BusyBeaverWiki - Recent changes [en]

BB(2,7)

Tjligocki — Tue, 01 Sep 2026 08:44:38 GMT

Stage 1: Consolidated 70-79xxxx

@@ Line 34: / Line 34: @@
 |-
 |00-09xxxx
-|rowspan='24'|Terry Ligocki
+|rowspan='15'|Terry Ligocki
 |Yes
 |309,868,865
-|rowspan='24'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
+|rowspan='15'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
 |-
 |10-19xxxx
@@ Line 63: / Line 63: @@
 |309,917,121
 |-
-|70xxxx
+|70-79xxxx
 |Yes
-|35,159,738
+|315,355,885
 |-
 |80xxxx

TMBR: August 2026

Tjligocki — Tue, 01 Sep 2026 08:39:52 GMT

Holdouts: Update BB(2,7) enumeration

← Older revision		Revision as of 08:39, 1 September 2026
Line 19:		Line 19:
	** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.		** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
	*[[BB(2,7)]]:		*[[BB(2,7)]]:
	**Terry Ligocki enumerated ~~130K~~ more subtasks, increasing the number of holdouts to '''2,~~587~~,~~752~~,~~107~~'''. A total of ~~830K~~ subtasks out of the 1 million subtasks (or '''83%''') have been enumerated.		**Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''2,615,659,487'''. A total of 840K subtasks out of the 1 million subtasks (or '''84%''') have been enumerated.

	[[Category:This Month in Beaver Research\|2026-08]]		[[Category:This Month in Beaver Research\|2026-08]]

BB(2,7)

Tjligocki — Tue, 01 Sep 2026 08:39:09 GMT

Stage 1: Added 79xxxx

← Older revision		Revision as of 08:39, 1 September 2026
Line 34:		Line 34:
	\|-		\|-
	\|00-09xxxx		\|00-09xxxx
	\|rowspan='23'\|Terry Ligocki		\|rowspan='24'\|Terry Ligocki
	\|Yes		\|Yes
	\|309,868,865		\|309,868,865
	\|rowspan='23'\|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]		\|rowspan='24'\|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
	\|-		\|-
	\|10-19xxxx		\|10-19xxxx
Line 100:		Line 100:
	\|-		\|-
	\|79xxxx		\|79xxxx
	\|No		\|Yes
	\|~~TBD~~		\|27,907,380
	\|-		\|-
	\|80xxxx		\|80xxxx
Line 127:		Line 127:
	\|TBD		\|TBD
	\|-		\|-
	\|86-99xxxx		\|86xxxx
			\|No
			\|TBD
			\|-
			\|87-99xxxx
	\|TBD		\|TBD
	\|No		\|No
Line 136:		Line 140:
	\| ---		\| ---
	\| ---		\| ---
	\|'''2,~~587~~,~~752~~,~~107~~'''		\|'''2,615,659,487'''
	\| ---		\| ---
	\|-		\|-

TMBR: August 2026

Tjligocki — Tue, 01 Sep 2026 00:17:43 GMT

Holdouts: Update BB(2,7) enumeration

← Older revision		Revision as of 00:17, 1 September 2026
Line 19:		Line 19:
	** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.		** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
	*[[BB(2,7)]]:		*[[BB(2,7)]]:
	**Terry Ligocki enumerated ~~100K~~ more subtasks, increasing the number of holdouts to '''2,~~499~~,~~145~~,~~299~~'''. A total of ~~800K~~ subtasks out of the 1 million subtasks (or '''80%''') have been enumerated.		**Terry Ligocki enumerated 130K more subtasks, increasing the number of holdouts to '''2,587,752,107'''. A total of 830K subtasks out of the 1 million subtasks (or '''83%''') have been enumerated.

	[[Category:This Month in Beaver Research\|2026-08]]		[[Category:This Month in Beaver Research\|2026-08]]

BB(2,7)

