BusyBeaverWiki - User contributions [en]

Register machine

2026-09-19T13:58:02Z

A6P457: /* Analysis */ fixed

'''Register machines''', also known as '''Minsky machines''', are a [[Turing complete|Turing-complete]] model of computation.

Register machines contain a set of instructions and a set of registers. The instructions are labelled A, B, C, and so on. The registers are numbered 0, 1, 2, and so on. There are 2 types of instructions:

* inc(c, n) adds 1 to the register c then jumps to instruction n.
* dec(c, n, m) jumps to instruction m if register c equals 0, else subtract 1 to the register c then jump to instruction n.

The program halts if it reaches an undefined instruction. Here we label an undefined instruction with <code>*</code>.

== Register Busy Beaver ==
The Register Busy Beaver function, denoted MBB(n,r), returns the maximum number of instructions executed by a register machine with n instructions and r registers when started in instruction A and all registers initialized to 0. MBB(n) = MBB(n,n) (unlimited registers).
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+*</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|-
|MBB(7)
|≥ 231
|<code>0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*</code>
|-
|MBB(8)
|≥ 3394
|<code>0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*</code>
|-
|MBB(9)
|≥ 124686
|<code>0+B_0+C_0+D_1-IE_1+F_1+G_1+H_0-EC_1-A*</code>
|}

== Analysis ==
MBB(7):
<pre>Let A(x) = A:[x, 0]

A(2x) -> 9x+20 -> A(3x+4)
A(2x+1) -> 9x+27 -> halt

A(0) -> 20 -> A(4) -> 38 -> A(10) -> 65 -> A(19) -> 108 -> halt</pre>
MBB(8):
Let S(z) = F:[0, 2z+1, z]

S(2k) -> 44k+19 -> S(5k+2)
S(2k+1) -> 4k+2 -> halt

A:[0, 0, 0] -> 44 -> S(4) -> 107 -> S(12) -> 283 -> S(32) -> 723 -> S(82) -> 1823 -> S(207) -> 414 -> halt
MBB(9): (analysis by @Sheep)
<pre>
Let A(n) = E:[n, 0, 0]

A(2n) —(23n + 12)—> *:[9n + 4, 0, 0]
A(2n + 1) —(23n + 25)—> A(9n + 10)

A:[0, 0, 0]
—(4)—> A(3)
—(48)—> A(19)
—(232)—> A(91)
—(1.060)—> A(415)
—(4.786)—> A(1.873)
—(21.553)—> A(8.434)
—(97.003)—> *:[37.957, 0, 0]</pre>

== Cryptids ==
No [[Cryptids]] have been found via exhaustive search, but [[Hydra]] has been hand coded into a 10-instruction, 3-register Minsky machine: <code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code> which can be interpreted the following way:[https://discord.com/channels/960643023006490684/1243312334907375676/1446694179337011325]
Let S(h,w) = A:[h-3,0,w]

Start: A:[0,0,0] = S(3,0)
S(2k,0) = A:[2k-3,0,0] -> Halt
S(2k,w+1) = A:[2k-3,0,w+1] -> A:[3k-3,0,w] = S(3k,w)
S(2k+1,w) = A:[2k-2,0,w] -> A:[3k-2,0,w+2] = S(3k+1,w+2)
[[Category:functions]]

Fractran

2026-09-10T13:53:34Z

A6P457: /* BBf(24) */ updated

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_k</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

TODO: create pages about the deciders.

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, and all other machines for BBf(22) are solved.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|No holdouts remain. Claude Opus 4.6's proof of nonhalting of all the 140 holdouts: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]
|3 holdouts remain. Claude Opus 4.6 gave a proof of all machines but the 3 Fenrir Cryptids, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

The holdouts list whose elements are exactly the 3 Fenrir Cryptids on GitHub: [https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1510781736374763702 1 Jun 2026]
|21,295 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1511579969825013811 2 Jun 2026]
By August 5th, 2026, the unofficial holdouts count had been reduced to 13. [https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
Known [[Cryptid|Cryptids]]:

# 11 Hydra-like Cryptids (including Frankenstein's Monster and Antihydra-like Cryptid)
|-
|24
|<math>> 9.000 \times 10^{14448}</math>
|<code>[14/15, 1/6, 11/3, 441/2, 5/7, 15/11]</code>
|<math>\begin{bmatrix}
1 & -1 & -1 & 1 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 1 \\
-1 & 2 & 0 & 2 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 1 & 0 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383 9 Sep 2026]
|231,444 holdouts remain. [https://github.com/int-y1/BBFractran/blob/main/holdout/sz24_231444.txt 6 Sep 2026]
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{k+5} & \xrightarrow{3} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{k+7} & \xrightarrow{4} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

 
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This pseudo-random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>
If it were a truly random walk, there would be a 5.9% chance that it takes at least 275 steps to reach 0. So this program is mildly lucky.
==== BBf(23) ====
The BBf(23) champion
<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk:
let <math>A(x,y)=[0,0,0,x,y]</math>

<math>[1,0,0,0,0] \to A(0,2)</math>

<math>A(2x,y) \xrightarrow{12x+4} A(5x+1,y+1)</math> if y>0

<math>A(2x+1,y) \xrightarrow{12x+10} A(5x+4,y-1)</math> if y>0

<math>A(x,0) \to halt</math>

<math>A(0,2) \to A(1,3) \to A(4,2) \to A(11,3) \to A(29,2) \to A(74,1) \to A(186,2) \to A(466,3) \to A(1166,4) \to A(2916,5) \to A(7291,6) \to \dots</math>
==== BBf(24) ====
The BBf(24) champion
<math>\begin{bmatrix}
1 & -1 & -1 & 1 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 1 \\
-1 & 2 & 0 & 2 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 1 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk: let <math>A(x,y) = [0,0,0,x,y]</math>

<math>[1,0,0,0,0] \xrightarrow{3} A(2,2)</math>

<math>A(x,0) \xrightarrow{x} halt</math>

<math>A(3x,y) \xrightarrow{9x+5} A(7x+3,y+1)</math> if y>0

<math>A(3x+1,y) \xrightarrow{9x+6} A(7x+4,y-1)</math> if y>0

<math>A(3x+2,y) \xrightarrow{9x+10} A(7x+7,y)</math> if y>0

<math>A(2,2) \to A(7,2) \to A(18,1) \to A(45,2) \to A(108,3) \to A(255,4) \to A(598,5) \to A(1397,4) \to A(3262,4) \to \dots</math>

== Cryptids ==

=== Size 22: Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 2003 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 1997 holdouts were non-halting and 3 holdouts were halting; the remaining 3 holdouts are the Fenrir family.[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179] Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|Nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Size 23: 11 Hydra-like Cryptids ===

Define <math>Hydra(r_\mathrm{num}, r_\mathrm{den}, x_\mathrm{offset}, y_\mathrm{offset}, (x_\mathrm{init}, y_\mathrm{init}))</math> to be the problem as follows:

# The initial state is <math>(x_\mathrm{init}, y_\mathrm{init})</math>.
# The iteration <math>(x, y) \mapsto (r_\mathrm{num} \times \lfloor x/r_\mathrm{den} \rfloor + x_\mathrm{offset}[x \bmod r_\mathrm{den}], y + y_\mathrm{offset}[x \bmod r_{den}])</math> is repeated. Here, <math>x_\mathrm{offset}, y_\mathrm{offset}</math> are 0-indexed.
# Are all the values of y non-negative?

Furthermore, the Hydra-like problem is considered a [[Cryptids|Cryptid]] if it also satisfies:

# <math>y_\mathrm{offset}</math> contains a negative number, and the average is positive.
# <math>x \bmod r_\mathrm{den}</math> is a pseudorandom sequence.
#There are no negative values of <math>y</math> early on.

For example, Fenrir is non-halting if and only if Hydra(5, 2, [2, 0], [-1, 2], (1, 0)). In this Hydra problem, the first few visited states are <math>(1, 0) \to (0, 2) \to (2, 1) \to (7, 0) \to (15, 2) \to (35, 4)</math>.

There are 11 Hydra-like Cryptids of size exactly 23, listed in the table below. All 11 Cryptids are not correlated with each other. Fenrir is included in the table as a reference.
{| class="wikitable sortable"
|+
!Domain: Holdout Number
!Holdout
!Hydra problem
!Notes
|-
|BBf(23): #11/694
|<code>[1/135, 25/21, 33/5, 2/3, 7/11, 5/2]</code>
|Hydra(7, 4, [1, 3, 5, 6], [0, 1, 2, -1], (1, 0))
|
|-
|BBf(22): #29/2003
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|Hydra(5, 2, [2, 0], [-1, 2], (1, 0))
|Fenrir
|-
|BBf(23): #26/694
|<code>[1/15, 49/3, 81/77, 10/49, 33/2]</code>
|Hydra(7, 2, [-1, 8], [3, -1], (1, 2))
|
|-
|BBf(23): #47/694
|<code>[1/18, 4/15, 21/2, 121/3, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 3, 4], [1, 3, -1], (1, 1))
|If #47 doesn't halt then Frankenstein's Monster doesn't halt
|-
|BBf(23): #77/694
|<code>[1/54, 4/15, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(7, 4, [0, 2, 4, 5], [0, 1, 2, -1], (1, 0))
|Same ratio 7/4 as #11
|-
|BBf(23): #151/694
|<code>[14/15, 1/12, 11/3, 63/2, 5/7, 2/11]</code>
|Hydra(8, 5, [1, 2, 4, 6, 7], [2, -1, 1, 3, 0], (0, 1))
|
|-
|BBf(23): #159/694
|<code>[14/15, 1/6, 121/3, 63/2, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 2, 4], [3, -1, 1], (1, 3))
|Same ratio 5/3 as #47
|-
|BBf(23): #207/694
|<code>[2/15, 1/12, 441/2, 11/3, 5/7, 2/11]</code>
|Hydra(6, 5, [1, 3, 3, 5, 5], [1, 3, 0, 2, -1], (1, 1))
|
|-
|BBf(23): #218/694
|<code>[2/15, 1/6, 441/2, 121/3, 5/7, 2/11]</code>
|Hydra(4, 3, [1, 2, 3], [3, 1, -1], (1, 3))
|
|-
|BBf(23): #317/694
|<code>[4/15, 1/18, 63/2, 11/3, 5/7, 2/11]</code>
|Hydra(5, 4, [1, 2, 3, 5], [1, 0, -1, 2], (1, 1))
|
|-
|BBf(23): #319/694
|<code>[4/15, 1/24, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(5, 4, [0, 2, 4, 3], [0, 1, 2, -1], (1, 0))
|Same ratio 5/4 as #317
|-
|BBf(23): #323/694
|<code>[4/15, 1/6, 21/2, 1331/3, 5/7, 2/11]</code>
|Hydra(3, 2, [1, 2], [2, -1], (1, 2))
|If #323 doesn't halt then Antihydra-like Cryptid doesn't halt
|}

As of August 2026, the 13 unofficial holdouts are as follows: Fenrir, 11 Hydra-like Cryptids of size 23, and #601 <code>[9/10, 1/42, 22/3, 49/2, 5/11, 3/7]</code>.

