%I #111 Jul 03 2024 03:26:04
%S 1,6,21,107,47176870
%N Busy Beaver problem: a(n) is the maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting.
%C "In 1965 Tibor Rado, together with Shen Lin, proved that a(3) is 21. (...) Next, in 1983, Allan Brady proved that a(4) is 107. (...) Then in 1989 Heiner Marxen and Jürgen Buntrock discovered that a(5) is at least 47176870. (...) As for a(6), Marxen and Buntrock set another record in 1997 by proving that it is at least 8690333381690951." [Based on Aaronson's web page.]
%C The function Sigma(n) = Sigma(n, 2) (A028444) denotes the maximal number of tape marks (1's) that a Turing Machine with n internal states (plus the Halt state), 2 symbols, and a two-way infinite tape can write on an initially blank tape (all 0's) and then halt. The function a(n) (the present sequence) denotes the maximal number of steps S(n) = S(n, 2) (thus shifts, since direction NONE is excluded) that such a machine can make (not necessarily the same Turing machine producing a maximum number of 1s and need not even produce many tape marks).
%C Given that 5-state 2-symbol halting Turing machines can compute Collatz-like congruential functions (see references), it may be very hard to find the next term.
%C The sequence grows faster than any computable function of n and so is noncomputable.
%C From _Daniel Forgues_, Jun 05-06 2011: (Start)
%C A more precise definition might be as follows:
%C Busy Beaver Problem: a(n) is the maximal number of steps that an n-state, 2-symbol, d+ in {LEFT, RIGHT}, 5-tuple (q, s, q+, s+, d+) Turing machine can make on an initially blank tape and then halt.
%C Further comments:
%C H in H_(n, k) is a halting* Turing machine with n states and k symbols;
%C * (on a blank tape (all 0's) as input)
%C States q, q+ in set Q_n of n distinct states (plus the Halt state);
%C Symbols s, s+ in set S_k of k distinct symbols (0 as the blank symbol);
%C Shift direction d+ in {LEFT, RIGHT} (NONE is excluded here);
%C sigma(H) is the number of non-blank symbols left on the tape by H;
%C s(H) is the number of steps (or shifts in our case) taken by H;
%C Sigma(n, k) = max {sigma(H) : H is a halting Turing machine with n states and k symbols}
%C S(n, k) = max {s(H) : H is a halting Turing machine with n states and k symbols}
%C a(n) is S(n) = S(n, 2) since a 2-symbol BB-class Turing machine is assumed.
%C For all n, S(n, k) >= Sigma(n, k), k >= 2. (End)
%C a(6) > 10^^15, a tower of 10's of height 15 [Pavel Kropitz] - see Shawn Ligocki's blog. - _N. J. A. Sloane_, Jun 22 2022
%C It is conjectured that a(5) = 47176870 (Conjecture 10 of the Scott Aaronson link "The busy beaver frontier"). - _Jianing Song_, Feb 21 2024
%C On July 02, 2024 cosmo (Tristan Stérin) on behalf of the Busy Beaver Challenge group announced: "We have proved 'BB(5) = 47,176,870'". - _Peter Luschny_, Jul 02 2024
%D A. H. Brady, The busy beaver game and the meaning of life, in Herken, R. (Ed) The Universal Turing Machine: A Half-Century Survey, pp. 259-277, Oxford Univ Press 1988. Reprinted by Springer-Verlag, 1995 (see pages 237-254).
%D J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see p. 33.
%D Yu. V. Rogozhin, Seven universal Turing machines (Russian), abstract, Fifth All-Union Conference on Math. Logic, Akad. Nauk. SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1979, p. 127.
%D Yu. V. Rogozhin, Seven universal Turing machines (Russian), Systems and Theoretical Programming, Mat. Issled. no. 69, Akademiya Nauk Moldavskoi SSSR, Kishinev, 1982, pp. 76-90.
%D Jeffrey Shallit, A second course in formal languages and automata theory. Cambridge University Press, 2008. See Fig. 6.2, p. 185.
%H Scott Aaronson, <a href="http://www.scottaaronson.com/writings/bignumbers.html">Who Can Name the Bigger Number?</a>
%H Scott Aaronson, <a href="https://www.scottaaronson.com/papers/bb.pdf">The busy beaver frontier</a>, ACM SIGACT News 51.3 (2020): 32-54.
%H A. H. Brady, <a href="https://doi.org/10.1090/S0025-5718-1983-0689479-6">The determination of Rado's noncomputable function Sigma(k) for four-state Turing machines</a>, Math. Comp. 40 #62 (1983) 647-665.