Tjligocki — Tue, 01 Sep 2026 00:16:11 GMT

Stage 1: Added 81xxxx, 82xxxx, and 84xxxx

← Older revision		Revision as of 00:16, 1 September 2026
(2 intermediate revisions by the same user not shown)
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	\|-		\|-
	\|00-09xxxx		\|00-09xxxx
	\|rowspan='21'\|Terry Ligocki		\|rowspan='23'\|Terry Ligocki
	\|Yes		\|Yes
	\|309,868,865		\|309,868,865
	\|rowspan='21'\|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]		\|rowspan='23'\|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
	\|-		\|-
	\|10-19xxxx		\|10-19xxxx
Line 108:		Line 108:
	\|-		\|-
	\|81xxxx		\|81xxxx
	\|No		\|Yes
	\|~~TBD~~		\|35,313,171
	\|-		\|-
	\|82xxxx		\|82xxxx
			\|Yes
			\|26,627,542
			\|-
			\|83xxxx
	\|No		\|No
	\|TBD		\|TBD
	\|-		\|-
	\|~~83xxxx~~		\|84xxxx
			\|Yes
			\|26,666,095
			\|-
			\|85xxxx
	\|No		\|No
	\|TBD		\|TBD
	\|-		\|-
	\|84-99xxxx		\|86-99xxxx
	\|TBD		\|TBD
	\|No		\|No
Line 128:		Line 136:
	\| ---		\| ---
	\| ---		\| ---
	\|'''2,~~499~~,~~145~~,~~299~~'''		\|'''2,587,752,107'''
	\| ---		\| ---
	\|-		\|-

Cyclic Tag

Sligocki — Mon, 31 Aug 2026 15:44:41 GMT

Busy Beaver Function: Clarify size calculation

← Older revision		Revision as of 15:44, 31 August 2026
Line 18:		Line 18:
	The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.		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~~.		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"		{\| class="wikitable"
	\|+		\|+

Cyclic Tag

A6P457 — Mon, 31 Aug 2026 15:31:41 GMT

Execution rules:: fixed string change rule

← Older revision		Revision as of 15:31, 31 August 2026
(One intermediate revision by the same user not shown)
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	* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first 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.		* 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.		* '''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 ~~has less than two symbol~~.		* The cyclic tag system halts when it reaches the empty string.

	== Busy Beaver Function ==		== Busy Beaver Function ==

Cyclic Tag

Azerty — Mon, 31 Aug 2026 14:52:07 GMT

Show changes

Cyclic Tree Rewriting System

Azerty — Mon, 31 Aug 2026 14:30:53 GMT

Collection of Champions: Consistency fixes

← Older revision		Revision as of 14:30, 31 August 2026
Line 176:		Line 176:
	** "((()))" -> "((()))()()"		** "((()))" -> "((()))()()"
	** "()" -> "D		** "()" -> "D
	</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(6) ~~Champion~~'''<div class="mw-collapsible-content">		</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(6)'''<div class="mw-collapsible-content">
	Pair found by Moja and Racheline:		Pair found by Moja and Racheline:

Line 187:		Line 187:
	** ()()() -> D		** ()()() -> D
	** () -> D		** () -> D
	</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(7) ~~Champion~~'''<div class="mw-collapsible-content">		</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(7)'''<div class="mw-collapsible-content">
	Pair found by Moja and Racheline:		Pair found by Moja and Racheline:

Cyclic Tree Rewriting System

Azerty — Mon, 31 Aug 2026 14:27:27 GMT

Renamed the page and removed unecessary spacings.

New page

== Cyclic Tree Rewriting System ==
This variant of Cyclic Tag operates upon a Dyck String instead of a binary string. A Dyck String is a string of parentheses “(“ and “)” of length 2n, that consists of an equal number of symbols such that for all prefixes, “(“ occurs at least as many times as the other “)”. In shorter terms, a Dyck String is a balanced parenthetical string of length 2n. There does not exist a Dyck String (in this format) with an odd length (hence the 2n).

=== Rooted Ordered Trees and Dyck Strings ===
Dyck Strings (pronounced as: DYE-K) encode finite rooted ordered trees through a well-known bijection that maps the structure of a tree to a balanced sequence of parentheses. The most common method involves a depth-first traversal of the tree. A rooted ordered tree is a tree with these key features:

* One vertex is distinguished as the root,

* All edges are directed away from the root (conceptually, though not always drawn with arrows),

* For each node, the children are arranged in a left-to-right order.

=== Introduction to Δ and P ===
Define an ordered pair [Δ,Ρ] where Δ is an initial finite non-empty Dyck String & P are production rules P=(R₁,R₂,…,Rₖ), each in the form “X→Y” where X and Y are finite non-empty Dyck Strings. Y has the ability to be the symbol D. When Y is instead D, this means: in Δ, delete X.

