA---3RA4RB|non-halt}} and {{TM|1RB1RB3LA4LA2RA_2LB3RA---3RA4RB|non-halt}} ===

Let <math>v_2(n)</math> be the largest integer <math>k</math> such that <math>2^k</math> divides <math>n</math>. Let <math>(a_n)_{n \ge 0}</math> be a sequence such that

<math display="block">a_n = \begin{cases}
2 & \text{if } n=0 \\
a_{n-1}+2^{v_2(a_{n-1})+2}-1 & \text{if } n \ge 1
\end{cases}</math>

for all non-negative integers <math>n</math>. Is there an integer <math>n</math> such that <math>a_n=4^k</math> for some positive integer <math>k</math>?

Link to Discord discussion: https://discord.com/channels/960643023006490684/1084047886494470185/1252634913220591728

Formalised solution: [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Initial announcement], [https://discord.com/channels/960643023006490684/1259770421046411285/1488743526882738276 Lean proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488781537699696821 LLM-translated Rocq proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488898995865784442 Proof of closure of existing mid-level rules].

=== 4. {{TM|1RB3RB---1LB0LA_2LA4RA3LA4RB1LB|non-halt}} ===

Bonnie the beaver was bored, so she tried to construct a sequence of integers <math>\{a_n\}_{n \ge 0}</math>. She first defined <math>a_0=2</math>, then defined <math>a_{n+1}</math> depending on <math>a_n</math> and <math>n</math> using the following rules:

* If <math>a_n \equiv 0\text{ (mod 3)}</math>, then <math>a_{n+1}=\frac{a_n}{3}+2^n+1</math>.
* If <math>a_n \equiv 2\text{ (mod 3)}</math>, then <math>a_{n+1}=\frac{a_n-2}{3}+2^n-1</math>.

With these two rules alone, Bonnie calculates the first few terms in the sequence: <math>2, 0, 3, 6, 11, 18, 39, 78, 155, 306, \dots</math>. At this point, Bonnie plans to continue writing terms until a term becomes <math>1\text{ (mod 3)}</math>. If Bonnie sticks to her plan, will she ever finish?

<div class="toccolours mw-collapsible mw-collapsed">'''Solution'''<div class="mw-collapsible-content">
How to guess the closed-form solution: Firstly, notice that <math>a_n \approx \frac{3}{5} \times 2^n</math>. Secondly, calculate the error term <math>a_n - \frac{3}{5} \times 2^n</math>. The error term appears to have a period of 4. This leads to the following guess:

<math display="block">a_n=\frac{3}{5}\begin{cases}
2^n+\frac{7}{3} &\text{if } n\equiv 0 \pmod{4}\\
2^n-2 &\text{if } n\equiv 1 \pmod{4}\\
2^n+1 &\text{if } n\equiv 2 \pmod{4}\\
2^n+2 &\text{if } n\equiv 3 \pmod{4}
\end{cases}</math>

This closed-form solution can be proven correct by induction. Unfortunately, the induction may require a lot of tedious calculations.

For all <math>k</math>, we have <math>a_{4k} \equiv 2\text{ (mod 3)}</math> and <math>a_{4k+1} \equiv a_{4k+2} \equiv a_{4k+3} \equiv 0\text{ (mod 3)}</math>. Therefore, Bonnie will never finish.
</div></div>

=== 7. {{TM|1RB1RF_1RC0RA_1LD1RC_1LE0LE_0RA0LD_0RB---|non-halt}} ===
Let <math>v_2(n)</math> be the largest integer <math>k</math> such that <math>2^k</math> divides <math>n</math>.

Let <math>f(n) = n+1+(v_2(n+1) \bmod 2)</math>.

Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b)))</math>, <math>f^0(b)=b</math>.

Let <math>(a_n)_{n \ge 0}</math> be a sequence such that <math>a_0=1</math> and <math>a_{n+1} = f^{n+2}\left(\left\lfloor\frac{a_n}{2}\right\rfloor\right)</math> for all non-negative integers <math>n</math>.

Does there exist a non-negative integer <math>k</math> such that <math>a_k</math> is even?

(for simplicity, this question is slightly stronger than the halting problem of this TM)

Link to Discord discussion: https://discord.com/channels/960643023006490684/1421782442213376000/1431483206208852001

== Practice Problems ==
Problems that are not BMO-level, but provide counter-examples to certain [[probvious]] intuition:

* {{TM|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD}}
* {{TM|1RB0RD_0LC1RA_0RA1LB_1RE1LB_1LF1LB_---1LE}}

[[Category:Individual machines]]

1RB0LD 0RC0RF 0RD0RA 1LE0RD 1LF--- 0LA1LA

2026-04-28T16:55:43Z

Sligocki: Redirected page to 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA

#REDIRECT [[1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA]]

1RB0LD 0RC0RF 0RD0RA 1LE0RD 1LF--- 0LA1LA

2026-04-28T16:55:26Z

Sligocki: Redirect

{{machine|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA}}
#REDIRECT [[1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA]]

1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA

2026-04-28T16:54:37Z

Sligocki: Removed redirect to Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)

{{machine|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
{{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}} (and its equivalent {{TM|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA}}) is an unsolved [[BB(6)]] TM. It is the 8th [[Beaver Math Olympiad]] problem.