==== Frankenstein's Monster ====
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the <math>y</math> values now change by <math>\{+1,+3,-1\}</math> depending on the value of <math>x\bmod 3</math> (instead of <math>\{0,+1,-1\}</math> in the original size 22 program). The <math>x</math> values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

==== Antihydra-like Cryptid ====
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Size 25: Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== Size 36: BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \xrightarrow{4}& A(1, 2) \\
A(a, b) & \xrightarrow{5b+4}& A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \xrightarrow{5a+2} & A(2a+1, b-a) & \text{if } a 

=== Size 48: BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

Post Tag System

2026-09-10T12:53:17Z

A6P457: Undo revision 8459 by Sligocki (talk) see results.md(the size definition doesn't add the symbols)

{{Stub}}
A '''Post Tag System''' is a [[Turing-complete]] formal system invented by Emil Post in 1920 (but not published until 1943). Post explored tag systems before [[Turing machines]] were invented and thus before the solution to the Entscheidungsproblem, the Church-Turing thesis and the undecidability of the Halting problem were known. According to Wolfram, Post studied systems where no more than 2 symbols were ever deleted or added and found none produced complicated behavior, but that a rule with up to 3 symbols deleted or added could produce chaotic and seemingly intractable behavior. Tag systems were proven to be Turing-complete by Marvin Minsky in 1961.

== Tag System ==
A tag system is defined by:

* A positive integer ''v'', called the deletion number
* A finite alphabet of symbols: ''A''
* One "production rules" for each symbol: <math>w_a \in A^*</math> (for each <math>a \in A</math>)

We will often write a v-tag system to specify that it is a tag system with deletion number v (ex: a 2-tag system).

A tag system operates on a tape (a possibly empty sequence of symbols). At each step:

* If the tape is shorter than v, it halts.
* Otherwise, it reads the first symbol (a), deletes the first v symbols from the tape and appends <math>w_a</math>

It repeats this process until it halts.

1-tag systems are decidable, but v-tag systems for any v ≥ 2 are undecidable and (in fact) Turing complete.

== BB_PT ==
Let us define the size of a tag system to be the <math>S = |A| + \sum_{a \in A} |w_a| = \sum_{a \in A} (|w_a| + 1)</math>.

Let BB_PT(v, S) = max steps across all v-tag systems of size S which halt when started on tape <math>0^v</math>. We default to BB_PT(S) = BB_PT(2,S) since 2-tags is Turing complete.

Starting with tape <math>0^v</math> means that after the first step the tape is <math>w_0</math>.

== Champions ==
{| class="wikitable"
|+2-tag Champions
!Size
!BB_PT(n)
!Champion
!Holdouts
|-
|1
|= 1
|<code>0 -> eps</code>
|0
|-
|2
|= 1
|<code>0 -> 0</code>
|0
|-
|3
|= 1
|<code>0 -> 0, 1 -> eps</code>
|0
|-
|4
|= 2
|<code>0 -> 11, 1 -> eps</code>
|0
|-
|5
|= 4
|<code>0 -> 011, 1 -> eps</code>
|0
|-
|6
|= 5
|<code>0 -> 011, 1 -> 1</code>
|0
|-
|7
|= 10
|<code>0 -> 0111, 1 -> 1</code>
|0
|-
|8
|≥ 19
|<code>0 -> 111, 1 -> 20, 2 -> eps</code>
|7
|-
|9
|≥ 49
|<code>0 -> 11, 1 -> 021, 2 -> 2</code>
|154
|-
|10
|≥ 779
|<code>0 -> 112, 1 -> 1, 2 -> 002</code>
|1,933
|-
|11
|≥ 196,841
|<code>0 -> 120221, 1 -> 0, 2 -> 2</code>
|11,759
|}
"eps" is the empty string

== Cryptids ==
realsheepthe constructed a size 11 [[Cryptid]] (originally mislabeled as a near-Cryptid) <code>0 —> 11111, 1 —> 20, 2 —> 1</code> which simulates the [[Collatz-like]] iteration:<pre>S(n) = 1^n
S(0) —(0)-> (halt)
S(1) —(0)-> 1 (halt)
S(2k + 2) -(2k + 2)-> S(k + 1)
S(2k + 3) -(2k + 4)-> S(5k + 9)

00 -(1)-> S(5)
-(6)-> S(14)
-(14)-> S(7)
-(8)-> S(19)
-(20)-> S(49)
-(50)-> S(124)
...</pre>
== Analysis ==
BB_PT(7) champion analysis:<syntaxhighlight line="1">
0 -> 0111, 1 -> 1

S(n) = 1^n 0 111
S(2k) —(k + 1)—> S(k + 2)
S(2k + 1) —(2k + 4)—> 1 (halt)

00 —(1)—> S(0)
—(1)—> S(2)
—(2)—> S(3)
—(6)—> 1 (halt)
</syntaxhighlight>BB_PT(8) champion analysis:<syntaxhighlight line="1" start="1">
0 -> 111, 1 -> 20, 2 ->

S(n) = 1^(n + 2)
S(2k) —(2k + 2)—> (halt)
S(2k + 1) —(2k + 4)—> S(3k + 3)

00 —(1)—> S(1)
—(4)—> S(3)
—(6)—> S(6)
—(8)—> (halt)
</syntaxhighlight>

== Code ==
GitHub repositories:

* https://github.com/sligocki/etc/tree/main/post_tag
* https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Post%20Tag%20System
[[Category:Functions]]
[[Category:Stubs]]

CA

2026-09-06T10:02:42Z

A6P457: Redirected page to Cellular automaton

#REDIRECT
[[Cellular automaton]]

Cyclic Tag

2026-09-06T09:57:28Z

A6P457: /* Busy Beaver Function */ BBCT(7) proved

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]] where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''delete''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it reaches the empty string.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the lengths of all the production rules plus the number of production rules (same as for [[Post Tag System|BB_PT]]).
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|= 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 1,626,862
|<code>1001_1_00</code>
|-
|11
|≥ 1,859,166
|<code>10_10010_0</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 5,217,735
|<code>001110_1_0_0</code>
|-
|14
|≥ 154,171,565
|<code>01011__0_0100</code>
|}

==See also==
* [[Tiny Tag]]

== Sources ==
GitHub repository (with documentation, deciders and holdout lists): https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Cyclic%20Tag%20System
[[Category:functions]]

BBCT

2026-09-06T09:55:58Z

A6P457: Redirected page to Cyclic Tag

#REDIRECT
[[Cyclic Tag]]

Post Tag System

2026-09-06T09:40:17Z

A6P457: /* Champions */ BB_PT(7) proven

{{Stub}}
A '''Post Tag System''' is a [[Turing-complete]] formal system invented by Emil Post in 1920 (but not published until 1943). Post explored tag systems before [[Turing machines]] were invented and thus before the solution to the Entscheidungsproblem, the Church-Turing thesis and the undecidability of the Halting problem were known. According to Wolfram, Post studied systems where no more than 2 symbols were ever deleted or added and found none produced complicated behavior, but that a rule with up to 3 symbols deleted or added could produce chaotic and seemingly intractable behavior. Tag systems were proven to be Turing-complete by Marvin Minsky in 1961.

== Tag System ==
A tag system is defined by:

* A positive integer ''v'', called the deletion number
* A finite alphabet of symbols: ''A''
* One "production rules" for each symbol: <math>w_a \in A^*</math> (for each <math>a \in A</math>)

We will often write a v-tag system to specify that it is a tag system with deletion number v (ex: a 2-tag system).

A tag system operates on a tape (a possibly empty sequence of symbols). At each step:

* If the tape is shorter than v, it halts.
* Otherwise, it reads the first symbol (a), deletes the first v symbols from the tape and appends <math>w_a</math>

It repeats this process until it halts.

1-tag systems are decidable, but v-tag systems for any v ≥ 2 are undecidable and (in fact) Turing complete.

== BB_PT ==
Let us define the size of a tag system to be the <math>S = |A| + \sum_{a \in A} |w_a| = \sum_{a \in A} (|w_a| + 1)</math>.

Let BB_PT(v, S) = max steps across all v-tag systems of size S which halt when started on tape <math>0^v</math>. We default to BB_PT(S) = BB_PT(2,S) since 2-tags is Turing complete.

Starting with tape <math>0^v</math> means that after the first step the tape is <math>w_0</math>.

== Champions ==
{| class="wikitable"
|+2-tag Champions
!Size
!BB_PT(n)
!Champion
!Holdouts
|-
|1
|= 1
|<code>0 -> eps</code>
|0
|-
|2
|= 1
|<code>0 -> 0</code>
|0
|-
|3
|= 1
|<code>0 -> 0, 1 -> eps</code>
|0
|-
|4
|= 2
|<code>0 -> 11, 1 -> eps</code>
|0
|-
|5
|= 4
|<code>0 -> 011, 1 -> eps</code>
|0
|-
|6
|= 5
|<code>0 -> 011, 1 -> 1</code>
|0
|-
|7
|= 10
|<code>0 -> 0111, 1 -> 1</code>
|0
|-
|8
|≥ 19
|<code>0 -> 111, 1 -> 20, 2 -> eps</code>
|8
|-
|9
|≥ 49
|<code>0 -> 11, 1 -> 021, 2 -> 2</code>
|326
|-
|10
|≥ 779
|<code>0 -> 112, 1 -> 1, 2 -> 002</code>
|2,491
|-
|11
|≥ 196,841
|<code>0 -> 120221, 1 -> 0, 2 -> 2</code>
|14,972
|}
"eps" is the empty string

== Cryptids ==
realsheepthe constructed a size 11 [[Cryptid]] (originally mislabeled as a near-Cryptid) <code>0 —> 11111, 1 —> 20, 2 —> 1</code> which simulates the [[Collatz-like]] iteration:<pre>S(n) = 1^n
S(0) —(0)-> (halt)
S(1) —(0)-> 1 (halt)
S(2k + 2) -(2k + 2)-> S(k + 1)
S(2k + 3) -(2k + 4)-> S(5k + 9)

00 -(1)-> S(5)
-(6)-> S(14)
-(14)-> S(7)
-(8)-> S(19)
-(20)-> S(49)
-(50)-> S(124)
...</pre>
== Analysis ==
BB_PT(7) champion analysis:<syntaxhighlight line="1">
0 -> 0111, 1 -> 1

S(n) = 1^n 0 111
S(2k) —(k + 1)—> S(k + 2)
S(2k + 1) —(2k + 4)—> 1 (halt)

00 —(1)—> S(0)
—(1)—> S(2)
—(2)—> S(3)
—(6)—> 1 (halt)
</syntaxhighlight>BB_PT(8) champion analysis:<syntaxhighlight line="1" start="1">
0 -> 111, 1 -> 20, 2 ->

S(n) = 1^(n + 2)
S(2k) —(2k + 2)—> (halt)
S(2k + 1) —(2k + 4)—> S(3k + 3)

00 —(1)—> S(1)
—(4)—> S(3)
—(6)—> S(6)
—(8)—> (halt)
</syntaxhighlight>

== Sources ==
GitHub repository (with documentation, deciders and holdout lists): https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Post%20Tag%20System
[[Category:Functions]]
[[Category:Stubs]]

1RB0RF 1RC1RF 1LD0LE ---1LC 1RA1LE 1LC1RB

2026-09-03T17:46:53Z

A6P457: Adder domain

{{machine|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}}
{{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}} is a probviously non-halting [[BB(6)]] cryptid similar to [[Space Needle]].
== Analysis ==
<pre>A(x,y) = 0^inf <E 1^x 0 1^y 0^inf

A(2x,y) -> A(3x+y+1,2)
A(2x+3,y) -> A(x,x+y+4)
A(1,y) -> halt

A(x) = A(x-3,2)
p(x) = The largest n such that 2^n divides x

A(x) -> A(x+p(x)+(x/2^p(x)+3)/2) if x/2^p(x)>1
A(2^x) -> halt

A(3) -> A(6) -> A(10) -> A(15) -> A(24) -> A(30) -> A(40) -> A(47) -> A(72) -> ...
[[Category:BB(6)]]
[[Category:Cryptids]]

BB(6)

