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The [[Busy Beaver function]] BB (called S originally) was introduced by [https://en.wikipedia.org/wiki/Tibor_Rad%C3%B3 Tibor Radó] in 1962 <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> for 2-symbol [[Turing machines]] and later generalised<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> to m-symbol Turing machines:
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"
{| 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
| 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, and few of its values are known:
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"
{| class="wikitable"
|+ Small busy beaver values <ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>  <ref name="bbchallenge">https://bbchallenge.org/</ref>  
|+ 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  
|  || 2-state || 3-state || 4-state || 5-state || 6-state  
|-   
|-   
| 2-symbol  
! 2-symbol  
| [[BB(2)]] = 6  
| [[BB(2)]] = 6  
| [[BB(3)]] = 21
| [[BB(3)]] = 21
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| style="background: orange;" | [[BB(6)]] > <math>10 \uparrow \uparrow 15</math>  
| style="background: orange;" | [[BB(6)]] > <math>10 \uparrow \uparrow 15</math>  
|-
|-
| 3-symbol || [[BB(2,3)]] = 38  
! 3-symbol
| [[BB(2,3)]] = 38  
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| 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;" | [[BB(4,3)]] > <math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
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| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
|-
| 4-symbol   
! 4-symbol   
| [[BB(2,4)]] = 3,932,964
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
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| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
|-
| 5-symbol  
! 5-symbol  
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
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| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}
|}


In the above table, <span style="background: orange">cells are highlighted in orange</span> when there are known [[Cryptids]] (mathematically-hard machines) in that class, and <span style="background: #ffe4b2">cells are highlighted in light orange</span> when the existence of a Cryptid is given by using a known one with less states or symbols.
In the above table, <span style="background: orange">cells are highlighted in orange</span> when there are known [[Cryptids]] (mathematically-hard machines) in that class, and <span style="background: #ffe4b2">cells are highlighted in light orange</span> when the existence of a Cryptid is given by using a known one with less states or symbols.


== bbchallenge.org ==
== About bbchallenge ==
[[bbchallenge.org]] <ref name="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:
[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])
* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Coq_(software) Coq])
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* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]


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


== Contribute to this wiki ==
== Contribute to this wiki ==

Revision as of 17:36, 12 March 2025

The Busy Beaver function BB (called S originally) was introduced by Tibor Radó in 1962 for 2-symbol Turing machines and later generalised to m-symbol Turing machines:[1][2]

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:

Small busy beaver values[3]
2-state 3-state 4-state 5-state 6-state
2-symbol BB(2) = 6 BB(3) = 21 BB(4) = 107 BB(5) = 47,176,870 BB(6) >
3-symbol BB(2,3) = 38 BB(3,3) > BB(4,3) >
4-symbol BB(2,4) = 3,932,964 BB(3,4) >
5-symbol BB(2,5) >

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

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:

In June 2024, bbchallenge achieved a significant milestone by proving in Coq that the 5th busy beaver value, BB(5), is equal to the lower bound found in 1989: 47,176,870.[4]

Contribute to this wiki

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

Notes

  1. ↑ 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
  2. ↑ 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.
  3. ↑ P. Michel, "Historical survey of Busy Beavers".
  4. ↑ 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
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