== Analysis by mxdys ==
https://discord.com/channels/960643023006490684/1239205785913790465/1402211374565953609
<pre>
1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA

(n+a,b,2n+c) --> (a,n+b,c)

(0,b,c) --> (5+3b,0,c)

(1,10+b,0) --> halt
(1,10+b,1) --> halt

(2,b,0) --> (2,0,8+3b)
(2,b,1) --> (2,0,11+3b)

(3+a,b,0) --> (a,0,6+3b)
(3+a,b,1) --> (a,0,9+3b)

start: (10,0,12)

this machine is weird
it does't halt in 8.9e12 rules (the first rule doesn't count)
for small (a,0,c), none of them halts between 1e5 and 1e8 rules (use the simplified model that (1,b,0) and (1,b,1) halts)
for each visited (a,0,c), log(c/a) seems to have a good distribution, which means that it doesn't fall into a fixpoint quickly
and the (2,b,0),(2,b,1) rules are used for several times:
T=462: 2 673 1
T=41231: 2 59502 0
T=6056795: 2 8748186 1
T=45863796: 2 66242895 0
T=2012212661: 2 2906495658 1
T=10335020887: 2 14928329409 1
T=468871071868: 2 677258129269 1
</pre>

== Analysis by dyuan ==
https://discord.com/channels/960643023006490684/1468065303278911613
<pre>
1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA

A(a, b, c) = 1^4 0^a <A 1^3b 0 1^c 0 1^2
A(a+1, b, c+2) → A(a, b+1, c)
A(0, b, c) → A(3b+3, 0, c+2)
A(a+2, b, 0) → A(a, 0, 3b+5)
A(a+2, b, 1) → A(a, 0, 3b+8)
A(1, b+5, 0) → Halt (after some time)
A(1, b+4, 1) → Halt (after some time)

Start: A(4, 0, 11)

B(a, b) = A(a, 0, b)

B(a+b+2, 2b+1) → B(a, 3b+8)
B(b+1, 2b+1) → Halt
B(a+b+2, 2b) → B(a, 3b+5)
B(b+1, 2b) → Halt
B(a, 2a+b) → B(3a+3, b+2)

Start: B(4, 11)
</pre>

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-28T15:48:34Z

Sligocki: /* Probabilistic Model */ Fix exponent in survival function

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a chaotic [[probviously]] halting [[BB(6)]] TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407

Low level rules:
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

High level rules:<pre>if a ≥ floor(c/3):
A(a, b, 3k) --> A(4k+2b+1, 1, a-k+1) X -> X+4
A(a, b, 3k+1) --> A(4k+2b+3, 1, a-k+1) X -> X+5
A(a, b, 3k+2) --> A(4k+2b+5, 1, a-k+1) X -> X+6
if a < floor(c/3):
A(k, b, 3k+3) --> Halt
A(k, b, 3k+4) --> Halt
A(k, b, 3k+5+c) --> A(4k+2b+4, 2, c) X -> X+3

with X=a+2b+c</pre>where <math>X = a+2b+c</math> is a sort of norm on these configs that we can see grows only linearly.

Simulated out to 1T high level rules:
<pre>
1_000_000_000 A( 1_449_166_375, 1, 3_050_820_388)
10_000_000_000 A( 39_348_725_977, 1, 5_651_355_297)
100_000_000_000 A( 150_379_323_247, 1, 299_620_772_649)
500_000_000_000 A(2_123_188_460_901, 1, 126_811_502_474)
1_000_000_000_000 A(2_018_953_178_979, 1, 2_481_046_137_608)
</pre>
=== Probabilistic Model ===
[[File:Tent Distr 5B.png|thumb|Distribution of ''r'' values over the first 5B steps (1000 buckets).]]
[[File:Skewed tent map.png|thumb|Map governing update of ''r'' stat in the limit. It is a Skewed or [[wikipedia:Tent_map#Asymmetric_tent_map|Asymmetric Tent Map]].]]
Consider <math>r = \frac{a}{a+c}</math>. This value seems empirically to be be extremely uniform on the range [0,1]. And for large values of a,c there is a clear reason why: The update function is a [[wikipedia:Tent_map#Asymmetric_tent_map|Skewed Tent Map]], a map which is known to have long-run time average distribution completely uniform.

If we assume that this sequence has (pseudo-)random uniform distribution, then we can calculate an exact probability of halting. Let <math>S = a+c</math> and consider at each step that the model has a fixed value of S and b, but the value of ''a'' (and consequently also ''c'') are chosen uniformly at random. If S = 4k+3, there is exactly 1/S choices that will lead to halt: A(k, b, 3k+3). The same is true for S = 4k+4. But for S = 4k+{0,1}, all choices must reset (hit a non-halt rule). Assuming that S is equally likely to have each remainder, we can say that there is a <math>\frac{1}{2S}</math> chance of halting at each high level rule. We can see above that X grows linearly with each rule application, the same is true for S since <math>b \in \{1,2\}</math>. Thus <math>\sum_t \frac{1}{2S(t)} \to \infty</math> and thus <math>P(\text{never halt}) = \prod_t \left( 1 - \frac{1}{2S(t)} \right) = 0</math>. In other words, this random model is guaranteed to eventually halt.

This model supports a survival function <math>V(T) = \prod_{t=0}^T \left( 1 - \frac{1}{2(S_0 + \mu t)} \right) </math> for <math>\mu \approx 4.5 </math> the average increase in S per step. With a bit of approximation you can get <math>V(t) \approx \left( 1 + \frac{\mu t}{S_0} \right)^{-\frac{1}{2\mu}} </math> and solving for ''t'' (to find the time when only P = V(t) have survived) you get <math>t(P) = \frac{S_0}{\mu} (P^{2\mu} - 1) </math>, if you further assume that <math>S_0 \approx N\mu </math> (the sim started from S=0, and after N steps was at <math>N\mu </math>), then you see that the median halting time (given that you have already simulated for N steps without halt) is 512N.

I (Shawn) simulated starting from all a,c such that S=a+c=2048 (and all <math>b \in \{1,2\} </math> and there were the empirical survival rates:<pre>S=2048: 4098 configs, 2264 halted within 20_000_000 steps

Survival at multiples of S:
step 2_048: 3491/4098 (85.2%)
step 4_096: 3407/4098 (83.1%)
step 8_192: 3205/4098 (78.2%)
step 16_384: 3205/4098 (78.2%)
step 32_768: 2704/4098 (66.0%)
step 65_536: 2704/4098 (66.0%)
step 131_072: 2350/4098 (57.3%)
step 262_144: 2235/4098 (54.5%)
step 524_288: 2070/4098 (50.5%)
step 1_048_576: 2070/4098 (50.5%)
step 2_097_152: 2070/4098 (50.5%)
step 4_194_304: 1834/4098 (44.8%)
step 8_388_608: 1834/4098 (44.8%)
step 16_777_216: 1834/4098 (44.8%)</pre>These seem to roughly match the model which expects that the median halting time would have been 1,048,576.
[[Category:BB(6)]]

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-28T02:22:15Z

Sligocki: /* Probabilistic Model */ Add more details and empirical result.