2026-09-03T17:15:28Z

A6P457: /* Cryptids */ added similar to Space Needle

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== History ==
* In 1964, Green established Σ(5) ≥ 35.<ref name=":PMH">Pascal Michel. (last updated 2026). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm62 </ref>
* In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.<ref name=":PMH" />
* In 1983, Brady established S(6) ≥ 13,488 and Σ(6) ≥ 117.<ref name=":PMH" />
* In 1982, Schult established S(6) ≥ 4,208,824 and Σ(6) ≥ 2,075, although these bounds were only published later.<ref name=":PMH" />
* In 1990, Heiner Marxen and Jürgen Buntrock established first S(6) ≥ 13,122,572,797 and Σ(6) ≥ 136,612, then S(6) ≥ 8,690,333,381,690,951 and Σ(6) ≥ 95,524,079.<ref name=":PMH" />
* In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 1042 and Σ(6) > 2.5 × 1021.<ref name=":PMH" />
* In August 2000, Heiner Marxen and Jürgen Buntrock established first S(6) > 6.1 × 10119 and Σ(6) > 1.4 × 1060, then S(6) > 6.1 × 10925 and Σ(6) > 6.4 × 10462.<ref name=":PMH" />
* In February 2001, Heiner Marxen and Jürgen Buntrock established S(6) > 3.0 × 101730 and Σ(6) > 1.2 × 10865.<ref name=":PMH" />
* In November 2007, Terry and Shawn Ligocki established S(6) > 8.9 × 101762 and Σ(6) > 2.5 × 10881.<ref name=":PMH" />
* In December 2007, Terry and Shawn Ligocki established S(6) > 2.5 × 102879 and Σ(6) > 4.6 × 101439.<ref name=":PMH" />
* In May 2010, Pavel Kropitz established S(6) > 3.8 × 1021132 and Σ(6) > 3.1 × 1010566.<ref name=":PMH" />
* In June 2010, Pavel Kropitz established S(6) > 7.4 × 1036534 and Σ(6) > 3.5 × 1018267.<ref name=":PMH" />
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1003 machines up to equivalence and 2190 machines not considering equivalence as of August 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet]. The informal holdout count is 1101.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 to 1e15. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet].

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}, Space Needle
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}, similar to Space Needle
* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}, similar to Space Needle
* {{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB|undecided}}, similar to Space Needle

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}
* {{TM|1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF|undecided}}
* {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA|halt}}
|<math>10^{1\,400\,000\,000}</math>(runtime)
|Racheline<ref>https://discord.com/channels/960643023006490684/1345502880727040091/1345502880727040091</ref>
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub) upon initial application.

== References ==
<references />

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

1RB0RF 1RC1RF 1LD0LE ---1LC 1RA1LE 1LC1RB

2026-09-03T17:10:51Z

A6P457: Created page with "{{machine|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}} {{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}} is a probviously non-halting cryptid similar to Space Needle. == Analysis == <pre>A(x,y) = 0^inf <E 1^x 0 1^y 0^inf A(2x,y) -> A(3x+y+1,2) A(2x+3,y) -> A(x,x+y+4) A(1,y) -> halt A(x) = A(x-3,2) p(x) = The largest n such that 2^n divides x A(x) -> A(x+p(x)+(x/2^p(x)+3)/2) if x/2^p(x)>1 A(2^x) -> halt A(3) -> A(6) -> A(10) -> A(15) -> A(24) -> A(30) -> A(40) -> A(..."

{{machine|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}}
{{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB}} is a probviously non-halting cryptid similar to [[Space Needle]].
== Analysis ==
<pre>A(x,y) = 0^inf <E 1^x 0 1^y 0^inf

A(2x,y) -> A(3x+y+1,2)
A(2x+3,y) -> A(x,x+y+4)
A(1,y) -> halt

A(x) = A(x-3,2)
p(x) = The largest n such that 2^n divides x

A(x) -> A(x+p(x)+(x/2^p(x)+3)/2) if x/2^p(x)>1
A(2^x) -> halt

A(3) -> A(6) -> A(10) -> A(15) -> A(24) -> A(30) -> A(40) -> A(47) -> A(72) -> ...
[[Category:BB(6)]]
[[Category:Cryptids]]

Cyclic Tag

2026-08-31T15:31:41Z

A6P457: /* Execution rules: */ fixed string change rule

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]] where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''delete''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it reaches the empty string.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the symbols count and production rules count.
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 209
|<code>11_0_0100</code>
|-
|11
|≥ 2,269
|<code>1011_1_00_</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 50,906
|<code>0011_010_1_0</code>
|}

==See also==
* [[Tiny Tag]]
[[Category:functions]]

Cyclic Tag

2026-08-31T15:29:30Z

A6P457: /* Execution rules: */ fixed halt rule

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]] where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''read''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it reaches the empty string.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the symbols count and production rules count.
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 209
|<code>11_0_0100</code>
|-
|11
|≥ 2,269
|<code>1011_1_00_</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 50,906
|<code>0011_010_1_0</code>
|}

==See also==
* [[Tiny Tag]]
[[Category:functions]]

Fractran

2026-08-30T09:51:26Z

A6P457: /* Size 36: BMO1 */ added step counts

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_k</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

TODO: create pages about the deciders.

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, and all other machines for BBf(22) are solved.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|No holdouts remain. Claude Opus 4.6's proof of nonhalting of all the 140 holdouts: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]
|3 holdouts remain. Claude Opus 4.6 gave a proof of all machines but the 3 Fenrir Cryptids, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

The holdouts list whose elements are exactly the 3 Fenrir Cryptids on GitHub: [https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1510781736374763702 1 Jun 2026]
|21,295 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1511579969825013811 2 Jun 2026]
By August 5th, 2026, the unofficial holdouts count had been reduced to 13. [https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
Known [[Cryptid|Cryptids]]:

# 11 Hydra-like Cryptids (including Frankenstein's Monster and Antihydra-like Cryptid)
|-
|24
|<math>> 9.263 \times 10^{9595}</math>
|<code>[18/35, 1/10, 11/5, 75/2, 49/3, 5/11]</code>
|<math>\begin{bmatrix}
1 & 2 & -1 & -1 & 0 \\
-1 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 \\
-1 & 1 & 2 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 0 & 1 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1540835921182728272 22 Aug 2026]
|No holdouts list yet.

An informal list from a 20%-complete enumeration: [https://discord.com/channels/960643023006490684/1438019511155691521/1540831684071530637 22 Aug 2026]
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{k+5} & \xrightarrow{3} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{k+7} & \xrightarrow{4} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

 
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This pseudo-random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>
If it were a truly random walk, there would be a 5.9% chance that it takes at least 275 steps to reach 0. So this program is mildly lucky.
==== BBf(23) ====
The BBf(23) champion
<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk:
let <math>A(x,y)=[0,0,0,x,y]</math>

<math>[1,0,0,0,0] \to A(0,2)</math>

<math>A(2x,y) \xrightarrow{12x+4} A(5x+1,y+1)</math> if y>0

<math>A(2x+1,y) \xrightarrow{12x+10} A(5x+4,y-1)</math> if y>0

<math>A(x,0) \to halt</math>

<math>A(0,2) \to A(1,3) \to A(4,2) \to A(11,3) \to A(29,2) \to A(74,1) \to A(186,2) \to A(466,3) \to A(1166,4) \to A(2916,5) \to A(7291,6) \to \dots</math>
==== BBf(24) ====
The BBf(24) champion
<math>\begin{bmatrix}
1 & 2 & -1 & -1 & 0 \\
-1 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 \\
-1 & 1 & 2 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 0 & 1 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk: let <math>A(x,y) = [0,x,0,0,y]</math>

<math>[1,0,0,0,0] \xrightarrow{3} A(1,2)</math>

<math>A(x,0) \xrightarrow{x} halt</math>

<math>A(3x,y) \xrightarrow{15x+2} A(16x,y)</math> if y>0

<math>A(3x+1,y) \xrightarrow{15x+6} A(16x+5,y-1)</math> if y>0

<math>A(3x+2,y) \xrightarrow{15x+13} A(16x+11,y+1)</math> if y>0

<math>A(1,2) \to A(5,1) \to A(27,2) \to A(144,2) \to A(768,2) \to A(4096,2) \to A(21845,1) \to A(116507,2) \to A(621371,3) \to \dots</math>

== Cryptids ==

=== Size 22: Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 2003 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 1997 holdouts were non-halting and 3 holdouts were halting; the remaining 3 holdouts are the Fenrir family.[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179] Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|Nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Size 23: 11 Hydra-like Cryptids ===

Define <math>Hydra(r_\mathrm{num}, r_\mathrm{den}, x_\mathrm{offset}, y_\mathrm{offset}, (x_\mathrm{init}, y_\mathrm{init}))</math> to be the problem as follows:

# The initial state is <math>(x_\mathrm{init}, y_\mathrm{init})</math>.
# The iteration <math>(x, y) \mapsto (r_\mathrm{num} \times \lfloor x/r_\mathrm{den} \rfloor + x_\mathrm{offset}[x \bmod r_\mathrm{den}], y + y_\mathrm{offset}[x \bmod r_{den}])</math> is repeated. Here, <math>x_\mathrm{offset}, y_\mathrm{offset}</math> are 0-indexed.
# Are all the values of y non-negative?

Furthermore, the Hydra-like problem is considered a [[Cryptids|Cryptid]] if it also satisfies:

# <math>y_\mathrm{offset}</math> contains a negative number, and the average is positive.
# <math>x \bmod r_\mathrm{den}</math> is a pseudorandom sequence.
#There are no negative values of <math>y</math> early on.

For example, Fenrir is non-halting if and only if Hydra(5, 2, [2, 0], [-1, 2], (1, 0)). In this Hydra problem, the first few visited states are <math>(1, 0) \to (0, 2) \to (2, 1) \to (7, 0) \to (15, 2) \to (35, 4)</math>.

There are 11 Hydra-like Cryptids of size exactly 23, listed in the table below. All 11 Cryptids are not correlated with each other. Fenrir is included in the table as a reference.
{| class="wikitable sortable"
|+
!Domain: Holdout Number
!Holdout
!Hydra problem
!Notes
|-
|BBf(23): #11/694
|<code>[1/135, 25/21, 33/5, 2/3, 7/11, 5/2]</code>
|Hydra(7, 4, [1, 3, 5, 6], [0, 1, 2, -1], (1, 0))
|
|-
|BBf(22): #29/2003
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|Hydra(5, 2, [2, 0], [-1, 2], (1, 0))
|Fenrir
|-
|BBf(23): #26/694
|<code>[1/15, 49/3, 81/77, 10/49, 33/2]</code>
|Hydra(7, 2, [-1, 8], [3, -1], (1, 2))
|
|-
|BBf(23): #47/694
|<code>[1/18, 4/15, 21/2, 121/3, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 3, 4], [1, 3, -1], (1, 1))
|If #47 doesn't halt then Frankenstein's Monster doesn't halt
|-
|BBf(23): #77/694
|<code>[1/54, 4/15, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(7, 4, [0, 2, 4, 5], [0, 1, 2, -1], (1, 0))
|Same ratio 7/4 as #11
|-
|BBf(23): #151/694
|<code>[14/15, 1/12, 11/3, 63/2, 5/7, 2/11]</code>
|Hydra(8, 5, [1, 2, 4, 6, 7], [2, -1, 1, 3, 0], (0, 1))
|
|-
|BBf(23): #159/694
|<code>[14/15, 1/6, 121/3, 63/2, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 2, 4], [3, -1, 1], (1, 3))
|Same ratio 5/3 as #47
|-
|BBf(23): #207/694
|<code>[2/15, 1/12, 441/2, 11/3, 5/7, 2/11]</code>
|Hydra(6, 5, [1, 3, 3, 5, 5], [1, 3, 0, 2, -1], (1, 1))
|
|-
|BBf(23): #218/694
|<code>[2/15, 1/6, 441/2, 121/3, 5/7, 2/11]</code>
|Hydra(4, 3, [1, 2, 3], [3, 1, -1], (1, 3))
|
|-
|BBf(23): #317/694
|<code>[4/15, 1/18, 63/2, 11/3, 5/7, 2/11]</code>
|Hydra(5, 4, [1, 2, 3, 5], [1, 0, -1, 2], (1, 1))
|
|-
|BBf(23): #319/694
|<code>[4/15, 1/24, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(5, 4, [0, 2, 4, 3], [0, 1, 2, -1], (1, 0))
|Same ratio 5/4 as #317
|-
|BBf(23): #323/694
|<code>[4/15, 1/6, 21/2, 1331/3, 5/7, 2/11]</code>
|Hydra(3, 2, [1, 2], [2, -1], (1, 2))
|If #323 doesn't halt then Antihydra-like Cryptid doesn't halt
|}

As of August 2026, the 13 unofficial holdouts are as follows: Fenrir, 11 Hydra-like Cryptids of size 23, and #601 <code>[9/10, 1/42, 22/3, 49/2, 5/11, 3/7]</code>.

