自动获胜班次
Automatic winning shifts
摘要：
对于每个一维子移位$X$，我们可以将获胜移位$W(X)$关联起来，该获胜移位产生于在$X$的语言上进行的组合游戏。先前已经研究了$W(X)$继承了$X$的哪些性质。例如，$X$和$W(X)$具有相同的因子复杂度，并且如果$X$是sofic子移位，则$W(X)$也是sofic。在本文中，我们开发了$W(X)$的自动化的概念，也就是说，我们提出了什么意思，$W(X)$的向量表示是接受的有限自动机。设$S$是一个抽象数制，使得关于$S$的加法是一个有理关系。设$X$是由$S$自动字生成的子移位。我们证明了，只要有一个约束的非零符号的数量在配置的$W(X)$（这是从$X$具有次线性因子复杂度），然后$W(X)$接受的有限自动机，它可以有效地构造从描述$X$。当$X$由某些自动词（如Thue-Morse词）生成时，我们提供了一个显式自动机。
英文摘要：
To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from a combinatorial game played on the language of $X$. Previously it has been studied what properties of $X$ does $W(X)$ inherit. For example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of automaticity for $W(X)$, that is, we propose what it means that a vector representation of $W(X)$ is accepted by a finite automaton. Let $S$ be an abstract numeration system such that addition with respect to $S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of $W(X)$ (which follows from $X$ having sublinear factor complexity), then $W(X)$ is accepted by a finite automaton, which can be effectively constructed from the description of $X$. We provide an explicit automaton when $X$ is generated by certain automatic words such as the Thue-Morse word.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