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a chaotic [[probviously]] halting [[BB(6)]] TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407

Low level rules:
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

High level rules:<pre>if a ≥ floor(c/3):
A(a, b, 3k) --> A(4k+2b+1, 1, a-k+1) X -> X+4
A(a, b, 3k+1) --> A(4k+2b+3, 1, a-k+1) X -> X+5
A(a, b, 3k+2) --> A(4k+2b+5, 1, a-k+1) X -> X+6
if a < floor(c/3):
A(k, b, 3k+3) --> Halt
A(k, b, 3k+4) --> Halt
A(k, b, 3k+5+c) --> A(4k+2b+4, 2, c) X -> X+3

with X=a+2b+c</pre>where <math>X = a+2b+c</math> is a sort of norm on these configs that we can see grows only linearly.

Simulated out to 1T high level rules:
<pre>
1_000_000_000 A( 1_449_166_375, 1, 3_050_820_388)
10_000_000_000 A( 39_348_725_977, 1, 5_651_355_297)
100_000_000_000 A( 150_379_323_247, 1, 299_620_772_649)
500_000_000_000 A(2_123_188_460_901, 1, 126_811_502_474)
1_000_000_000_000 A(2_018_953_178_979, 1, 2_481_046_137_608)
</pre>
=== Probabilistic Model ===
[[File:Tent Distr 5B.png|thumb|Distribution of ''r'' values over the first 5B steps (1000 buckets).]]
[[File:Skewed tent map.png|thumb|Map governing update of ''r'' stat in the limit. It is a Skewed or [[wikipedia:Tent_map#Asymmetric_tent_map|Asymmetric Tent Map]].]]
Consider <math>r = \frac{a}{a+c}</math>. This value seems empirically to be be extremely uniform on the range [0,1]. And for large values of a,c there is a clear reason why: The update function is a [[wikipedia:Tent_map#Asymmetric_tent_map|Skewed Tent Map]], a map which is known to have long-run time average distribution completely uniform.

If we assume that this sequence has (pseudo-)random uniform distribution, then we can calculate an exact probability of halting. Let <math>S = a+c</math> and consider at each step that the model has a fixed value of S and b, but the value of ''a'' (and consequently also ''c'') are chosen uniformly at random. If S = 4k+3, there is exactly 1/S choices that will lead to halt: A(k, b, 3k+3). The same is true for S = 4k+4. But for S = 4k+{0,1}, all choices must reset (hit a non-halt rule). Assuming that S is equally likely to have each remainder, we can say that there is a <math>\frac{1}{2S}</math> chance of halting at each high level rule. We can see above that X grows linearly with each rule application, the same is true for S since <math>b \in \{1,2\}</math>. Thus <math>\sum_t \frac{1}{2S(t)} \to \infty</math> and thus <math>P(\text{never halt}) = \prod_t \left( 1 - \frac{1}{2S(t)} \right) = 0</math>. In other words, this random model is guaranteed to eventually halt.

This model supports a survival function <math>V(T) = \prod_{t=0}^T \left( 1 - \frac{1}{2(S_0 + \mu t)} \right) </math> for <math>\mu \approx 4.5 </math> the average increase in S per step. With a bit of approximation you can get <math>V(t) \approx \left( 1 + \frac{\mu t}{S_0} \right)^\frac{1}{2\mu} </math> and solving for ''t'' (to find the time when only P = V(t) have survived) you get <math>t(P) = \frac{S_0}{\mu} (P^{2\mu} - 1) </math>, if you further assume that <math>S_0 \approx N\mu </math> (the sim started from S=0, and after N steps was at <math>N\mu </math>), then you see that the median halting time (given that you have already simulated for N steps without halt) is 512N.

I (Shawn) simulated starting from all a,c such that S=a+c=2048 (and all <math>b \in \{1,2\} </math> and there were the empirical survival rates:<pre>S=2048: 4098 configs, 2264 halted within 20_000_000 steps

Survival at multiples of S:
step 2_048: 3491/4098 (85.2%)
step 4_096: 3407/4098 (83.1%)
step 8_192: 3205/4098 (78.2%)
step 16_384: 3205/4098 (78.2%)
step 32_768: 2704/4098 (66.0%)
step 65_536: 2704/4098 (66.0%)
step 131_072: 2350/4098 (57.3%)
step 262_144: 2235/4098 (54.5%)
step 524_288: 2070/4098 (50.5%)
step 1_048_576: 2070/4098 (50.5%)
step 2_097_152: 2070/4098 (50.5%)
step 4_194_304: 1834/4098 (44.8%)
step 8_388_608: 1834/4098 (44.8%)
step 16_777_216: 1834/4098 (44.8%)</pre>These seem to roughly match the model which expects that the median halting time would have been 1,048,576.
[[Category:BB(6)]]

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-27T14:04:11Z

Sligocki: 1T rules sim

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-26T03:03:41Z

Sligocki: Sim to 500B steps

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:54:24Z

Sligocki: Add 10B result

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:40:45Z

Sligocki: /* Probabilistic Model */

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:40:14Z

Sligocki: /* Probabilistic Model */ Add some more explanation about 100% chance to halt.