=== Solving a Given Δ and P: ===
We follow these steps:

* '''(1)''' Begin with R₁,
* '''(2)''' Scan Δ from left to right,
* '''(3)''' Find the leftmost instance of X in Δ (according to said R)
* '''(4)''' Rewrite X as Y and leave the rest of Δ unchanged (or delete X if Y is the symbol D)
* '''(5)''' Move to the next R in cyclic order (1, 2,…,n,1,2,…)
* '''(6)''' If a rule doesn’t apply, simply skip it and move to the immediate next one in P,
* '''(7)''' Repeat from (2) on the altered Δ each time.

'''''NOTE:''''' Duplicate rules in the same P are allowed.

=== Halting Conditions ===
Some given pairs [Δ,P] never halt. However, some do! Here are two halting conditions we can define:

'''Halt if:'''

Δ becomes “∅” (empty), or if there does not exist a rule in P that can transform Δ any further, meaning that Δ is what we can call “stuck”.

== Example Run-Through ==
(Initial Dyck String) Δ: ()()(())

(Production Rules) P: R₁=()→(()), R₂=(())()→D

* ()()(()) initial String Δ,
* (())()(()) as per R₁, (()) replaces (),
* (()) as per R₂, (())() is deleted,
* ((())) as per R₁, (()) replaces (),
* '''SKIP R₂, IT DOES NOT APPLY,'''
* …

This example doesn't halt as ((…()…)) will keep expanding indefinitely.

== Correlation to BB(n) ==
From here, we can define a function analogous to the Busy Beaver function by taking the maximal finite halting times for all pairs. Here’s how we can do it:

Let |x| denote the length of x.

A pair [Δ,P] of “size n” consists of n rules, where for each Rᵢ ∈ P, |X|≤2n and |Y|≤2n ('''NOTE''': the lengths of X and Y for each Rᵢ do NOT have to be the same, but they must be ≤2n). We also assume that |Δ|≤2n. We say that skipping a rule counts as a step, and a “step” is defined as a singular rule application.

== Function ==
The “Cyclic Tree Busy Beaver” function CTBB(n) is therefore defined as follows:

* '''(1)''' Run all pairs [Δ,P] of size n,
* '''(2)''' Let S be the set of all pairs S=(P₁,P₂,…,Pₘ) that halted, and filter out (discard) all pairs that do not,
* '''(3)''' Let S’ be the set of all steps for every pair in S to halt,
* '''(4)''' CTBB(n) outputs max(S’).

==== Notes ====
Because Tag Systems in this fashion are Turing-Complete, we can expect CTBB(n) to be on par with the classic BB(n).

We can also correctly say that CTBB(n)>*f(n) where f(n) is any computable function (where >* represents eventual domination).

== Known Values ==
{| class="wikitable"
|+Small CTBB values
|CTBB
|Value / Lower bound
|Discoverer
|Date
|Source
|-
|2
|CTBB(2) = 5
|Jack
|16 Nov 2025
|[https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233 Link]
|-
|3
|CTBB(3) <u>></u> 38
|aparker
|18 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517 Link]
|-
|4
|CTBB(4) ≥ 672
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043 Link]
|-
|5
|CTBB(5) ≥ 2^2^2^2^182
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063 Link]
|-
|6
|CTBB(6) <u>></u> 2↑↑↑131
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479 Link]
|-
|7
|CTBB(7) <u>></u> 4↑↑↑↑(4↑↑↑3)
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281 1], [https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217 2]
|}

== Collection of Champions ==
<div class="toccolours mw-collapsible mw-collapsed"><div class="mw-collapsible-content"><div class="toccolours mw-collapsible mw-collapsed">'''CCTB(1)'''<div class="mw-collapsible-content">
The following was given by Jack: We are only allowed:

* 1 rule

* For each rule part (X,Y) can only contain at most 2 symbols (because 2(1)=2)

* The initial string Δ also only contains at most 2 symbols

There is only one possible Dyck string with at most 2 symbols, it is: (). Here are all possible P (rulesets) and Δ given these constraints:

* Δ: (), single rule: ()→()
* Δ: (), single rule: ()→D

'''Remember''': Δ must be non-empty. It’s easy to tell that for each step, the first ruleset results in an infinite loop. The second one halts after 1 step (it immediately deletes () ).

S is the set of all halting pairs (only the second one) and S’ is the halting times for said pair(s) (only 1).

max(S’)=1.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(2)'''<div class="mw-collapsible-content">
The domain was solved by Jack.

