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We introduce  rational complexity , a new complexity measure for binary sequences. The sequence  s  ∈  B   ω   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.

Rational complexity of binary sequences, F$\mathbb {Q}$SRs, and pseudo-ultrametric continued fractions in $\mathbb {R}$