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:23:14Z

Sligocki: /* Analysis by Shawn Ligocki */ Update to 100B resets

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:17:53Z

Sligocki:

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T16:17:26Z

Sligocki: Add graphs and start of Probabilistic model section

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a [[probviously]] halting [[BB(6)]] TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

Simulated out to 1B "resets" (rule count ignoring the first rule):
<pre>
1_000_000_000 A(1_449_166_375, 1, 3_050_820_388)
</pre>
Looks kind of like BMO1 where it only halts if a and c are "close" in some sense. Configs like <code>A(k,b,3k+{3,4})</code>

=== Probabilistic Model ===
[[File:Tent Distr 5B.png|thumb|Distribution of ''r'' values over the first 5B steps (1000 buckets).]]
[[File:Skewed tent map.png|thumb|Map governing update of ''r'' stat in the limit. It is a Skewed or [[wikipedia:Tent_map#Asymmetric_tent_map|Asymmetric Tent Map]].]]
Consider <math>r = \frac{a}{a+c}</math>. This value seems empirically to be be extremely uniform on the range [0,1]. And for large values of a,c there is a clear reason why: The update function is a [[wikipedia:Tent_map#Asymmetric_tent_map|Skewed Tent Map]], a map which is known to have long-run time average distribution completely uniform.

[[Category:BB(6)]]

File:Skewed tent map.png

2026-04-24T16:08:26Z

Sligocki:

Map governing update of {{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} r stat in the limit. It is a Skewed or [https://en.wikipedia.org/wiki/Tent_map#Asymmetric_tent_map Asymmetric Tent Map].

File:Tent Distr 5B.png

2026-04-24T15:18:18Z

Sligocki:

Distribution of "r" values for {{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} over the first 5B steps.

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-24T04:47:16Z

Sligocki: Add lede

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}{{TM|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} appears to be a [[probviously]] halting TM, but with no estimate for halting time. It is still under analysis as of 24 Apr 2026.

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

Simulated out to 1B "resets" (rule count ignoring the first rule):
<pre>
1_000_000_000 A(1_449_166_375, 1, 3_050_820_388)
</pre>
Looks kind of like BMO1 where it only halts if a and c are "close" in some sense. Configs like <code>A(k,b,3k+{3,4})</code>

1RB--- 0RC0RD 1LD1RB 0LE0LC 1RA0LF 1LD1LE

2026-04-23T22:41:02Z

Sligocki: Created page with "{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}} == Analysis by Shawn Ligocki == https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407 <pre> 1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf A(a+1,b,c+3) --> A(a,b+2,c) A(a,b,0) --> A(2b+1,1,a+1) A(a,b,1) --> A(2b+3,1,a+1) A(a,b,2) --> A(2b+5,1,a+1) A(0,b,c+5) --> A(2b+4,2,c) A(0,b,4) --> Halt A(0,b,3) --> Halt Start: A(0,1,0) </pre> Simulate..."

{{machine|1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE}}

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1239205785913790465/1497001816741646407
<pre>
1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE

Let A(a,b,c) = 0^inf 1^a 10^b C> 1^c 0^inf

A(a+1,b,c+3) --> A(a,b+2,c)

A(a,b,0) --> A(2b+1,1,a+1)
A(a,b,1) --> A(2b+3,1,a+1)
A(a,b,2) --> A(2b+5,1,a+1)

A(0,b,c+5) --> A(2b+4,2,c)
A(0,b,4) --> Halt
A(0,b,3) --> Halt

Start: A(0,1,0)
</pre>

Simulated out to 1B "resets" (rule count ignoring the first rule):
<pre>
1_000_000_000 A(1_449_166_375, 1, 3_050_820_388)
</pre>
Looks kind of like BMO1 where it only halts if a and c are "close" in some sense. Configs like <code>A(k,b,3k+{3,4})</code>

Fast-Growing Hierarchy

2026-04-22T19:28:56Z

Sligocki: /* Examples */ Fix f_omega(0) = 1 issue

A '''Fast-Growing Hierarchy''' (FGH) is an ordinal-indexed hierarchy of functions satisfying certain restrictions. FGHs are used for assigning growth rates to fast computable functions, and are useful for approximating scores and halting times of [[Turing machine|Turing machines]].
[[Category:Functions]]
== Definition ==

A fundamental sequence for an ordinal <math>\alpha</math> is an increasing sequence of ordinals <math><\alpha</math> which is unbounded in <math>\alpha</math>. The <math>\beta</math>-th element of <math>\alpha</math>'s fundamental sequence is denoted by <math>\alpha[\beta]</math>. In the context of FGHs, there is usually a restriction that the sequence's length must be as small as possible (that is, the length is the [https://en.wikipedia.org/wiki/Cofinality cofinality] of <math>\alpha</math>). A system of fundamental sequences for a set of ordinals is a function which assigns a fundamental sequence to each ordinal in the set.

Given a system of fundamental sequences for limit ordinals below <math>\lambda</math>, its corresponding FGH is defined by

<math display="block">\begin{array}{l}
f_0(n) & = & n+1 \\
f_{\alpha+1}(n) & = & f_\alpha^n(n) & \text{for }\alpha<\lambda \\
f_\alpha(n) & = & f_{\alpha[n]}(n) & \text{for limit ordinals }\alpha<\lambda
\end{array}</math>

Most natural fundamental sequence systems almost exactly agree on the growth rates in their corresponding FGHs. Specifically, if <math>f,f'</math> are FGHs given by natural fundamental sequence systems, it is usually the case that <math>f_\alpha(n+1)>f'_\alpha(n)</math> and <math>f'_\alpha(n+1)>f_\alpha(n)</math> for all natural <math>n</math> and all successor ordinals <math>\alpha</math>. For this reason, the specific choice of a fundamental sequence system often doesn't matter for large ordinals. For small ordinals (below <math>\varepsilon_0</math>), a common choice of fundamental sequences is given by

<math display="block">\begin{array}{l}
(\alpha+\omega^{\beta+1})[n] & = & \alpha+\omega^\beta n & \text{if }\alpha\text{ is a multiple of }\omega^{\beta+1} \\
(\alpha+\omega^\beta)[n] & = & \alpha+\omega^{\beta[n]} & \text{if }\alpha\text{ is a multiple of }\omega^\beta\text{ and }\beta\text{ is a limit ordinal}
\end{array}</math>

The FGH given by these fundamental sequences is sometimes called the Wainer hierarchy. Above <math>\varepsilon_0</math>, a relatively elegant choice is the expansion associated to the [https://apeirology.com/wiki/Bashicu_matrix_system Bashicu matrix system], which has the [https://en.wikipedia.org/wiki/Fundamental_sequence_(set_theory)#Additional_conditions Bachmann property].