Δ: (()) (Initial String)

P: R₁=()()→D, R₂=()→()()

* (()) (Δ)
* Skip rule 1 (skipping counts as a step)
* (()()) (apply rule 2)
* () (apply rule 1)
* ()() (apply rule 2)
* ∅ (apply rule 1)

All 432 pairs were brute-forced.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(3)'''<div class="mw-collapsible-content">
Pair found by aparker314159:<pre>
Δ: [][[]]
Rule 0: [[]] => [[[]]]
Rule 1: [][][] => D
Rule 2: [] => [][]
</pre></div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(4)'''<div class="mw-collapsible-content">
Pair found by Moja:

CTBB(4)>=672, by the following pair:

* Δ: ()()()()
* P:
** ()() -> (()())()
** ()() -> ()()(())
** () -> (())
** (((()))) -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(5)'''<div class="mw-collapsible-content">
Pair found by Moja

* Δ: "(())(()())"
* P:
** "(()())" -> "(()())(())"
** "(()())" -> "(()())(())"
** "()(())" -> "(())((()))"
** "((()))" -> "((()))()()"
** "()" -> "D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(6) Champion'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()))
* P:
** ((())) -> ((()))(()())
** (()()) -> (()())(())
** (()) -> (())(()()())
** (()()()) -> (()()())()()
** ()()() -> D
** () -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(7) Champion'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()()))
* P:
** ((()())) -> ((()()))((()))
** ((())) -> ((()))(()()())
** ((())) -> ((()))(()()())
** (()()()) -> (()()())(()())
** (()()) -> (()())(())(())
** (()) -> (())()()()()()
** () -> D
</div></div></div></div>

== Total Number of Pairs ==
The number of pairs [Δ,Ρ] grows rather quick. For CTBB(n), there are C(1)+C(2)+…+C(n) total Δ’s (where C(n)=n-th Catalan Number). The first few values are:

* CTBB(1) has 1 possible Δ,
* CTBB(2) has 3 possible Δ,
* CTBB(3) has 8 possible Δ,
* …
* CTBB(10) has 23713 possible Δ.

=== Let’s Dive Deeper… ===
Number of possible X: Sum of 1st,2nd,…,n-th Catalan Numbers

Number of possible Y: ((Sum of 1st,2nd,…,n-th Catalan Numbers)+1) (the “+1” represents the optional “D” symbol)

Each R in P is technically an ordered pair [x,y] so we multiply the counts together:

'''(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)'''

There are n rules, so we exponentiate this by n:

'''[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

We have our initial Dyck string Δ of length at most 2n (as mentioned earlier). This is also equal to the sum of the 1st,2nd,…,n-th Catalan numbers. This forms another pair [Δ,Ρ] by multiplying our Catalan sum with our formula from before:

'''Pair(n)=(Sum of 1st,2nd,…,n-th Catalan Numbers)×[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

=== Values of the Pair Function ===

* Pair(1) = b2 (as shown earlier)
* Pair(2) = 432
* Pair(3) = 2985984
* Pair(4) ≈ 1.44×10¹²
* Pair(5) ≈ 7.97×10¹⁷

== Turing-Completeness of Tag Systems in General ==
Cocke and Minsky's construction of a Universal Turing Machine within a tag system can be found [https://dl.acm.org/doi/pdf/10.1145/321203.321206 here].

It is worth mentioning that this variant of cyclic tag does not involve appending, only rewriting.

Post Tag

Azerty — Mon, 31 Aug 2026 14:15:06 GMT

Added redirect

New page

#REDIRECT [[Post Tag System]]

BB(6)

Polygon — Mon, 31 Aug 2026 09:35:51 GMT

History: Updated holdout counts

← Older revision		Revision as of 09:35, 31 August 2026
Line 26:		Line 26:
	* In June 2010, Pavel Kropitz established S(6) > 7.4 × 10<sup>36534</sup> and Σ(6) > 3.5 × 10<sup>18267</sup>.<ref name=":PMH" />		* In June 2010, Pavel Kropitz established S(6) > 7.4 × 10<sup>36534</sup> and Σ(6) > 3.5 × 10<sup>18267</sup>.<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.]]		[[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 ~~1016~~ machines up to equivalence and ~~2203~~ machines not considering equivalence as of August 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].		@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.		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.

