Busy beaver lack of hope recurrence
People working on BB(n-1) believed BB(n) was impossible:
BB(2) to BB(3)
Second-hand report: "Suppose that, just to gain experience, we simplify the situation by merely asking whether a given 3-card machine will ever stop if started (with its card 1) on an all-zero tape. This particular question has been studied extensively by the authors in connection with the subject of sequential circuits. Many computer programs were written to answer this question; these programs grew larger and larger as more and more criteria for stoppers were covered. These programs were the results of co-operative efforts of experienced mathematicians and skilled programmers, and were run on some of the finest existing computers. Yet this extremely primitive-looking problem was still unsolved when this paper was presented, and probably most of the participants in the studies felt that perhaps it would always remain so. But since then, this problem has been solved by T. Rado and one of his graduate students, S. Lin."
Source: R. W. House & T. Rado, An Approach to Artificial Intelligence, IEEE Special Publication S-142, January 1963.
BB(3) to BB(4)
Second-hand report: "In any case, even though skilled mathematicians and experienced programmers attempted to evaluate 2(3) and 5(3), there is no evidence that any known approach will yield the answer, even if we avail ourselves of high-speed computers and elaborate programs. As regards \Sigma(4), S(4), the situation seems to be entirely hopeless at present."—Tibor Rado, 1963.
Source: Brady, Allen H.. “The determination of the value of Rado’s noncomputable function Σ(4) for four-state Turing machines.” Mathematics of Computation 40 (1983): 647-665.
BB(4) to BB(5)
Second-hand report: "Brady predicted that there will never be a proof of the values of Sigma(5) and S(5). We are just slightly more optimistic, and are lead to recast a parable due to Erdos (who spoke in the context of determining Ramsey numbers): suppose a vastly superior alien force lands and announces that they will destroy the planet unless we provide a value of the S function, along with a proof of its correctness. If they ask for S(5) we should put all of our mathematicians, computer scientists, and computers to the task, but if they ask for S(6) we should immediately attack because the task is hopeless."
Source: From Machlin & Stout (1990) https://web.eecs.umich.edu/~qstout/abs/busyb.html
Original source: 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 26 Sept. 2024
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