== Examples ==
* <math>f_{\alpha+1}(0) = 0</math>
* <math>f_\alpha(1) = 2</math>

* <math>f_0(n) = n+1</math>
* <math>f_1(n) = 2n</math>
* <math>f_2(n) = n \cdot 2^n</math>
* <math>f_3(n) = f_2^n(n) > (2 \uparrow)^n(n + \log_2 n) > (2 \uparrow)^n n \ge 2 \uparrow\uparrow (n+1)</math> for <math>n \ge 2</math>
* <math>f_3(n) > 10 \uparrow\uparrow n</math> for <math>n \ge 4</math>
* <math>f_k(n) > 10 \uparrow^{k-1} n \; \text{ for } k \ge 4 \text{ and } n \ge 2 </math>
* <math>f_\omega(n) = f_n(n) > 10 \uparrow^{n-1} n \; \text{ for } n \ge 4</math>
* <math>f_{\omega+1}(64) = f_\omega^{64}(64) > \text{Graham's number}</math>

*

=== Table of Small Values ===
{| class="wikitable"
|+
!
! colspan="8" |n
|-
!
!0
!1
!2
!3
!4
!5
!...
!n
|-
|<math>f_0(n)</math>
|1
|2
|3
|4
|5
|6
|
|n+1
|-
|<math>f_1(n)</math>
|0
|2
|4
|6
|8
|10
|
|2n
|-
|<math>f_2(n)</math>
|0
|2
|8
|24
|64
|160
|
|<math>n \, 2^n</math>
|-
|<math>f_3(n)</math>
|0
|2
|2048
|<math>>10^{10^8}</math>
|<math>> 10^{10^{10^{20}}}</math>
|<math>> (10 \uparrow)^4 49 > 10 \uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow n</math>
|-
|<math>f_4(n)</math>
|0
|2
|<math>> 10 \uparrow\uparrow 2048</math>
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^8} > 10 \uparrow\uparrow\uparrow 3</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow\uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow\uparrow n</math>
|-
|...
|
|
|
|
|
|
|
|
|-
|<math>f_\omega(n)</math>
|1
|2
|8
|<math>>10^{10^8}</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow^4 5</math>
|
|<math>> 10 \uparrow^{n-1} n</math>
|-
|<math>f_{\omega+1}(n)</math>
|0
|2
|<math>> 10 \uparrow^7 8</math>
|
|
|
|
|
|}

== Growth Bound Theorem ==
The ''Fast-Growing Hierarchy Growth Bound Theorem'' is an important result in mathematical logic that has significant implications for unprovability results. The theorem highlights a relationship between computable functions that are provably total in first-order Peano Arithmetic (PA) and the fast-growing functions in the [[Fast-Growing Hierarchy|Wainer hierarchy]].

The theorem is based on work by several mathematicians. Georg Kreisel laid the groundwork in 1952 by investigating connections between recursions over well ordered sets and proofs in PA. These results were subsequently extended by many others; the following form is based on the presentation by Buchholz and Wainer.

=== Statement ===
Let <math>T</math> be a Turing machine that computes a function <math>g:\N\to\N</math>, terminating on every input. Suppose that PA can prove the statement «<math>T</math> terminates on every input.» Then <math>g</math> cannot grow too fast: There exist <math>\alpha < \varepsilon_0</math> and <math>n_0\in\N</math> such that <math>g(n) < F_\alpha(n)</math> for every <math>n\ge n_0</math>.

== References ==
Wilfried Buchholz and Stan S. Wainer. Provably computable functions and the fast growing hierarchy. In S. G. Simpson, editor, Logic and Combinatorics, volume 65 of Contemporary Mathematics, pages 179–198. AMS, 1987. [https://epub.ub.uni-muenchen.de/3843/1/3843.pdf]

General Recursive Function

2026-04-22T18:07:55Z

Sligocki: /* Champions */

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (defined below). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 7
|<math>C(AckDiag[0, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492246885329670184 10 Apr 2026]
|
|-
|15
|≥ 7
|<math>K^0[7]</math>
|
|
|-
|16
|≥ 10
|<math>C(AckDiag[0, Plus[2]], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492394919728517160 11 Apr 2026]
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|...
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|96
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|98
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]

General Recursive Function

2026-04-22T18:04:32Z

Sligocki: /* Champions */ Add Omega champions

'''General recursive functions''' ('''GRFs'''), also called '''µ-recursive functions''' or '''partial recursive functions''', are the collection of [[Wikipedia:partial functions|partial functions]] <math>\N^k \rightharpoonup \N</math> that are computable. This definition is equivalent using any [[Turing complete]] system of computation. See [[Wikipedia:general recursive function]] for background.

Historically it was defined as the smallest class of partial functions <math>\N^k \rightharpoonup \N</math> that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections (see formal definitions below). In the rest of this article, this is the formulation that we focus on exclusively. In this way, it can be considered to be a Turing complete model of computation. In fact, it is one of the oldest Turing complete models, first formalized by Kurt Gödel and Jacques Herbrand in 1933, 3 years before λ-calculus and Turing machines.

'''BBµ'''(n) is a [[Busy Beaver function]] for GRFs:<math display="block">BB \mu (n) = \max \{ f() | f \in \text{GRF}_0 , |f| = n \}</math>

where <math>f \in GRF_k</math> means that <math>f: \N^k \rightharpoonup \N</math> is a k-ary GRF and <math>|f|</math> is the "structural size" of ''f'' (defined below). In other words, it is the largest number computable via a 0-ary function (a constant) with a limited "program" size. It is more akin to the traditional [[Sigma score]] for a Turing machine rather than the Step function in the sense that it maximizes over the produced value, not the number of steps needed to reach that value.

== Definition ==

=== Structure ===
Define <math>GRF_k</math> inductively based on the following construction rules, start with Atoms and combine them using Combinators.