==== Frankenstein's Monster ====
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the <math>y</math> values now change by <math>\{+1,+3,-1\}</math> depending on the value of <math>x\bmod 3</math> (instead of <math>\{0,+1,-1\}</math> in the original size 22 program). The <math>x</math> values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

==== Antihydra-like Cryptid ====
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Size 25: Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== Size 36: BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \xrightarrow{4}& A(1, 2) \\
A(a, b) & \xrightarrow{5b+4}& A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \xrightarrow{5a+2} & A(2a+1, b-a) & \text{if } a 

=== Size 48: BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

MBB

2026-08-30T09:12:46Z

A6P457: Redirected page to Minsky machine

#REDIRECT [[Minsky machine]]

Bell eats counter

2026-08-26T16:37:44Z

A6P457: Added TM template

{{Stub}}
'''Bell eats counter''' is an informal class of Turing machines. A typical Turing machine in this class has the following behavior:

* It has both a bell and a counter on the tape.
* Increment: when the bouncer in the bell finishes a period, the counter is increased by one.
* Overflow: when the bouncer in the bell overflows, the bell eats the lowest digit of the counter (the counter is halved), and the bouncer in the bell is reset.
[https://github.com/ccz181078/busycoq/blob/BB6/verify/BECv1.v A Rocq proof of a kind of typical behavior doesn't halt.]

== Examples ==
{{TM|1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA}}

{{TM|1RB---_1RC0RA_1LD1RA_1LE0LD_0RE1RF_0RB0LF}}

{{TM|1RB0LE_0RC---_1LC0RD_0RB1RA_1LF1LE_0LA1LF}} (more complex than typical ones)

{{TM|1RB3LB---3RA0LA_2LA3LB4RB1RB2RA}}

{{TM|1RB0RB_0RC1RB_1LD1RC_0LE0LD_1RA1LE}}

[[Category:Zoology]]

Bell eats counter

2026-08-26T16:09:55Z

A6P457: Added 5 state example

{{Stub}}
'''Bell eats counter''' is an informal class of Turing machines. A typical Turing machine in this class has the following behavior:

* It has both a bell and a counter on the tape.
* Increment: when the bouncer in the bell finishes a period, the counter is increased by one.
* Overflow: when the bouncer in the bell overflows, the bell eats the lowest digit of the counter (the counter is halved), and the bouncer in the bell is reset.
[https://github.com/ccz181078/busycoq/blob/BB6/verify/BECv1.v A Rocq proof of a kind of typical behavior doesn't halt.]

== Examples ==
[https://bbchallenge.org/1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA 1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA]

[https://bbchallenge.org/1RB---_1RC0RA_1LD1RA_1LE0LD_0RE1RF_0RB0LF 1RB---_1RC0RA_1LD1RA_1LE0LD_0RE1RF_0RB0LF]

[https://bbchallenge.org/1RB0LE_0RC---_1LC0RD_0RB1RA_1LF1LE_0LA1LF 1RB0LE_0RC---_1LC0RD_0RB1RA_1LF1LE_0LA1LF] (more complex than typical ones)

[https://bbchallenge.org/1RB3LB---3RA0LA_2LA3LB4RB1RB2RA 1RB3LB---3RA0LA_2LA3LB4RB1RB2RA]

[https://bbchallenge.org/1RB0RB_0RC1RB_1LD1RC_0LE0LD_1RA1LE 1RB0RB_0RC1RB_1LD1RC_0LE0LD_1RA1LE]

[[Category:Zoology]]

Fractran

2026-08-23T06:46:23Z

A6P457: /* Behavior of Champions */ Added BBf(24)

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_k</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

TODO: create pages about the deciders.

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, and all other machines for BBf(22) are solved.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|No holdouts remain. Claude Opus 4.6's proof of nonhalting of all the 140 holdouts: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]
|3 holdouts remain. Claude Opus 4.6 gave a proof of all machines but the 3 Fenrir Cryptids, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

The holdouts list whose elements are exactly the 3 Fenrir Cryptids on GitHub: [https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1510781736374763702 1 Jun 2026]
|21,295 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1511579969825013811 2 Jun 2026]
By August 5th, 2026, the unofficial holdouts count had been reduced to 13. [https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
Known [[Cryptid|Cryptids]]:

# 11 Hydra-like Cryptids (including Frankenstein's Monster and Antihydra-like Cryptid)
|-
|24
|<math>> 9.263 \times 10^{9595}</math>
|<code>[18/35, 1/10, 11/5, 75/2, 49/3, 5/11]</code>
|<math>\begin{bmatrix}
1 & 2 & -1 & -1 & 0 \\
-1 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 \\
-1 & 1 & 2 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 0 & 1 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1540835921182728272 22 Aug 2026]
|No holdouts list yet.

An informal list from a 20%-complete enumeration: [https://discord.com/channels/960643023006490684/1438019511155691521/1540831684071530637 22 Aug 2026]
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{k+5} & \xrightarrow{3} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{k+7} & \xrightarrow{4} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

 
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This pseudo-random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>
If it were a truly random walk, there would be a 5.9% chance that it takes at least 275 steps to reach 0. So this program is mildly lucky.
==== BBf(23) ====
The BBf(23) champion
<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk:
let <math>A(x,y)=[0,0,0,x,y]</math>

<math>[1,0,0,0,0] \to A(0,2)</math>

<math>A(2x,y) \xrightarrow{12x+4} A(5x+1,y+1)</math> if y>0

<math>A(2x+1,y) \xrightarrow{12x+10} A(5x+4,y-1)</math> if y>0

<math>A(x,0) \to halt</math>

<math>A(0,2) \to A(1,3) \to A(4,2) \to A(11,3) \to A(29,2) \to A(74,1) \to A(186,2) \to A(466,3) \to A(1166,4) \to A(2916,5) \to A(7291,6) \to \dots</math>
==== BBf(24) ====
The BBf(24) champion
<math>\begin{bmatrix}
1 & 2 & -1 & -1 & 0 \\
-1 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 \\
-1 & 1 & 2 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 0 & 1 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk: let <math>A(x,y) = [0,x,0,0,y]</math>

<math>[1,0,0,0,0] \xrightarrow{3} A(1,2)</math>

<math>A(x,0) \xrightarrow{x} halt</math>

<math>A(3x,y) \xrightarrow{15x+2} A(16x,y)</math> if y>0

<math>A(3x+1,y) \xrightarrow{15x+6} A(16x+5,y-1)</math> if y>0

<math>A(3x+2,y) \xrightarrow{15x+13} A(16x+11,y+1)</math> if y>0

<math>A(1,2) \to A(5,1) \to A(27,2) \to A(144,2) \to A(768,2) \to A(4096,2) \to A(21845,1) \to A(116507,2) \to A(621371,3) \to \dots</math>

== Cryptids ==

=== Size 22: Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 2003 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 1997 holdouts were non-halting and 3 holdouts were halting. The remaining 3 holdouts are the Fenrir family.[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179] Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Size 23: 11 Hydra-like Cryptids ===

Define <math>Hydra(r_\mathrm{num}, r_\mathrm{den}, x_\mathrm{offset}, y_\mathrm{offset}, (x_\mathrm{init}, y_\mathrm{init}))</math> to be the problem as follows:

# The initial state is <math>(x_\mathrm{init}, y_\mathrm{init})</math>.
# The iteration <math>(x, y) \mapsto (r_\mathrm{num} \times \lfloor x/r_\mathrm{den} \rfloor + x_\mathrm{offset}[x \bmod r_\mathrm{den}], y + y_\mathrm{offset}[x \bmod r_{den}])</math> is repeated. Here, <math>x_\mathrm{offset}, y_\mathrm{offset}</math> are 0-indexed.
# Are all the values of y non-negative?

Furthermore, the Hydra-like problem is considered a [[Cryptids|Cryptid]] if it also satisfies:

# <math>y_\mathrm{offset}</math> contains a negative number, and the average is positive.
# <math>x \bmod r_\mathrm{den}</math> is a pseudorandom sequence.
#There are no negative values of <math>y</math> early on.

For example, Fenrir is non-halting if and only if Hydra(5, 2, [2, 0], [-1, 2], (1, 0)). In this Hydra problem, the first few visited states are <math>(1, 0) \to (0, 2) \to (2, 1) \to (7, 0) \to (15, 2) \to (35, 4)</math>.

There are 11 Hydra-like Cryptids of size exactly 23, listed in the table below. All 11 Cryptids are not correlated with each other. Fenrir is included in the table as a reference.
{| class="wikitable sortable"
|+
!Domain: Holdout Number
!Holdout
!Hydra problem
!Notes
|-
|BBf(23): #11/694
|<code>[1/135, 25/21, 33/5, 2/3, 7/11, 5/2]</code>
|Hydra(7, 4, [1, 3, 5, 6], [0, 1, 2, -1], (1, 0))
|
|-
|BBf(22): #29/2003
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|Hydra(5, 2, [2, 0], [-1, 2], (1, 0))
|Fenrir
|-
|BBf(23): #26/694
|<code>[1/15, 49/3, 81/77, 10/49, 33/2]</code>
|Hydra(7, 2, [-1, 8], [3, -1], (1, 2))
|
|-
|BBf(23): #47/694
|<code>[1/18, 4/15, 21/2, 121/3, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 3, 4], [1, 3, -1], (1, 1))
|If #47 doesn't halt then Frankenstein's Monster doesn't halt
|-
|BBf(23): #77/694
|<code>[1/54, 4/15, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(7, 4, [0, 2, 4, 5], [0, 1, 2, -1], (1, 0))
|Same ratio 7/4 as #11
|-
|BBf(23): #151/694
|<code>[14/15, 1/12, 11/3, 63/2, 5/7, 2/11]</code>
|Hydra(8, 5, [1, 2, 4, 6, 7], [2, -1, 1, 3, 0], (0, 1))
|
|-
|BBf(23): #159/694
|<code>[14/15, 1/6, 121/3, 63/2, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 2, 4], [3, -1, 1], (1, 3))
|Same ratio 5/3 as #47
|-
|BBf(23): #207/694
|<code>[2/15, 1/12, 441/2, 11/3, 5/7, 2/11]</code>
|Hydra(6, 5, [1, 3, 3, 5, 5], [1, 3, 0, 2, -1], (1, 1))
|
|-
|BBf(23): #218/694
|<code>[2/15, 1/6, 441/2, 121/3, 5/7, 2/11]</code>
|Hydra(4, 3, [1, 2, 3], [3, 1, -1], (1, 3))
|
|-
|BBf(23): #317/694
|<code>[4/15, 1/18, 63/2, 11/3, 5/7, 2/11]</code>
|Hydra(5, 4, [1, 2, 3, 5], [1, 0, -1, 2], (1, 1))
|
|-
|BBf(23): #319/694
|<code>[4/15, 1/24, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(5, 4, [0, 2, 4, 3], [0, 1, 2, -1], (1, 0))
|Same ratio 5/4 as #317
|-
|BBf(23): #323/694
|<code>[4/15, 1/6, 21/2, 1331/3, 5/7, 2/11]</code>
|Hydra(3, 2, [1, 2], [2, -1], (1, 2))
|If #323 doesn't halt then Antihydra-like Cryptid doesn't halt
|}

As of August 2026, the 13 unofficial holdouts are as follows: Fenrir, 11 Hydra-like Cryptids of size 23, and #601 <code>[9/10, 1/42, 22/3, 49/2, 5/11, 3/7]</code>.