%H Ben Brubaker, <a href="https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/">With Fifth Busy Beaver, Researchers Approach Computation’s Limits</a>, Quanta Magazine (2024).
%H Busy Beaver Challenge, <a href="https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237">We have proved "BB(5) = 47,176,870"</a>, July 02, 2024.
%H Bill Dubuque, <a href="http://groups.google.com/groups?selm=WGD.96Feb13081831%40berne.ai.mit.edu">Re: Halting is weak</a>
%H A. Gravell and U. Ultes-Nitsche, <a href="http://www.ecs.soton.ac.uk/~uun/CM219/HTML/sld169.htm">BB(n) Grows Faster Than Any Computable Function</a> [dead link]
%H Maja Kądziołka, <a href="https://github.com/ccz181078/Coq-BB5">Coq-BB5</a> (proof that a(5) = 47176870)
%H Shawn Ligocki, <a href="https://www.sligocki.com//2022/06/21/bb-6-2-t15.html">BB(6,2) > 10^^15</a>, Jun 21 2022.
%H Shen Lin and T. Rado, <a href="https://doi.org/10.1145/321264.321270">Computer Studies of Turing Machine Problems</a>, J. ACM 12 (1965), 196-212.
%H Rona Machlin (nee Kopp) and Quentin F. Stout, <a href="https://doi.org/10.1016/0167-2789(90)90068-Z">The Complex Behavior of Simple Machines</a>, Physica D 42 (1990) 85-98.
%H Heiner Marxen and Jürgen Buntrock, <a href="https://turbotm.de/~heiner/BB/mabu90.html">Attacking the Busy Beaver 5</a>, Bulletin of the EATCS, Number 40, February 1990, pp. 247-251. (See Table 1.)
%H Pascal Michel, <a href="https://doi.org/10.1007/BF01409968">Busy beaver competition and Collatz-like problems</a>, Arch. Math. Logic (1993) 32:351-367.
%H Pascal Michel, <a href="https://bbchallenge.org/~pascal.michel/ha.html">Historical survey of Busy Beavers</a>, 2011.
%H Pascal Michel, <a href="https://bbchallenge.org/~pascal.michel/beh.html">Behavior of busy beavers</a>, 2010.
%H Pascal Michel, <a href="https://bbchallenge.org/~pascal.michel/bbc.html">The Busy Beaver Competitions</a>, 2010.
%H Pascal Michel, <a href="http://arxiv.org/abs/0906.3749">The Busy Beaver Competition: a historical survey</a>, arXiv, 2010.
%H John Pavlus, <a href="https://www.quantamagazine.org/the-busy-beaver-game-illuminates-the-fundamental-limits-of-math-20201210/">How the Slowest Computer Programs Illuminate Math's Fundamental Limits</a>, Quanta Magazine, Dec 10 2020.
%H T. Rado, <a href="https://doi.org/10.1002/j.1538-7305.1962.tb00480.x">On Non-Computable Functions</a>, Bell System Technical J. 41 (1962), 877-884.
%H Raphael M. Robinson, <a href="https://doi.org/10.1142/S0129167X91000302">Minsky's small universal Turing machine</a>, Int'l Jnl. Math, 2 #5 (1991) 551-562.
%H Claude E. Shannon, <a href="https://doi.org/10.1515/9781400882618-007">A universal Turing machine with two internal states</a>, Automata Studies, Ann. of Math. Stud. 34 (1956) 157-165.
%H Michael Somos, <a href="https://grail.eecs.csuohio.edu/~somos/bb.html">Busy Beaver Turing Machine</a>
%H Michael Somos, <a href="https://grail.eecs.csuohio.edu/~somos/busy.html">Busy Beaver</a>
%H Q. F. Stout, <a href="http://www.eecs.umich.edu/~qstout/abs/busyb.html">The Complex Behavior of Simple Machines</a>
%H <a href="https://bbchallenge.org/story">The Busy Beaver Challenge</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BusyBeaver.html">Busy Beaver</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Busy_beaver">Busy beaver</a>
%H <a href="/index/Br#beaver">Index entries for sequences related to Busy Beaver problem</a>
%Y Cf. A028444, A362199.
%K hard,nice,nonn,more
%O 1,2
%A _Jud McCranie_ and _N. J. A. Sloane_, May 02 2001
%E Additional references from Bill Dubuque (wgd(AT)martigny.ai.mit.edu)
%E Edited by _N. J. A. Sloane_, Aug 30 2011
%E Additional links from _Robert C. Lyons_, Jun 22 2022
%E a(5) added by _Charles R Greathouse IV_, Jul 02 2024