====== Atoms ======

* Zero: <math>\forall k \in \N, Z^k \in GRF_k</math> is the constant 0 function <math>Z^k(x_1, \dots, x_k) = 0</math>
* Successor: <math>S \in GRF_1</math> is the successor function <math>S(x) = x+1</math>
* Projection: <math>\forall 1 \le i \le k \in \N, P^k_i \in GRF_k</math> is a projection function <math>P^k_i(x_1, \dots x_k) = x_i</math>

===== Combinators =====

* Composition: <math>\forall k,m \in \N, \forall h \in GRF_m, \forall g_1, \dots g_m \in GRF_k, C(h, g_1, \dots g_m) \in GRF_k</math> is the composition or substitution of the ''g''s into ''h'': <math>C(h, g_1, \dots g_m)(x_1, \dots x_k) = h(g_1(x_1, \dots x_k), \dots g_m(x_1, \dots x_k))</math>
* Primitive Recursion: <math>\forall k \in \N, \forall g \in GRF_k, \forall h \in GRF_{k+2}, R(g, h) \in GRF_{k+1}</math> is primitive recursion using ''g'' as the base case and ''h'' as the inductive step.
* Minimization / Unlimited Search: <math>\forall k \in \N, \forall f \in GRF_{k+1}, M(f) \in GRF_k</math> is the µ-operator which allows unlimited search.

=== Primitive Recursion ===
<math>R</math> models a typical iterative function definition over ℕ.

Base case: <math>R^k(g, h)(0, x_2, x_3, ..., x_k) = g(x_2, x_3, ..., x_k)</math>

Iterative case (for <math>x_1 > 0</math>): <math>R^k(g, h)(x_1, x_2, ..., x_k) = h(x_1-1, v, x_2, x_3, ..., x_k)</math> where <math>v = R(g, h)(x_1-1, x_2, x_3, ..., x_k)</math>.

<math>R</math> can be recursively evaluated following its definition directly. Or it can be iterated over its first argument, starting with 0 (and thus a call to <math>g</math>), then 1, 2, 3, etc. until <math>x_1</math> is reached, each time calling <math>h</math> with the prior iteration count for its first argument and the result of the prior call for its second.

=== Minimization ===
<math>M^k(f)(x_1, ..., x_k) \triangleq \min \{i \in \N: f(i, x_1, ..., x_k) = 0\}</math>

In computational language, when ''M(f)'' is evaluated it can be considered to calculate <math>f(i, x_1, ..., x_k)</math> with <math>i=0</math>, then <math>i=1</math>, then <math>i=2</math> etc. until one of the calls to <math>f</math> returns 0, at which point it returns the value of <math>i</math> which first gave a result of 0. If no first argument causes <math>f</math> to return 0, <math>M(f)</math> doesn't return. (This is the only way for a GRF to not halt.)

== Macros ==
In order to improve readability we define the following macros. For all <math>f \in GRF_1</math>
{| class="wikitable"
|+
!
!Macro
!arity
!Definition
!Size
!Function
|-
|Constant
|<math>K^k[n]</math>
|k
|<math>\begin{array}{lcl}
K^k[0] & := & Z^k \\
K^k[n] & := & C(Plus[n], Z^k)
\end{array}</math>
|<math>2n+1</math>
|<math>\lambda x_1 \dots x_k. n</math>
|-
|Plus constant
|<math>Plus[n]</math>
|1
|<math>\begin{array}{lcl}
Plus[1] & := & S \\
Plus[n+1] & := & C(S, Plus[n])
\end{array}</math>
|<math>2n-1</math>
|<math>\lambda x. x+n</math>
|-
|Triangular numbers
|<math>Tri</math>
|1
|<math>Tri := R(Z^0, R(S, C(Plus[2], P^3_2)))</math>
|<math>7</math>
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|Iteration
|<math>RepSucc[f]</math>
|2
|<math>RepSucc[f] := R(S, C(f, P^3_2))</math>
|<math>|f| + 4</math>
|<math>\lambda x\ y. f^x(y+1)</math>
|-
|Diagonalization & Iteration
|<math>DiagRep[f]</math>
|2
|<math>DiagRep[f] := R(S, C(f, P^3_2, P^3_2))</math>
|<math>|f| + 5</math>
|<math>\lambda x\ y. (\lambda z. f(z,z))^x (y+1)</math>
|-
|Diagonalization
|<math>DiagS[f]</math>
|1
|<math>DiagS[f] := C(f, S, S)</math>
|<math>|f| + 3</math>
|<math>\lambda x. f(x+1, x+1)</math>
|-
| rowspan="2" |Ackermann iteration
|<math>AckDiag2[n,f]</math>
|2
|<math>\begin{array}{lcl}
AckDiag2[0,f] & := & RepSucc[f] \\
AckDiag2[n+1,f] & := & DiagRep[AckDiag2[n,f]]
\end{array}</math>
|<math>5n + 4 + |f|</math>
|
|-
|<math>AckDiag[n,f]</math>
|1
|<math>AckDiag[n,f] := DiagS[AckDiag2[n,f]]</math>
|<math>5n + 7 + |f|</math>
|
|}

