Friedman's "Long Finite Sequences”: The End of the Busy Beaver Contest
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Harvey Friedman gives a comparatively short description of an “unimaginably large” number n(3) , beyond, e.g. the values A(7,184)< A(7198,158386) < n(3) of Ackermann's function - but finite. We implement Friedman's combinatorial problem about subwords of words over a 3-letter alphabet on a family of Turing machines, which, starting on empty tape, run (more than) n(3) steps, and then halt. Examples include a (44,8) (symbol,state count) machine as well as a (276,2) and a (2,1840) one. In total, there are at most 37022 non-trivial pairs (n,m) with Busy Beaver values BB(n,m) < A(7198,158386). We give algorithms to map any (|Q|,|E|) TM to another, where we can choose freely either |Q'|≥ 2 or |E'|≥ 2 (the case |Q'|=2 for empty initial tape is the tricky one). Given the size of n(3) and the fact that these TMs are not holdouts, but assured to stop, Friedman's combinatorial problem provides a definite upper bound on what might ever be possible to achieve in the Busy Beaver contest. We also treat n(4)> A^(A(187196))(1).
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