BB(5): Difference between revisions
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* In 1974, Lynn, cited by Brady (1983)<ref>Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. ''Mathematics of Computation, 40'', 647-665.</ref>, establishes S(5) ≥ 7,707 and Σ(5) ≥ 112. | * In 1974, Lynn, cited by Brady (1983)<ref>Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. ''Mathematics of Computation, 40'', 647-665.</ref>, establishes S(5) ≥ 7,707 and Σ(5) ≥ 112. | ||
* In 1983, the [https://docs.bbchallenge.org/other/lud20.pdf Dortmund contest]is organised to find new 5-state champions, winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501. | * In 1983, the [https://docs.bbchallenge.org/other/lud20.pdf Dortmund contest]is organised to find new 5-state champions, winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501. | ||
* In 1989, Heiner and Buntrock find the [[5-state busy beaver winner]], establishing S(5) ≥ 47,176,870 and Σ(5) ≥ 4,098.<ref name=":0" /> They do not prove that the machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they present some ideas for [[Deciders]].<ref name=":0" /> | |||
=== Proving that the winner is actually the winner === | === Proving that the winner is actually the winner === | ||
Revision as of 16:51, 14 June 2024
BB(5) refers to the 5th value of the Busy Beaver function. In 1989, the 5-state busy beaver winner was found: a 5-state Turing machine halting after 47,176,870 giving the lower bound BB(5) ≥ 47,176,870.[1]
In 2024, BB(5) = 47,176,870 was proven by the bbchallenge.org massively collaborative research project.
History
In this Section, we use Radó's original S (number of steps) and Σ (number of ones on the final tape) notations, see Busy Beaver Functions.
Finding the 5-state winner
- In 1964, Green establishes Σ(5) ≥ 17.[2]
- In 1972, Lynn establishes S(5) ≥ 435 and Σ(5) ≥ 22.[2]
- In 1973, Weimann establishes S(5) ≥ 556 and Σ(5) ≥ 40.[2]
- In 1974, Lynn, cited by Brady (1983)[3], establishes S(5) ≥ 7,707 and Σ(5) ≥ 112.
- In 1983, the Dortmund contestis organised to find new 5-state champions, winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501.
- In 1989, Heiner and Buntrock find the 5-state busy beaver winner, establishing S(5) ≥ 47,176,870 and Σ(5) ≥ 4,098.[1] They do not prove that the machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they present some ideas for Deciders.[1]
Proving that the winner is actually the winner
- In 2003, Georgi Georgiev (Skelet) publishes a list of Skelet's 43 holdouts, based on bbfind, a collection of Deciders written in Pascal[4].
References
- ↑ 1.0 1.1 1.2 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
- ↑ 2.0 2.1 2.2 Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52
- ↑ Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. Mathematics of Computation, 40, 647-665.
- ↑ https://skelet.ludost.net/bb/index.html