==== Frankenstein's Monster ====
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the <math>y</math> values now change by <math>\{+1,+3,-1\}</math> depending on the value of <math>x\bmod 3</math> (instead of <math>\{0,+1,-1\}</math> in the original size 22 program). The <math>x</math> values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

==== Antihydra-like Cryptid ====
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Size 25: Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== Size 36: BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \to^* & A(1, 2) \\
A(a, b) & \to^* & A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \to^* & A(2a+1, b-a) & \text{if } a 

=== Size 48: BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

Semi Infinite Tape Turing Machine

2026-08-21T10:15:22Z

A6P457: Added links

The function BBt(n,m) returns the maximum number of right steps a Turing Machine with a semi infinite tape performs before its head leaves the tape. The head starts in the first cell and halting transition is not required.
{| class="wikitable"
|+
!Domain
!Value
!Champion
|-
|BBt(2)
|≥ 3
|<code>{{TM|1RB---_1LB0RB}}</code>
|-
|BBt(3)
|≥ 12
|<code>{{TM|0RB0LB_0RC1RC_1LA0LC}}</code>
|-
|BBt(4)
|≥ 168
|<code>{{TM|1RB1LD_1RC0RC_1RD0LA_1LA0RD}}</code>
|-
|BBt(5)
|≥ 4,300
|<code>{{TM|0RB0RC_0RC0RE_0RD1LE_1LA1RD_1LC1RB}}</code>
|-
|BBt(2,3)
|≥ 17
|<code>{{TM|1RB0RA0RB_2LA2RB1LB}}</code>
|-
|BBt(3,3)
|≥ 10,053
|<code>{{TM|0RB1LA1LB_0RC2LB1RB_1LA2LC2RC}}</code>
|-
|BBt(2,4)
|≥ 88
|<code>{{TM|1RB2RB3LA2RA_1LA3LB0RB3RB}}</code>
|-
|BBt(2,5)
|≥ 415
|<code>{{TM|0RB2LA1LB4RA2RB_1LA1RB3RB4LA1LA}}</code>
|-
|BBt(2,6)
|≥ 1,429
|<code>{{TM|0RB4RB3LA2RA2LB2LA_1LA2RB5RA1RA3RA5RA}}</code>
|}

BBrev

2026-08-19T08:36:49Z

A6P457: Redirected page to Reversible Turing Machine

#REDIRECT [[Reversible Turing Machine]]

1RB1RF 1RC1RA 1LD1LF 0RB0RE ---1LC ---0LC

2026-08-18T13:18:49Z

A6P457: Added link

{{machine|1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC}}
{{TM|1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC|halt}} is the current [[BBu]](6) champion halting after 16088 steps
== Analysis ==
<pre>
A(x) = 0^inf <C 1^x 0^inf

A(3x) -> A(4x+3)
A(3x+1) -> halt
A(3x+2) -> A(4x+5)

A(2) -> A(5) -> A(9) -> A(15) -> A(23) -> A(33) -> A(47) -> A(65) -> A(89) -> A(121) -> halt

If B(x) = A(2x+1), it's trajectory is equivalent to the BBf(20) champion
[[Category:BB(6)]]

1RB1RF 1RC1RA 1LD1LF 0RB0RE ---1LC ---0LC

2026-08-18T13:18:22Z

A6P457: Created page with "{{machine|1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC}} 1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC is the current BBu(6) champion halting after 16088 steps == Analysis == <pre> A(x) = 0^inf <C 1^x 0^inf A(3x) -> A(4x+3) A(3x+1) -> halt A(3x+2) -> A(4x+5) A(2) -> A(5) -> A(9) -> A(15) -> A(23) -> A(33) -> A(47) -> A(65) -> A(89) -> A(121) -> halt If B(x) = A(2x+1), it's trajectory is equivalent to the BBf(20) champion Category:BB(6)"

{{machine|1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC}}
1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC is the current [[BBu]](6) champion halting after 16088 steps
== Analysis ==
<pre>
A(x) = 0^inf <C 1^x 0^inf

A(3x) -> A(4x+3)
A(3x+1) -> halt
A(3x+2) -> A(4x+5)

A(2) -> A(5) -> A(9) -> A(15) -> A(23) -> A(33) -> A(47) -> A(65) -> A(89) -> A(121) -> halt

If B(x) = A(2x+1), it's trajectory is equivalent to the BBf(20) champion
[[Category:BB(6)]]

BBu

2026-08-18T12:54:58Z

A6P457: Redirected page to Uniform-action Turing Machine

#REDIRECT [[uniform-action Turing Machine]]

Uniform-action Turing Machine

2026-08-18T09:57:27Z

A6P457: Added links

A Turing Machine is uniform-action is every state write the same symbol and move in the same direction. This means that the cell color affects only the next state.

The busy beaver function for UATM is BBu(n).
{| class="wikitable"
|+
!BBu(n)
!Value
!Champion
|-
|2
|6
|<code>{{TM|1RB---_1LB1LA|halt}}</code>
|-
|3
|17
|<code>{{TM|1RB---_0RC0RC_1LC1LA|halt}}</code>
|-
|4
|29
|<code>{{TM|1RB1RD_0LC0LA_1LC1LA_0RC---|halt}}</code>
|-
|5
|≥ 441
|<code>{{TM|1RB1RA_0RC0RE_1LC1LD_1LA1LD_---1RB|halt}}</code>
|-
|6
|> 16,088
|<code>{{TM|1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC|halt}}</code>
|}

== Source ==
Discord thread: https://discord.com/channels/960643023006490684/1377330731222564904/threads/1458010522967609425

Register machine

2026-08-12T06:47:07Z

A6P457: /* Analysis */ added MBB(9)

'''Register machines''', also known as '''Minsky machines''', are a [[Turing complete|Turing-complete]] model of computation.

Register machines contain a set of instructions and a set of registers. The instructions are labelled A, B, C, and so on. The registers are numbered 0, 1, 2, and so on. There are 2 types of instructions:

* inc(c, n) adds 1 to the register c then jumps to instruction n.
* dec(c, n, m) jumps to instruction m if register c equals 0, else subtract 1 to the register c then jump to instruction n.

The program halts if it reaches an undefined instruction. Here we label an undefined instruction with <code>*</code>.

== Register Busy Beaver ==
The Register Busy Beaver function, denoted MBB(n,r), returns the maximum number of instructions executed by a register machine with n instructions and r registers when started in instruction A and all registers initialized to 0. MBB(n) = MBB(n,n) (unlimited registers).
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+*</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|-
|MBB(7)
|≥ 231
|<code>0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*</code>
|-
|MBB(8)
|≥ 3394
|<code>0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*</code>
|-
|MBB(9)
|≥ 9870
|<code>0+B_0+C_0+D_1-IE_1+F_0-GI_0-HC_0-E*_0+A</code>
|}

MBB(9) is likely unoptimal.

== Analysis ==
MBB(7):
<pre>Let A(x) = A:[x, 0]

A(2x) -> 9x+20 -> A(3x+4)
A(2x+1) -> 9x+27 -> halt

A(0) -> 20 -> A(4) -> 38 -> A(10) -> 65 -> A(19) -> 108 -> halt</pre>
MBB(8):
Let S(z) = F:[0, 2z+1, z]

S(2k) -> 44k+19 -> S(5k+2)
S(2k+1) -> 4k+2 -> halt

A:[0, 0, 0] -> 44 -> S(4) -> 107 -> S(12) -> 283 -> S(32) -> 723 -> S(82) -> 1823 -> S(207) -> 414 -> halt
MBB(9):
<pre>
Let A(x) = I:[0,x]
B(x) = I:[1,x]

A(3x) -> 31x+14 -> B(4x+1)
A(3x+1) -> 31x+22 -> halt
A(3x+2) -> 31x+33 -> A(4x+5)
B(3x) -> 31x+13 -> halt
B(3x+1) -> 31x+24 -> A(4x+4)
B(3x+2) -> 31x+36 -> B(4x+4)

A:[0,0] -> 10 -> A(2) -> 33 -> A(5) -> 64 -> A(9) -> 107 -> B(13) -> 148 -> A(20) -> 219 -> A(29) -> 312 -> A(41) -> 436 -> A(57) -> 603 -> B(77) -> 811 -> B(104) -> 1090 -> B(140) -> 1462 -> B(188) -> 1958 -> B(252) -> 2616 -> halt</pre>

== Cryptids ==
No [[Cryptids]] have been found via exhaustive search, but [[Hydra]] has been hand coded into a 10-instruction, 3-register Minsky machine: <code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code> which can be interpreted the following way:[https://discord.com/channels/960643023006490684/1243312334907375676/1446694179337011325]
Let S(h,w) = A:[h-3,0,w]

Start: A:[0,0,0] = S(3,0)
S(2k,0) = A:[2k-3,0,0] -> Halt
S(2k,w+1) = A:[2k-3,0,w+1] -> A:[3k-3,0,w] = S(3k,w)
S(2k+1,w) = A:[2k-2,0,w] -> A:[3k-2,0,w+2] = S(3k+1,w+2)
[[Category:functions]]

Register machine

2026-08-11T18:40:25Z

A6P457: /* Analysis */ added MBB(8) step counts

'''Register machines''', also known as '''Minsky machines''', are a [[Turing complete|Turing-complete]] model of computation.

Register machines contain a set of instructions and a set of registers. The instructions are labelled A, B, C, and so on. The registers are numbered 0, 1, 2, and so on. There are 2 types of instructions:

* inc(c, n) adds 1 to the register c then jumps to instruction n.
* dec(c, n, m) jumps to instruction m if register c equals 0, else subtract 1 to the register c then jump to instruction n.

The program halts if it reaches an undefined instruction. Here we label an undefined instruction with <code>*</code>.

== Register Busy Beaver ==
The Register Busy Beaver function, denoted MBB(n,r), returns the maximum number of instructions executed by a register machine with n instructions and r registers when started in instruction A and all registers initialized to 0. MBB(n) = MBB(n,n) (unlimited registers).
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+*</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|-
|MBB(7)
|≥ 231
|<code>0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*</code>
|-
|MBB(8)
|≥ 3394
|<code>0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*</code>
|-
|MBB(9)
|≥ 9870
|<code>0+B_0+C_0+D_1-IE_1+F_0-GI_0-HC_0-E*_0+A</code>
|}

MBB(9) is likely unoptimal.