== Champions ==
{| class="wikitable"
!n
!BBµ(n)
!Champion
!Champion Found
!Holdouts Proven
|-
|1
|= 0
|<math>Z^0</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|2
|= 0
|<math>M^0(Z^1), M^0(P^1_1), C^0(Z^0)</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|3
|= 1
|<math>K^0[1]</math>
|
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1447693296322215976 8 Dec 2025] By hand
|-
|4
|= 1
|<math>C^0(K^0[1])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|5
|= 2
|<math>K^0[2]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|6
|= 2
|<math>C^0(K^0[2])</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|7
|= 3
|<math>K^0[3]</math>
|
|Jacob Mandelson [https://mandelson.org/grf/ 3 Apr 2026]
|-
|8
|≥ 3
|<math>C^0(K^0[3])</math>
|
|
|-
|9
|≥ 4
|<math>K^0[4]</math>
|
|
|-
|10
|≥ 4
|<math>C^0(K^0[4])</math>
|
|
|-
|11
|≥ 5
|<math>K^0[5]</math>
|
|
|-
|12
|≥ 5
|<math>C^0(K^0[5])</math>
|
|
|-
|13
|≥ 6
|<math>K^0[6]</math>
|
|
|-
|14
|≥ 7
|<math>C(AckDiag[0, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492246885329670184 10 Apr 2026]
|
|-
|15
|≥ 7
|<math>K^0[7]</math>
|
|
|-
|16
|≥ 10
|<math>C(AckDiag[0, Plus[2]], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492394919728517160 11 Apr 2026]
|
|-
|17
|≥ 15
|<math>C(AckDiag[1, S], K^0[1])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 12 Apr 2026]
|
|-
|18
|≥ 21
|<math>C(AckDiag[0,Tri], K^0[1])</math>
|Jacob Mandelson [https://discord.com/channels/960643023006490684/1447627603698647303/1492021428546179182 9 Apr 2026]
|
|-
|19
|≥ 39
|<math>C(AckDiag[1, S], K^0[2])</math>
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1492997385247264941 12 Apr 2026]
|
|-
|20
|≥ 1540
|<math>C(AckDiag[0,Tri], K^0[2])</math>
|Jacob Mandelson 9 Apr 2026
|
|-
|21
|
|
|
|
|-
|22
|<math>> 10^{13.35}</math>
|<math>C(AckDiag[2, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|23
|<math>> 10^{10^{463}}</math>
|<math>C(AckDiag[1,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|24
|
|
|
|
|-
|25
|<math>> 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[1,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|26
|
|
|
|
|-
|27
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 3</math>
|<math>C(AckDiag[3, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|28
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[2,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|29
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 4</math>
|<math>C(AckDiag[3, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|30
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|<math>C(AckDiag[2,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|31
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>
|<math>C(AckDiag[3, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|
|
|
|
|
|-
|5k+17
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k+1, S], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+18
|> <math>10 \uparrow^k 10 \uparrow^k 3</math>
|<math>C(AckDiag[k,Tri], K^0[1])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+19
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 4</math>
|<math>C(AckDiag[k+1, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+20
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k,Tri], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|5k+21
|> <math>10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 10 \uparrow^k 5</math>
|<math>C(AckDiag[k+1, S], K^0[3])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|
|
|
|
|
|-
|94
|> <math>10 \uparrow^{15} 10 \uparrow^{15} 10 \uparrow^{15} 4</math>
|<math>C(AckDiag[16, S], K^0[2])</math>
|Shawn Ligocki 13 Apr 2026
|
|-
|95
|<math>> f_\omega(10^{13})</math>
|Omega() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|-
|
|
|
|
|
|-
|97
|<math>> f_\omega^3(3)</math>
|Omega3() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|
|
|
|
|
|-
|99
|<math>> f_\omega^4(3)</math>
|Omega4() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki 22 Apr 2026
|
|-
|100
|<math>> f_{\omega+1}(f_\omega^2(10^{13})) \gg \text{Graham's number}</math>
|Graham() from [https://github.com/sligocki/etc/blob/main/gen_rec/src/example_ack.rs example_ack.rs]
|Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 16 Apr 2026]
|
|}

== Cryptids ==
Jacob Mandelson constructed a size 141 [[Cryptid]] on 8 Apr 2026 [https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482], shrunk to 139 by 17 Apr 2026.[https://discord.com/channels/960643023006490684/1447627603698647303/1494889866704588983]

== Macro Bounds ==
Let <math>C = \frac{1}{\log_{10}(2)} \approx 3.32</math>

=== AckDiag[k, S] ===
Let <math>AS_k(n) := DiagRep^k[RepSucc[S]](n,n) = AckDiag[k,S](n-1)</math>

* <math>AS_0(n) = S^n(n+1) = 2n+1</math>
* <math>AS_1(n) = AS_0^n(n+1) = (n+2) 2^n - 1</math>
* <math>AS_1(n) > C \cdot 10^{n/C}</math> (for <math>n \ge 2</math>)
* <math>AS_2(n) = AS_1^n(n+1) > C \cdot (10 \uparrow)^n \left( \frac{n+1}{C} \right) > 10 \uparrow\uparrow n \left[ \uparrow \frac{n+1}{C} \right]</math> (for <math>n \ge 2</math>)
* <math>AS_k(n) = AS_{k-1}^n(n+1) > (10 \uparrow^{k-1})^n (n+1)</math> (for <math>k \ge 3</math> and <math>n \ge 1</math>)

=== AckDiag[k, Tri] ===
Let <math>AT_k(n) := DiagRep^k[RepSucc[Tri]](n,n) = AckDiag[k,Tri](n-1)</math>
* <math>Tri(n) = \frac{n(n+1)}{2} > \frac{1}{2} n^2</math>
* <math>AT_0(n) = Tri^n(n+1) > 2 \left( \frac{n+1}{2} \right)^{2^n}</math>
* <math>AT_0(2) = 21</math>, <math>AT_0(3) = 1540</math>, <math>AT_0(4) > 10^{7.42}</math>
* <math>AT_0(n) > C 10^{10^{\left(\frac{n-1}{C}\right)}} + 1</math> (for <math>n \ge 5</math>)
* <math>AT_1(n) = AT_0^n(n+1) > C (10 \uparrow)^{2n-2} \left( \frac{AT_0(n+1)}{C} \right)</math>
* <math>AT_1(2) > 10^{10^{AT_0(3) / C}} > 10^{10^{463}}</math>, <math>AT_1(3) > 10^{10^{10^{10^{AT_0(4) / C}}}} > 10 \uparrow\uparrow 5</math>
* <math>AT_1(n) > 10 \uparrow\uparrow 2n</math> (for <math>n \ge 4</math>)
* <math>AT_2(n) = AT_1^n(n+1) > (10 \uparrow\uparrow)^{n-1} (AT_1(n+1))</math>
* <math>AT_2(2) = AT_1^2(3) > AT_1(10 \uparrow\uparrow 5) > 10 \uparrow\uparrow 10 \uparrow\uparrow 5</math>, <math>AT_2(3) = AT_1^3(4) > 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
* <math>AT_2(n) > 10 \uparrow\uparrow\uparrow (n+1)</math> (for <math>n \ge 4</math>)
* <math>AT_3(n) = AT_2^n(n+1) > (10 \uparrow\uparrow\uparrow)^{n-1} (AT_2(n+1))</math>
* <math>AT_3(2) = AT_2^2(3) > AT_2(10 \uparrow\uparrow\uparrow 3) > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 3</math>

