Automatic complexity of shift register sequences

TL;DR

This paper proves that m-sequences have maximal subword complexity and nearly maximal nondeterministic automatic complexity, highlighting their high structural complexity compared to arbitrary sequences.

Contribution

It establishes that m-sequences possess maximal subword complexity and nearly maximal nondeterministic automatic complexity, advancing understanding of their combinatorial and computational properties.

Findings

m-sequences have maximal subword complexity

nondeterministic automatic complexity of m-sequences is close to maximal

contrast with general sequences where complexity is bounded by n/2+1

Abstract

Let $x$ be an $m$-sequence, a maximal length sequence produced by a linear feedback shift register. We show that $x$ has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity $A_N(x)$ is close to maximal: $n/2-A_N(x)=O(\log^2n)$, where $n$ is the length of $x$. In contrast, Hyde has shown $A_N(y)\le n/2+1$ for all sequences $y$ of length $n$.