== Analysis ==
MBB(7):
<pre>Let A(x) = A:[x, 0]

A(2x) -> 9x+20 -> A(3x+4)
A(2x+1) -> 9x+27 -> halt

A(0) -> 20 -> A(4) -> 38 -> A(10) -> 65 -> A(19) -> 108 -> halt</pre>
MBB(8):
Let S(z) = F:[0, 2z+1, z]

S(2k) -> 44k+19 -> S(5k+2)
S(2k+1) -> 4k+2 -> halt

A:[0, 0, 0] -> 44 -> S(4) -> 107 -> S(12) -> 283 -> S(32) -> 723 -> S(82) -> 1823 -> S(207) -> 414 -> halt

== Cryptids ==
No [[Cryptids]] have been found via exhaustive search, but [[Hydra]] has been hand coded into a 10-instruction, 3-register Minsky machine: <code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code> which can be interpreted the following way:[https://discord.com/channels/960643023006490684/1243312334907375676/1446694179337011325]
Let S(h,w) = A:[h-3,0,w]

Start: A:[0,0,0] = S(3,0)
S(2k,0) = A:[2k-3,0,0] -> Halt
S(2k,w+1) = A:[2k-3,0,w+1] -> A:[3k-3,0,w] = S(3k,w)
S(2k+1,w) = A:[2k-2,0,w] -> A:[3k-2,0,w+2] = S(3k+1,w+2)
[[Category:functions]]

Register machine

2026-07-29T15:29:54Z

A6P457: /* Analysis */ Added MBB(7)

'''Register machines''', also known as '''Minsky machines''', are a [[Turing complete|Turing-complete]] model of computation.

Register machines contain a set of instructions and a set of registers. The instructions are labelled A, B, C, and so on. The registers are numbered 0, 1, 2, and so on. There are 2 types of instructions:

* inc(c, n) adds 1 to the register c then jumps to instruction n.
* dec(c, n, m) jumps to instruction m if register c equals 0, else subtract 1 to the register c then jump to instruction n.

The program halts if it reaches an undefined instruction. Here we label an undefined instruction with <code>*</code>.

== Register Busy Beaver ==
The Register Busy Beaver function, denoted MBB(n,r), returns the maximum number of instructions executed by a register machine with n instructions and r registers when started in instruction A and all registers initialized to 0. MBB(n) = MBB(n,n) (unlimited registers).
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+*</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|-
|MBB(7)
|≥ 231
|<code>0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*</code>
|-
|MBB(8)
|≥ 3394
|<code>0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*</code>
|-
|MBB(9)
|≥ 9870
|<code>0+B_0+C_0+D_1-IE_1+F_0-GI_0-HC_0-E*_0+A</code>
|}

MBB(9) is likely unoptimal.

== Analysis ==
MBB(7):
<pre>
Let A(x) = A:[x, 0]

A(2x) -> 9x+20 -> A(3x+4)
A(2x+1) -> 9x+27 -> halt

A(0) -> 20 -> A(4) -> 38 -> A(10) -> 65 -> A(19) -> 108 -> halt
</pre>
MBB(8):
<pre>
Let S(z) = F:[0, 2z+1, z]

Start: F:[0, 9, 4] = S(4)
S(2k) = S(5k+2)
S(2k+1) = Halt

S(4) -> S(12) -> S(32) -> S(82) -> S(207)

== Cryptids ==
No [[Cryptids]] have been found via exhaustive search, but [[Hydra]] has been hand coded into a 10-instruction, 3-register Minsky machine: <code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code> which can be interpreted the following way:[https://discord.com/channels/960643023006490684/1243312334907375676/1446694179337011325]
Let S(h,w) = A:[h-3,0,w]

Start: A:[0,0,0] = S(3,0)
S(2k,0) = A:[2k-3,0,0] -> Halt
S(2k,w+1) = A:[2k-3,0,w+1] -> A:[3k-3,0,w] = S(3k,w)
S(2k+1,w) = A:[2k-2,0,w] -> A:[3k-2,0,w+2] = S(3k+1,w+2)

[[Category:functions]]

Lambda Calculus

2026-07-27T16:52:41Z

A6P457: BBl(40) solved

'''Lambda calculus''' is a model of computation developed by Alonzo Church (of Church-Turing thesis fame) in 1932. If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with [[wikipedia:Lambda_calculus|this article]].

'''BBλ''' is the [[Busy Beaver]] function for lambda calculus, where BBλ(n) is the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus], a binary prefix-free encoding for all closed lambda calculus terms. Pioneered by John Tromp, BBλ is uncomputable, and therefore grows faster than any computable function.

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

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if every beta-reduction sequence leads to this normal form and '''weakly normalizing''' if some sequence does (while others may diverge).

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

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

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

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

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

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

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

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

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

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

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer. The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
| 4 || = 4 || <math>\lambda 1</math>
|
|JT & BF
|-
| 6 || = 6 || <math>\lambda\lambda 1</math>
|
|JT & BF
|-
| 7 || = 7 || <math>\lambda\lambda 2</math>
|
|JT & BF
|-
| 8 || = 8 || <math>\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 9 || = 9 || <math>\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
|18
|= 18
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
|19
|= 19
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
|20
|= 20
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|}
</div></div>

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6 > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6 > 10^{19,729}</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>= 5\cdot{3^{3^{3^3}}} + 6 > 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles & 2014MELO & mammilaria
|-
|40 || <math> ≈ 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|- ||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|- ||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|- ||||||-
|331
| <math>> f_{lim(BMS)}^3(5)</math>
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|- ||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

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

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

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

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

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

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

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

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

== BBλ⁻¹ ==
BBλ⁻¹ is defined as the size of the smallest BLC program that outputs its input in church numerals. Unlike BBλ, here upper bounds are provided instead of lower bounds.
{| class="wikitable"
|+
!n
!Champion
!BBλ⁻¹
|-
|0
|λλ1
|≤ 6
|-
|1
|λ1
|≤ 4
|-
|2
|λλ2(21)
|≤ 16
|-
|3
|λλ2(2(21))
|≤ 21
|-
|4
|λλ2(2(2(21)))
|≤ 26
|-
|5
|λλ2(2(2(2(21))))
|≤ 31
|-
|6
|λλ2(2(2(2(2(21)))))
|≤ 36
|-
|7
|λλ2(2(2(2(2(2(21))))))
|≤ 41
|-
|8
|(λλ2(2(21)))(λλ2(21))
|≤ 39
|-
|9
|(λλ2(21))(λλ2(2(21)))
|≤ 39
|-
|10
|λ(λλ2(2(2(2(21)))))(λ2(21))
|≤ 49
|-
|11
|λλ2(2(2(2(2(2(2(2(2(2(21))))))))))
|≤ 61
|-
|12
|λ(λλ2(2(2(21))))(λ2(2(21)))
|≤ 49
|-
|13
|λλ2(2(2(2(2(2(2(2(2(2(2(21)))))))))))
|≤ 67
|-
|14
|λ(λλ2(2(2(2(2(2(21)))))))(λ2(21))
|≤ 59
|-
|15
|λ(λλ2(2(2(2(21)))))(λ2(2(21)))
|≤ 54
|-
|16
|(λ111)(λλ2(21))
|≤ 30
|-
|17
|λλ2((λ111)(λλ2(21))21)
|≤ 48
|-
|18
|λλ2(2((λ111)(λλ2(21))21))
|≤ 53
|-
|19
|λλ2(2(2((λ111)(λλ2(21))21)))
|≤ 58
|-
|20
|λ(λλ2(2(2(2(21)))))(λ2(2(2(21))))
|≤ 59
|-
|21
|λ(λλ2(2(2(2(2(2(21)))))))(λ2(2(21)))
|≤ 64
|-
|22
|λλ2(2(2(2(2(2((λ111)(λλ2(21))21))))))
|≤ 73
|-
|23
|λλ2(2(2(2(2(2(2((λ111)(λλ2(21))21)))))))
|≤ 78
|-
|24
|(λλ2(λλ2(21))(21))(λλ2(2(21)))
|≤ 57
|-
|25
|(λλ2(21))(λλ2(2(2(2(21)))))
|≤ 49
|-
|26
|λλ2(((λλ2(21))(λλ2(2(2(2(21))))))21)
|≤ 67
|-
|27
|(λ11)(λλ2(2(21)))
|≤ 31
|-
|28
|λλ2(((λ11)(λλ2(2(21))))21)
|≤ 49
|-
|29
|λλ2(2(((λ11)(λλ2(2(21))))21))
|≤ 54
|-
|30
|λλ2(2(2(((λ11)(λλ2(2(21))))21)))
|≤ 59
|-
|31
|λλ2(2(2(2(((λ11)(λλ2(2(21))))21))))
|≤ 64
|-
|32
|(λλ2(2221))(λλ2(21))
|≤ 44
|-
|33
|λλ2((λλ1(1112))2(λλ2(21))1)
|≤ 59
|-
|34
|(λλ2(λ2(33321)))(λλ2(21))
|≤ 61
|-
|35
|λλ2(2(2((λλ1(1112))2(λλ2(21))1)))
|≤ 69
|-
|36
|(λ1(λ2(λ2(2(21)))))(λλ2(21))
|≤ 50
|-
|37
|λλ2((λ1(λ2(λ2(2(21)))))(λλ2(21))21)
|≤ 68
|-
|38
|(λλ2(λ2(32(33321))))(λλ2(21))
|≤ 70
|-
|39
|(λλ2(λ2(32(3(32)1))))(λλ2(2(21)))
|≤ 69
|-
|40
|(λλ22(2(λ2(3321))))(λλ2(21))
|≤ 63
|-
|41
|λλ2((λλ11(1(λ3(2231))))2(λλ2(21))1)
|≤ 78
|-
|42
|(λ1(λλ2(λ2(321)))1)(λλ2(21))
|≤ 57
|-
|43
|(λλ1(2(11)2))(λλ2(21))(λλλ2(321))
|≤ 68
|-
|44
|(λλ1(1(2(11)2)))(λλ2(21))(λλλ2(321))
|≤ 72
|-
|45
|λ(λλ2(21))(λλ2(2(21)))(λ2(2(2(2(21)))))
|≤ 75
|-
|46
|(λλ2(λ2(32(332(33321)))))(λλ2(21))
|≤ 87
|}

== See Also ==
* [[Busy Beaver for SKI calculus|SKI calculus]]
* [[De Bruijn index]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://github.com/ccz181078/BLC.git
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
[[category:Functions]]

1RB1RF 0LC0RF 1RD1LC ---0LE 0RC1LF 1RA0LE

2026-07-27T16:43:20Z

A6P457:

{{machine|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}}
{{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}} is a [[BB(6)]] potential [[cryptid]] similar to [[BMO 1]]
== Analysis ==
<pre>
A(x,y,z) = 1^x 0 1^y F> 1^z

A(x,2y,z) -> A(x+4,2y-2,z-2) if y>0 and z>1
A(x,0,z) -> A(0,x+4,z-3) if z>2
A(x,0,1) -> halt
A(x,0,2) -> A(0,x+2,0)
A(2x,y,0) -> A(0,2x+4,y)
A(2x+1,y,0) -> A(4,2x,y) if x>0
A(1,y,0) -> A(4,0,y+1)
A(x,2y,1) -> A(x+4,2y-4,0) if y>1
A(x,2,1) -> A(0,4,x+2)
A(x,2y+1,1) -> A(0,x+2y+2,0)
A(x,2y+1,z) -> A(0,x+2y+4,z-2) if z>1

The A(2x+1,y,0) and A(1,y,0) rules are unreachable from the start(reaching them requires odd x)

B(x,y) = A(0,2x,y)