== Utilizing Minimization ==
All the current champions are primitive recursive functions. In other words none use the minimization combinator M. This fundamentally limits their growth rate. In fact, no primitive recursive function can grow faster than the Ackermann function and we can see that above where the assymtotic growth of the known BBµ bound is Ackermann growth: <math>BB\mu(6k+17) \ge 2 \uparrow^k 4</math>.

But, like the traditional BB function, BBµ grows uncomputably fast, so eventually it must surpass primitive recursive functions. In order to do that, it needs to use the M combinator. However, in order to do arbitrary computation, you need a way to store arbitrarily large amounts of data into a single integer and extract it back out. In other words, you need to implement a [[wikipedia:Pairing_function|pairing function]]. Thus there is value in finding small pairing/unpairing functions. A set of pairing functions is a triple Pair,Left,Right such that for all a,b: Left(Pair(a,b)) = a and Right(Pair(a,b)) = b. When functions consume both the left and right values, [[wikipedia:Common_subexpression_elimination|common subexpression elimination]] can be used to reduce the number of operations below that from calling Left and Right individually. The smallest known pairing functions are:
{| class="wikitable"
|+Smallest Pairing Functions
!Macro
!arity
!Definition
!Size
!Function
|-
|<math>Pair</math>
|2
|<math>Pair := C(AddS, C(Tri, Add), P^2_1)</math>
|20
|<math>\lambda xy. \frac{(x+y)(x+y+1)}{2} + x + 1</math>
|-
|<math>Left</math>
|1
|<math>Left := C(RMonus, C(TriP, InvTriCeil), Pred)</math>
|38
|<math>Left(Pair(x,y)) = x</math>
|-
|<math>Right</math>
|1
|<math>Right := C(RMonus, P^1_1, C(Tri, InvTriCeil))</math>
|36
|<math>Right(Pair(x,y)) = y</math>
|-
|<math>LRCall[f^2]</math>
|1
|<math>LRCall[f] := C(LRpart3[f], InvTriCeil, P^1_1)</math>
|<math>59 + |f|</math>
|<math>LRCall[f](Pair(x,y)) = f(x,y)</math>
|}
Where these are based on the following definitions:
{| class="wikitable"
|+Macros
!
!Macro
!arity
!Definition
!Size
!Function
|-
| rowspan="3" |Addition
|Add
|2
|<math>Add := R(P^1_1, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y</math>
|-
|AddXA
|3
|<math>AddXA := R(P^2_1, C(S, P^4_2))</math>
|5
|<math>\lambda xyz. x+y</math>
|-
|AddS
|2
|<math>AddS := R(S, C(S, P^3_2))</math>
|5
|<math>\lambda xy. x+y+1</math>
|-
|[[wikipedia:Primitive_recursive_function#Predecessor|Predecesor]]
|Pred
|1
|<math>Pred := R(Z^0, P^2_1)</math>
|3
|<math>\lambda x. x \dot - 1</math>
|-
|[[wikipedia:Monus#Natural_numbers|Monus]]
|RMonus
|2
|<math>RMonus := R(P^1_1, C(Pred, P^3_2))</math>
|7
|<math>\lambda xy. y \dot - x</math>
|-
| rowspan="3" |Triangular numbers
|Tri
|1
|<math>Tri := R(Z^0, AddS)</math>
|7
|<math>\lambda x. \frac{x(x+1)}{2}</math>
|-
|TriP
|1
|<math>TriP := R(Z^0, Add)</math>
|7
|<math>TriP(x+1) = Tri(x)</math>
|-
|TriPXA
|2
|<math>TriPXA := R(Z^1, AddXA)</math>
|7
|<math>TriPXA(x,y) = TriP(x)</math>
|-
| rowspan="2" |Inverting Tri
|RMonusTri
|2
|<math>RMonusTri := C(RMonus, C(Tri, P^2_1), P^2_2)</math>
|18
|<math>\lambda xy. y \dot - Tri(x)</math>
|-
|InvTriCeil
|1
|<math>InvTriCeil := M(RMonusTri)</math>
|19
|<math>\lambda y. \min \{x | Tri(x) \ge y \}</math>
|-
| rowspan="5" | Combined LRCall
|RightPiece
|3
|<math>RightPiece := R(P^2_2, C(Pred, P^4_2))</math>
|7
|<math>\lambda xyz. z \dot - x</math>
|-
|LeftPiece
|3
|<math>LeftPiece := C(RMonus, C(S, P^3_2), P^3_1)</math>
|12
|<math>\lambda xyz. x \dot - (y+1)</math>
|-
|LRpart1[f]
|3
|<math>LRpart1[f] := C(f, LeftPiece, RightPiece)</math>
|<math>20 + |f|</math>
|<math>\lambda xyz. f(x \dot - (y+1), z \dot - x)</math>
|-
|LRpart2[f]
|3
|<math>LRpart2[f] := C(LRpart1[f], P^3_3, P^3_1, AddXA)</math>
|<math>28 + |f|</math>
|<math>\lambda xyz. f(z \dot - (x+1), x+y\dot - z)</math>
|-
|LRpart3[f]
|2
|<math>LRpart3[f] := C(LRpart2[f], TriPXA, P^2_1, P^2_2)</math>
|<math>38 + |f|</math>
|<math>\lambda xy. f(y\dot - (TriP(x)+1), TriP(x)+x\dot - y)</math>
|}

[[Category:functions]]