Holdouts lists

Polygon — Mon, 31 Aug 2026 09:34:05 GMT

Updated BB(6) and BBlambda(42) holdout counts

← Older revision		Revision as of 09:34, 31 August 2026
Line 16:		Line 16:
	\|0		\|0
	\|0		\|0
	\|~~1016~~		\|1003
	\|15,743,264		\|15,743,264
	\|-		\|-
Line 83:		Line 83:
	!File		!File
	!Notes		!Notes
			\|-
			\|[https://discord.com/channels/960643023006490684/1239205785913790465/1543436534831906926 August 30th, 2026]
			\|@mxdys
			\|1003
			\|[https://wiki.bbchallenge.org/w/images/e/e0/BB6_holdouts_1003.txt BB6_holdouts_1003.txt]
	\|-		\|-
	\|[https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 August 15th, 2026]		\|[https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 August 15th, 2026]
Line 962:		Line 967:
	\|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]		\|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]
	\|@C++DSucker		\|@C++DSucker
	\|Variable (~~207~~)		\|Variable (158)
	\|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]		\|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]
	\|		\|

File:BB6 holdouts 1003.txt

Polygon — Mon, 31 Aug 2026 09:30:25 GMT

Polygon uploaded File:BB6 holdouts 1003.txt Holdouts list shared on August 30 2026 by mxdys over Discord.

New page

== Summary ==
Holdouts list shared on August 30 2026 by mxdys over Discord.

File:Mandelbrot set (ultrafractal).png

Jacobzheng — Mon, 31 Aug 2026 08:15:13 GMT

Jacobzheng uploaded File:Mandelbrot set (ultrafractal).png

New page

File:Golly.png

Jacobzheng — Mon, 31 Aug 2026 08:00:25 GMT

Jacobzheng uploaded File:Golly.png

New page

File:Powdertoy.png

Jacobzheng — Mon, 31 Aug 2026 07:53:35 GMT

Jacobzheng uploaded File:Powdertoy.png

New page

Fractran

A6P457 — Sun, 30 Aug 2026 09:51:26 GMT

Size 36: BMO1: added step counts

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

MBB

A6P457 — Sun, 30 Aug 2026 09:12:46 GMT

Redirected page to Minsky machine

New page

#REDIRECT [[Minsky machine]]

TMBR: August 2026

Tlonuqbar — Sun, 30 Aug 2026 07:27:33 GMT

new list

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	** prurq found three machines to halt.<sup>[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]</sup>		** prurq found three machines to halt.<sup>[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]</sup>
	** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.		** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
	** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts list.		** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts lists.
			** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
	*[[BB(2,7)]]:		*[[BB(2,7)]]:
	**Terry Ligocki enumerated 100K more subtasks, increasing the number of holdouts to '''2,499,145,299'''. A total of 800K subtasks out of the 1 million subtasks (or '''80%''') have been enumerated.		**Terry Ligocki enumerated 100K more subtasks, increasing the number of holdouts to '''2,499,145,299'''. A total of 800K subtasks out of the 1 million subtasks (or '''80%''') have been enumerated.

	[[Category:This Month in Beaver Research\|2026-08]]		[[Category:This Month in Beaver Research\|2026-08]]

Champions

Tlonuqbar — Sun, 30 Aug 2026 02:39:42 GMT

clarifying champion listing

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	== 2-Symbol TMs ==		== 2-Symbol TMs ==
	Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math>f_{x}(n)</math> used in the lower bounds represent the [[Fast-Growing Hierarchy]] while <math>\uparrow</math> represents [[wikipedia:Knuth's_up-arrow_notation\|Knuth's up-arrow notation]]. Note that most champions above 6 states are self-reported and have not been independently verified.		Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math>f_{x}(n)</math> used in the lower bounds represent the [[Fast-Growing Hierarchy]] while <math>\uparrow</math> represents [[wikipedia:Knuth's_up-arrow_notation\|Knuth's up-arrow notation]]. Machines are listed in [[Tree Normal Form#TNF-1RB\|TNF-1RB]] format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.

	{\| class="wikitable"		{\| class="wikitable"

User:CatalinaUlrich

Abuse filter — Fri, 28 Aug 2026 11:03:53 GMT

Abuse filter blocked CatalinaUlrich with an expiration time of indefinite (account creation disabled, autoblock disabled, cannot edit own talk page) Automatically blocked by abuse filter. Description of matched rule: External link spam

User:CatalinaUlrich

CatalinaUlrich — Fri, 28 Aug 2026 11:03:52 GMT

User account CatalinaUlrich was created

Cycler

Azerty — Thu, 27 Aug 2026 19:21:07 GMT

Created a section for cycler decider algorithm.

