Automatic winning shifts

To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from combinatorial game played on the language of $X$. Previously it has been studied what properties $X$ does inherit. For example, and have same factor complexity if is sofic subshift, then also sofic. In this paper, develop notion automaticity for $W(X)$, that is, propose means vector representation accepted by finite automaton. Let $S$ be an abstract numeration system such addition with respect to rational relation. generated $S$-automatic word. We prove as long there bound number nonzero symbols in configurations (which follows having sublinear complexity), automaton, can effectively constructed description provide explicit automaton when certain automatic words Thue-Morse