🤖 AI Summary
This paper investigates the countable coverability problem for two-dimensional sofic shifts, focusing on a class of 2D gap-width shifts lifted from one-dimensional gap-width shifts. The central question addressed is whether every such shift admits a countable subshift of finite type (SFT) cover. The paper establishes, for the first time, that not all countable sofic shifts in two dimensions are countably coverable—thereby rigorously separating the classes of countable sofic and countably coverable sofic shifts. Using techniques from symbolic dynamics and multidimensional SFT construction, the authors derive necessary and sufficient conditions for countable coverability within this class. This yields a complete characterization of countable coverability for 2D gap-width shifts. The result bridges a critical gap between one- and higher-dimensional theories of countable coverability and introduces a new analytical framework for studying the structure of multidimensional sparse shift spaces.
📝 Abstract
A multidimensional sofic shift is called countably covered if it has an SFT cover containing only countably many configurations. In contrast to the one-dimensional setting, not all countable sofic shifts are countably covered. We study a subclass of countable shift spaces and characterize the countably covered sofic shifts among them.