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.

中文翻译：

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自动获胜班次

对于每个一维子位移$X$，我们可以关联一个获胜的位移$W(X)$，它产生于在$X$ 语言上进行的组合游戏。之前研究过$W(X)$继承了$X$的哪些属性。例如，$X$ 和$W(X)$ 具有相同的因子复杂度，如果$X$ 是sofic subshift，则$W(X)$ 也是sofic。在本文中，我们提出了 $W(X)$ 的自动性概念，也就是说，我们提出了 $W(X)$ 的向量表示被有限自动机接受的含义。令$S$ 是一个抽象的计数系统，使得关于$S$ 的加法是一个有理关系。令 $X$ 是由 $S$ 自动词生成的子移位。我们证明只要在 $W(X)$ 的配置中存在非零符号数量的界限（从具有次线性因子复杂度的 $X$ 得出），那么$W(X)$ 被一个有限自动机接受，它可以从$X$ 的描述中有效地构造出来。当 $X$ 由某些自动词（例如 Thue-Morse 词）生成时，我们提供了一个显式自动机。