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	[[File:1RB0LB 1LB1LC 0RC1RA.png\|alt=1RB0LB_1LB1LC_0RC1RA\|thumb\|upright=1.5\|The record 3-state 2-symbol cycler {{TM\|1RB0LB_1LB1LC_0RC1RA}}, which has period 18 and preperiod 4.]]{{Stub}}		[[File:1RB0LB 1LB1LC 0RC1RA.png\|alt=1RB0LB_1LB1LC_0RC1RA\|thumb\|upright=1.5\|The record 3-state 2-symbol cycler {{TM\|1RB0LB_1LB1LC_0RC1RA}}, which has period 18 and preperiod 4.]]{{Stub}}
	A '''cycler''' is a Turing machine that eventually enters a repeating cycle. Such a Turing machine runs forever and thus is a [[non-halting Turing machine]]. A cycler may be seen as a special case of a [[translated cycler]] which has offset zero~~. The [[decider]] for cyclers works by memorizing all configurations which are visited by a TM. If a configuration is visited twice, the TM is shown to be non-halting~~.		A '''cycler''' is a Turing machine that eventually enters a repeating cycle. Such a Turing machine runs forever and thus is a [[non-halting Turing machine]]. A cycler may be seen as a special case of a [[translated cycler]] which has offset zero.

	The rule for identifying cyclers is simple:		The rule for identifying cyclers is simple:
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	where '''û''' is any headed tape segment.		where '''û''' is any headed tape segment.

			== Decider Algorithm ==
			The [[decider]] for cyclers works by memorizing all configurations which are visited by a TM. If a configuration is visited twice, the TM is shown to be non-halting.

			The algorithm can be optimized by checking a single configuration per step. The saved configuration is saved at a logarithmic tape (e.g. every power of 2 steps). This allows detecting cyclers of any period and preperiod while keeping the time complexity at O(n).

			== Other Models ==
	Most deterministic computational models have cyclers, like lambda calculus, fractran or register machines.		Most deterministic computational models have cyclers, like lambda calculus, fractran or register machines.

Diophantine Equation

Tlonuqbar — Thu, 27 Aug 2026 09:11:35 GMT

Cryptids

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	== Cryptids ==		== Cryptids ==
			As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/abs/2108.08705</ref>

	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.		There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.


			== References ==
			<references/>
	== See Also ==		== See Also ==

	* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]		* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]

	[[Category:Functions]]		[[Category:Functions]]

BB(6)

C7X — Thu, 27 Aug 2026 06:18:10 GMT

I guess "became known later" might be readable as either "discovered in computer output later" or "published later", the second one seems to be the situation

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	* In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.<ref name=":PMH" />		* 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 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 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 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 × 10<sup>42</sup> and Σ(6) > 2.5 × 10<sup>21</sup>.<ref name=":PMH" />		* In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 10<sup>42</sup> and Σ(6) > 2.5 × 10<sup>21</sup>.<ref name=":PMH" />

Bell eats counter

A6P457 — Wed, 26 Aug 2026 16:37:44 GMT

Added TM template

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	== Examples ==		== Examples ==
	~~[https://bbchallenge.org/~~1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA ~~1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA]~~		{{TM\|1RB1RE_0RC1RD_1LA1RC_1LC---_1LF0RE_0LF0LA}}

	~~[https://bbchallenge.org/~~1RB---_1RC0RA_1LD1RA_1LE0LD_0RE1RF_0RB0LF ~~1RB---_1RC0RA_1LD1RA_1LE0LD_0RE1RF_0RB0LF]~~		{{TM\|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)		{{TM\|1RB0LE_0RC---_1LC0RD_0RB1RA_1LF1LE_0LA1LF}} (more complex than typical ones)

	~~[https://bbchallenge.org/~~1RB3LB---3RA0LA_2LA3LB4RB1RB2RA ~~1RB3LB---3RA0LA_2LA3LB4RB1RB2RA]~~		{{TM\|1RB3LB---3RA0LA_2LA3LB4RB1RB2RA}}

			{{TM\|1RB0RB_0RC1RB_1LD1RC_0LE0LD_1RA1LE}}

	[[Category:Zoology]]		[[Category:Zoology]]