B(x,2y) -> B(2y+2,2x-2y) if x>=y
B(x,2y) -> B(2x+2,2y-2x-3) if x<y-1
B(x,2x+2) -> B(2x+1,0)
B(x,2y+1) -> B(2y+4,2x-2y-4) if x>y+1
B(x,2x-1) -> B(2,4x-2)
B(x,2y+1) -> B(2x+2,2y-2x-2) if x<y
B(x,2x+1) -> halt

B(1,0) -> B(2,2) -> B(4,2) -> B(4,6) -> B(8,2) -> B(4,14) -> B(10,3) -> B(6,14) -> B(13,0) -> B(2,26) -> B(6,19) -> B(14,4) -> B(6,24) -> B(14,9) -> B(12,16) -> B(18,8) -> B(10,28) -> B(22,5) -> B(8,36) -> B(18,17) -> B(20,16) -> B(18,24) -> ...
[[Category:BB(6)]]
[[Category:Potential Cryptids]]

BB(6)

2026-07-21T17:30:56Z

A6P457: Added potential cryptid

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== History ==
* In 1964, Green established Σ(5) ≥ 35.<ref name=":PMH">Pascal Michel. (last updated 2026). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm62 </ref>
* In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.<ref name=":PMH" />
* In 1983, Brady established S(6) ≥ 13,488 and Σ(6) ≥ 117.<ref name=":PMH" />
* In 1982, Schult established S(6) ≥ 4,208,824 and Σ(6) ≥ 2,075, although these bounds only became known later.<ref name=":PMH" />
* In 1990, Heiner Marxen and Jürgen Buntrock established first S(6) ≥ 13,122,572,797 and Σ(6) ≥ 136,612, then S(6) ≥ 8,690,333,381,690,951 and Σ(6) ≥ 95,524,079.<ref name=":PMH" />
* In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 1042 and Σ(6) > 2.5 × 1021.<ref name=":PMH" />
* In August 2000, Heiner Marxen and Jürgen Buntrock established first S(6) > 6.1 × 10119 and Σ(6) > 1.4 × 1060, then S(6) > 6.1 × 10925 and Σ(6) > 6.4 × 10462.<ref name=":PMH" />
* In February 2001, Heiner Marxen and Jürgen Buntrock established S(6) > 3.0 × 101730 and Σ(6) > 1.2 × 10865.<ref name=":PMH" />
* In November 2007, Terry and Shawn Ligocki established S(6) > 8.9 × 101762 and Σ(6) > 2.5 × 10881.<ref name=":PMH" />
* In December 2007, Terry and Shawn Ligocki established S(6) > 2.5 × 102879 and Σ(6) > 4.6 × 101439.<ref name=":PMH" />
* In May 2010, Pavel Kropitz established S(6) > 3.8 × 1021132 and Σ(6) > 3.1 × 1010566.<ref name=":PMH" />
* In June 2010, Pavel Kropitz established S(6) > 7.4 × 1036534 and Σ(6) > 3.5 × 1018267.<ref name=":PMH" />
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1094 machines up to equivalence and 2320 machines not considering equivalence as of June 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet]. The informal holdout count is 1101.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 to 1e15. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet].

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}, Space Needle
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}, similar to Space Needle
* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}, similar to Space Needle

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}
* {{TM|1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF|undecided}}
* {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA|halt}}
|<math>10^{1\,400\,000\,000}</math>(runtime)
|Racheline<ref>https://discord.com/channels/960643023006490684/1345502880727040091/1345502880727040091</ref>
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub) upon initial application.

== References ==
<references />

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

1RB1RF 0LC0RF 1RD1LC ---0LE 0RC1LF 1RA0LE

2026-07-21T16:10:23Z

A6P457: Created page with "{{machine|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}} {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}} is a BB(6) potential cryptid similar to BMO 1 == Analysis == <pre> A(x,y,z) = 1^x 0 1^y F> 1^z A(x,2y,z) -> A(x+4,2y-2,z-2) if y>0 and z>1 A(x,0,z) -> A(0,x+4,z-3) if z>2 A(x,0,1) -> halt A(x,0,2) -> A(0,x+2,0) A(2x,y,0) -> A(0,2x+4,y) A(2x+1,y,0) -> A(4,2x,y) if x>0 A(1,y,0) -> A(4,0,y+1) A(x,2y,1) -> A(x+4,2y-4,0) if y>1 A(x,2,1) -> A(0,4,x+2) A(x,2y+1,1)..."

{{machine|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}}
{{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE}} is a [[BB(6)]] potential [[cryptid]] similar to [[BMO 1]]
== Analysis ==
<pre>
A(x,y,z) = 1^x 0 1^y F> 1^z

A(x,2y,z) -> A(x+4,2y-2,z-2) if y>0 and z>1
A(x,0,z) -> A(0,x+4,z-3) if z>2
A(x,0,1) -> halt
A(x,0,2) -> A(0,x+2,0)
A(2x,y,0) -> A(0,2x+4,y)
A(2x+1,y,0) -> A(4,2x,y) if x>0
A(1,y,0) -> A(4,0,y+1)
A(x,2y,1) -> A(x+4,2y-4,0) if y>1
A(x,2,1) -> A(0,4,x+2)
A(x,2y+1,1) -> A(0,x+2y+2,0)
A(x,2y+1,z) -> A(0,x+2y+4,z-2) if z>1

The A(2x+1,y,0) and A(1,y,0) rules are unreachable from the start(reaching them requires odd x)

B(x,y) = A(0,x,y)

B(2x,2y) -> B(4y+4,2x-2y) if x>=y
B(2x,2y) -> B(4x+4,2y-2x-3) if x<y-1
B(2x,2x+2) -> B(4x+2,0)
B(2x,2y+1) -> B(4y+8,2x-2y-4) if x>y+1
B(2x+2,2x+1) -> B(4,4x+2)
B(2x,2y+1) -> B(4x+4,2y-2x-2) if x<y
B(2x,2x+1) -> halt
B(2x+1,2y) -> B(4y+4,2x-2y+1) if x>=y
B(2x+1,2y) -> B(4x+4,2y-2x-2) if x<y
B(2x+1,2y+1) -> B(2x+2y+2,0) if x>=y
B(2x+1,2y+1) -> B(4x+4,2y-2x-1) if x<y

The B(2x+1,y) rules can only be reached if they are the start

B(2,0) -> B(4,2) -> B(8,2) -> B(8,6) -> B(16,2) -> B(8,14) -> B(20,3) -> B(12,14) -> B(26,0) -> B(4,26) -> B(12,19) -> B(28,4) -> B(12,24) -> B(28,9) -> B(24,16) -> B(36,8) -> B(20,28) -> B(44,5) -> B(16,36) -> B(36,17) -> B(40,16) -> B(36,24) -> ...
[[Category:BB(6)]]
[[Category:Potential Cryptids]]

Instruction-Limited Busy Beaver

2026-07-14T16:05:55Z

A6P457: Added gBBi champions

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd<ref>Drozd, N. (Sept. 2025). "[https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html The Shape of a Turing Machine | Something Something Programming]"</ref>.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by [[Tibor Radó]], was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''')<ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|-
|9
|<math display="block">> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|
|
|-
|11
|<math> > 2 \uparrow^{15} 5</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|
|
|-
|14
|<math>> f_{\omega}(2 \uparrow^{15} 5)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|
|
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.

* [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|I-L Blanking Busy Beaver]].
{| class="wikitable"
|+
!n
!gBBi(n)
!Champions
|-
|1
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|2
|3
|{{TM|0RB---_1LA---|halt}}
|-
|3
|5
|{{TM|0RB---_1LA1RB|halt}} {{TM|0RB------_1LA2LA---|halt}} {{TM|0RB------_1LA2RB---|halt}}
|-
|4
|13
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|5
|19
|{{TM|0RB---1LB---_1LA2RB3LA---|halt}}
|-
|6
|25
|{{TM|0RB---1LB------_1LA2RB3LA4LA---|halt}} {{TM|0RB---1LB------_1LA2RB3LA4RB---|halt}} {{TM|0RB---1LB---_1LA2RB3LA1RB|halt}} {{TM|0RB---1LB---_1LA2RB3LA3RB|halt}}
|-
|7
|41
|{{TM|0RB---1LB---3LA_1LA2RB3LA4RB---|halt}} {{TM|0RB---1LB---3LB_1LA2RB3LA4RB---|halt}}
|-
|8
|55
|{{TM|0RB---1LB---3LA---_1LA2RB3LA4RB5RB---|halt}}
|-
|9
|238
|{{TM|0RB---1LB---3LA---_1LA2RB3LA4RB5RB1LB|halt}}
|-
|10
|941
|{{TM|0RB---1LB---3LA1RB_1LA2RB3LA4RB5RB1LB|halt}}
|-
|11
|1341
|{{TM|0RB6RB1LB---3LA1RB---_1LA2RB3LA4RB5RB1LB---|halt}}
|-
|12
|10465
|{{TM|0RB6RB1LB---3LA1RB---_1LA2RB3LA4RB5RB1LB5LA|halt}}
|-
|13
|10675
|{{TM|0RB6RB1LB---3LA1RB7RB---_1LA2RB3LA4RB5RB1LB5LA---|halt}} {{TM|0RB6RB1LB---3LA1RB3RB_1LA2RB3LA4RB5RB1LB5LA|halt}}
|-
|14
|>=9874580
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}
== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

1RB2RB1LA 2LC0LB2LB 2RC2RA0LC

2026-07-09T15:57:12Z

A6P457: Added category

{{machine|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}} is the current [[BLB]](3,3) champion blanking the tape after <math>10^{42745}</math> steps
== Analysis ==
<pre>
A(x,y)=2 C> 2^x 0 1^y

A(x,y) -> A(2x+5,y-3) if y>2
A(x,0) -> blank
A(x,1) -> A(8,2x+1)
A(x,2) -> blank

A(8,4) -> A(21,1) -> A(8,43) -> A(21,40) -> A(47,37) -> A(99,34) -> A(203,31) -> A(411,28) -> A(827,25) -> A(1659,22) -> A(3323,19) -> A(6651,16) -> A(13307,13) -> A(26619,10) -> A(53243,7) -> A(106491,4) -> A(212987,1) -> A(8,425975) -> ... -> A(13*2^141991-5,2) -> blank
</pre>
[[Category:BB(3,3)]]

1RB2RB1LA 2LC0LB2LB 2RC2RA0LC

2026-07-09T15:56:22Z

A6P457: Created page with "{{machine|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}} {{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}} is the current BLB(3,3) champion blanking the tape after <math>10^{42745}</math> steps == Analysis == <pre> A(x,y)=2 C> 2^x 0 1^y A(x,y) -> A(2x+5,y-3) if y>2 A(x,0) -> blank A(x,1) -> A(8,2x+1) A(x,2) -> blank A(8,4) -> A(21,1) -> A(8,43) -> A(21,40) -> A(47,37) -> A(99,34) -> A(203,31) -> A(411,28) -> A(827,25) -> A(1659,22) -> A(3323,19) -> A(6651,16) -> A(13307,13) -> A(26619,10) ->..."

