Uniformization Problems for Synchronizations of Automatic Relations on Words

有理关系的重同步一致化与可定义性问题

Resynchronized Uniformization and Definability Problems for Rational Relations

本文系统研究了基于正则同步语言定义的有理关系子类的一致化与可定义性问题的可判定性，重述已知结果并补充了若干新的可判定性与不可判定性结论。

Subword Complexity and k-Synchronization

We show that the subword complexity function p_x(n), which counts the number of distinct factors of length n of a sequence x, is k-synchronized in the sense of Carpi if x is k-automatic. As an application, we generalize recent results of Goldstein. We give analogous results for the number of distinct factors of length n that are primitive words or powers. In contrast, we show that the function that counts the number of unbordered factors of length n is not necessarily k-synchronized for k-automatic sequences.

Abelian Complexity and Synchronization

We present a general method for computing the abelian complexity $ρ^{\rm ab}_{\bf s} (n)$ of an automatic sequence $\bf s$ in the case where (a) $ρ^{\rm ab}_{\bf s} (n)$ is bounded by a constant and (b) the Parikh vectors of the length-$n$ prefixes of $\bf s$ form a synchronized sequence. We illustrate the idea in detail, using the free software Walnut to compute the abelian complexity of the Tribonacci word ${\bf TR} = 0102010\cdots$, the fixed point of the morphism $0 \rightarrow 01$, $1 \rightarrow 02$, $2 \rightarrow 0$. Previously, Richomme, Saari, and Zamboni showed that the abelian comp…