Automatic winning shifts

TL;DR

This paper explores the automaticity of winning shifts derived from one-dimensional subshifts, showing that under certain conditions, these shifts can be recognized by finite automata, with explicit constructions provided for specific cases.

Contribution

It introduces a notion of automaticity for winning shifts and proves that such shifts are automaton-recognizable when generated by automatic words with bounded nonzero symbols.

Findings

Winning shifts inherit automaton-recognizability under certain conditions

Explicit automata are constructed for shifts generated by specific automatic words like Thue-Morse

Winning shifts have bounded nonzero symbols when the original subshift has sublinear factor complexity

Abstract

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.