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 eitherGiven 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).
We introduce rational complexity, a new complexity measure for binary sequences. The sequence s ∈ Bω is considered as binary expansion of a real fraction $s \equiv {\sum }_{k\in \mathbb {N}}s_{k}2^{-k}\in [0,1] \subset \mathbb {R}$ s ≡ ∑ k ∈ ℕ s k 2 − k ∈ [ 0 , 1 ] ⊂ ℝ . We compute its continued fraction expansion (CFE) by the Binary CFE Algorithm, a bitwise approximation of s by binary search in the encoding space of partial denominators, obtaining rational approximations r of s with r → s. We introduce Feedback in$\mathbb {Q}$ ℚ Shift Registers (F$\mathbb {Q}$ ℚ SRs) as the analogue of Linear Feedback Shift Registers (LFSRs) for the linear complexity L, and Feedback with Carry Shift Registers (FCSRs) for the 2-adic complexity A. We show that there is a substantial subset of prefixes with “typical” linear and 2-adic complexities, around n/2, but low rational complexity. Thus the three complexities sort out different sequences as non-random.
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