论文标题
阿贝尔复杂性和同步
Abelian Complexity and Synchronization
论文作者
论文摘要
我们提出了一种计算Abelian复杂性$ρ^{\ rm ab} _ {\ bf s}（n）$的自动序列$ \ bf s $的$的通用方法$ \ bf s $形成同步序列。 我们详细说明了这个想法，使用免费软件胡桃木来计算tribonacci Word $ {\ bf tr} = 0102010 \ cdots $的ABELIAN复杂性，Mormphism $ 0 \ rightarrow 01 $，$ 1 \ rightArrow 02 $，$ 2 $，$ 2 $ 2 \ 2 \ rightarrow 0 $ 0 $ 0 $。以前，Richomme，Saari和Zamboni表明，这个词的Abelian复杂性在于$ \ {3,4,5,6,7 \} $，Turek给了Tribonacci Automaton对其进行计算。我们能够使用此处介绍的方法“自动”重新启动这些结果，以及更多。
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 complexity of this word lies in $\{ 3,4,5,6,7 \}$, and Turek gave a Tribonacci automaton computing it. We are able to "automatically" rederive these results, and more, using the method presented here.