1RB2RA1RA2RB 2LB3LA0RB0RA

2026-07-07T13:05:56Z

A6P457: Restored categories

{{machine|1RB2RA1RA2RB_2LB3LA0RB0RA}}
{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}} is the current [[BLB]](2,4) champion blanking the tape after 1,367,361,263,049 steps
== Analysis ==
<pre>
A(x,y,z) = 1^x 0 1^y B< 2^z

A(x,2y,z) -> A(0,x,5y+z+1) if x>0
A(0,2y,z) -> blank
A(x,2y+1,z) -> A(x+1,5y+z+1,1)

A(1,0,0) -> A(0,1,1) -> A(1,2,1) -> A(0,1,7) -> A(1,8,1) -> A(0,1,22) -> A(1,23,1) -> A(2,57,1) -> A(3,142,1) -> A(0,3,357) -> A(1,363,1) -> A(2,907,1) -> A(3,2267,1) -> A(4,5667,1) -> A(5,14167,1) -> A(6,35417,1) -> A(7,88542,1) -> A(0,7,221357) -> A(1,221373,1) -> A(2,553432,1) -> A(0,2,1383532) -> blank
</pre>
[[Category:BB(2,4)]]

1RB2RA1RA2RB 2LB3LA0RB0RA

2026-07-06T19:09:20Z

A6P457: Created page with "{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}} is the current BLB(2,4) champion blanking the tape after 1,367,361,263,049 steps == Analysis == <pre> A(x,y,z)=1^x 0 1^y B< 2^z A(x,2y,z)->A(0,x,5y+z+1) if x>0 A(0,2y,z)->blank A(x,2y+1,z)->A(x+1,5y+z+1,1) A(1,0,0)->A(0,1,1)->A(1,2,1)->A(0,1,7)->A(1,8,1)->A(0,1,22)->A(1,23,1)->A(2,57,1)->A(3,142,1)->A(0,3,357)->A(1,363,1)->A(2,907,1)->A(3,2267,1)->A(4,5667,1)->A(5,14167,1)->A(6,35417,1)->A(7,88542,1)->A(0,7,221357)->A(1,221373,1)->A..."

{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}} is the current [[BLB]](2,4) champion blanking the tape after 1,367,361,263,049 steps
== Analysis ==
<pre>
A(x,y,z)=1^x 0 1^y B< 2^z

A(x,2y,z)->A(0,x,5y+z+1) if x>0
A(0,2y,z)->blank
A(x,2y+1,z)->A(x+1,5y+z+1,1)

A(1,0,0)->A(0,1,1)->A(1,2,1)->A(0,1,7)->A(1,8,1)->A(0,1,22)->A(1,23,1)->A(2,57,1)->A(3,142,1)->A(0,3,357)->A(1,363,1)->A(2,907,1)->A(3,2267,1)->A(4,5667,1)->A(5,14167,1)->A(6,35417,1)->A(7,88542,1)->A(0,7,221357)->A(1,221373,1)->A(2,553432,1)->A(0,2,1383532)->blank
</pre>

Fractran

2026-06-29T07:53:15Z

A6P457: Added BBf(23)

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_n</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, and all other machines for BBf(22) are solved.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|140 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 proof of nonhalting of all 140: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]
|2003 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 gave a proof of all machines but the Fenrir-family, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1510781736374763702 1 Jun 2026]
|21,295 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1511579969825013811 2 Jun 2026]

Known [[Cryptid|Cryptids]]:

# Frankenstein's Monster
# Antihydra-like Cryptid
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{k+5} & \xrightarrow{3} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{k+7} & \xrightarrow{4} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

 
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This pseudo-random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>
If it were a truly random walk, there would be a 5.9% chance that it takes at least 275 steps to reach 0. So this program is mildly lucky.
==== BBf(23) ====
The BBf(23) champion
<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk:
let <math>A(x,y)=[0,0,0,x,y]</math>

<math>[1,0,0,0,0] \to A(0,2)</math>

<math>A(2x,y) \xrightarrow{12x+4} A(5x+1,y+1)</math> if y>0

<math>A(2x+1,y) \xrightarrow{12x+10} A(5x+4,y-1)</math> if y>0

<math>A(x,0) \to halt</math>

<math>A(0,2) \to A(1,3) \to A(4,2) \to A(11,3) \to A(29,2) \to A(74,1) \to A(186,2) \to A(466,3) \to A(1166,4) \to A(2916,5) \to A(7291,6) \to \dots</math>
== Cryptids ==

=== Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 2003 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 1997 holdouts were non-halting and 3 holdouts were halting. The remaining 3 holdouts are the Fenrir family.[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179] Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Frankenstein's Monster ===
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the y values now change by {+1,+3,-1} depending on the value of x mod 3 (instead of {0,+1,-1} in the original size 22 program). The x values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

=== Antihydra-like Cryptid ===
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \to^* & A(1, 2) \\
A(a, b) & \to^* & A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \to^* & A(2a+1, b-a) & \text{if } a 

=== BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

Brainfuck

2026-06-20T07:42:44Z

A6P457: /* Language Overview */ fixed typo

Brainfuck is a Turing complete language created in 1993 by Urban Muller.

The language was designed to have an extremely tiny compiler (on the order of 200 bytes), while remaining Turing-complete. For Busy Beaver and small Turing machine research, Brainfuck provides an interesting minimal model of computation distinct from the standard 2-symbol Turing machine formalism.

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

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

Programs are composed of commands, represented with symbols:
{| class="wikitable"
|+
!Command
!Operation
|-
|<code>></code>
|Move pointer right
|-
|<code><</code>
|Move pointer left
|-
|<code>+</code>
|Increment current cell (mod 256 or arbitrary integer)
|-
|<code>-</code>
|Decrement current cell
|-
|<code>[</code>
|Jump forward past matching <code>]</code> if current cell is zero
|-
|<code>]</code>
|Jump back to matching <code>[</code> if current cell is non-zero
|-
|<code>.</code>
|Output current cell as a character (often ASCII)
|-
|<code>,</code>
|Input a character and store in current cell
|}

== BB_brainf ==
'''BB_brain(n)''' returns the maximum cell value a program with n instructions can have when halting. Commands <code>.</code> and <code>,</code> are ignored.
{| class="wikitable"
!n
!BB_brainf(n)
!Champion
|-
|1
|1
|<code>+</code>
|-
|2
|2
|<code>++</code>
|-
|3
|3
|<code>+++</code>
|-
|4
|4
|<code>++++</code>
|-
|5
|≥ 5
|<code>+++++</code>
|-
|6
|≥ 6
|<code>++++++</code>
|-
|7
|≥ 7
|<code>+++++++</code>
|-
|8
|≥ 8
|<code>++++++++</code>
|-
|9
|≥ 9
|<code>+++++++++</code>
|-
|10
|≥ 10
|<code>++++++++++</code>
|-
|11
|≥ 11
|<code>+++++++++++</code>
|-
|12
|≥ 12
|<code>++++++++++++</code>
|-
|13
|≥ 16
|<code>++++[->++++<]</code>
|-
|14
|≥ 20
|<code>+++++[->++++<]</code>
|-
|15
|≥ 25
|<code>+++++[->+++++<]</code>
|-
|16
|≥ 30
|<code>++++++[->+++++<]</code>
|-
|17
|≥ 36
|<code>++++++[->++++++<]</code>
|-
|18
|≥ 42
|<code>+++++++[->++++++<]</code>
|-
|19
|≥ 49
|<code>+++++++[->+++++++<]</code>
|-
|20
|≥ 56
|<code>++++++++[->+++++++<]</code>
|-
|21
|≥ 64
|<code>++++++++[->++++++++<]</code>
|-
|22
|≥ 72
|<code>+++++++++[->++++++++<]</code>
|-
|23
|≥ 81
|<code>+++++++++[->+++++++++<]</code>
|-
|24
|≥ 90
|<code>+++++++++[->++++++++++<]</code>
|-
|25
|≥ 340
|<code>++++[>+[->++<]>[-<++>]<<]</code>
|-
|26
|≥ 1364
|<code>+++++[>+[->++<]>[-<++>]<<]</code>
|}

[page under construction]

[[Category:Functions]]

1RB0LE 1LC1RA ---1LD 0RB1LF 1RD1LA 0LA0RD

2025-07-16T11:40:26Z

A6P457: Added non-halting

{{machine|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD}}
{{TM|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD|non}} is a non-halting [[BB(6)]] Turing machine.

Analysis by @racheline on 29 July 2024 ([https://discord.com/channels/960643023006490684/1239205785913790465/1267551868997992652 Discord link]):

<pre>
1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD
A(n) = 0^inf <A 0 1^n

rules:
A(6n) -> A(12n+3)
A(6n+1) -> A(12n+6)
A(6n+2) -> halt
A(6n+3) -> A(9n+9)
A(6n+4) -> halt
A(6n+5) -> A(9n+12)
start from A(3)

as we can see, everything that doesn't halt goes to A(6m) or A(6m+3) for some m, so halting is unreachable

the next two (1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_1RD0LA and 1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0LA) are clearly equivalent to it, so also non-halting
</pre>

Champions

2025-07-15T13:40:58Z

A6P457: added links

Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] who maximize a [[Busy Beaver function]]. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for [https://bbchallenge.org/~pascal.michel/bbc Busy Beaver champions] and the [https://bbchallenge.org/~pascal.michel/ha History of Previous Champions].

== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the <math> f_{x}(n) </math> used in the lowerbounds represent the [[Fast-Growing Hierarchy]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2)]]
|<math> 6 </math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|Tibor Radó
|Direct Simulation
|-
|[[BB(3)]]
|<math> 21 </math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by Shen Lin
|Direct Simulation
|-
|[[BB(4)]]
|<math> 107 </math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math> > 2\uparrow\uparrow\uparrow 5 </math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math> > f_\omega(f_9(2)) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math> > f_\omega^2(25) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|
|
|
|
|-
|BB(14)
|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math> > f_{\omega + 1}(f_\omega(10^{57})) </math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math> > f_{\omega + 1}^2(10^{10^{57}}) </math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|
|
|
|
|-
|BB(19)
|<math> > f_{\omega + 1}^3(f_\omega(60)) </math>
|[[User:Jacobzheng/BB(19)]]
|Jacobzheng in 2024
|
|-
|BB(20)
|<math> > f_{\omega + 2}^2(21) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math> > f_{\omega^\omega}(75\,500) </math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math> > f_{\omega^\omega}^4(32) </math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,3)]]
|<math> 38 </math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|
|
|-
|[[BB(3,3)]]
|<math> > 10^{17} </math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|
|
|-
|[[BB(4,3)]]
|<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,4)]]
|<math> 3\,932\,964 </math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|Shawn & Terry Ligocki in 2005
|Pascal Michel, Heiner Marxen, Allen Brady
|-
|BB(3,4)
|<math> > 2 \uparrow^{15} 5 </math>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|
|
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,5)]]
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|
|
|-
|BB(3,5)
|<math> > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15) </math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|BB(2,6)
|<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|
|
|}

Holdouts lists

2025-07-06T16:54:32Z

A6P457: BB(6)#Holdouts

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are unable to decide.

Holdout lists are often shared by contributors. There is a [[#Downloadable Holdout Lists|Downloadable Holdout Lists]] table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.

The table with the "Number of holdouts" is based on the holdout lists listed in the table below it. Thus, some of these numbers have not been independently verified so they should be treated as such. All the zero entries, no holdouts, have been verified.

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|3335
|-
!3-symbol
|0
|6
|460,916,384
|
|
|-
!4-symbol
|0
|434,787,751
|
|
|
|-
!5-symbol
|83
|
|
|
|
|-
!6-symbol
|22,302,296
|
|
|
|
|}

== Downloadable Holdout Lists ==
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(2,5)]]
|[https://discordapp.com/channels/960643023006490684/1259770421046411285/1355593937531961365, March 29 2025]
|@mxdys
|83
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

Main Page

2025-07-06T16:25:07Z

A6P457: Champions

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [https://en.wikipedia.org/wiki/Tibor_Rad%C3%B3 Tibor Radó] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Rado, 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: #ffe4b2;" | [[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>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

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

== About bbchallenge ==
[https://www.bbchallenge.org 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/Coq_(software) Coq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* 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 Coq / Rocq 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>

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