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

Abstract

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 }_{kin mathbb {N}}s_{k}2^{-k}in [0,1] subset mathbb {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 inmathbb {Q}Shift Registers (Fmathbb {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.