This paper explores the fine-grained structure of classes of regular languages maintainable in fragments of first-order logic within the dynamic descriptive complexity framework of Patnaik and Immerman. A result by Hesse states that the class of regular languages is maintainable by first-order formulas even if only unary auxiliary relations can be used. Another result by Gelade, Marquardt,and Schwentick states that the class of regular languages coincides with the class of languages maintainable by quantifier-free formulas with binary auxiliary relations. We refine Hesse's result and show that with unary auxiliary data formulas with one quantifier alternation can maintain all regular languages. We then obtain precise algebraic characterizations of the classes of languages maintainable with quantifier-free formulas and positive existential formulas in the presence of unary auxiliary relations.

翻译：本文在 Patnaik 与 Immerman 的动态描述复杂性框架下，探索一阶逻辑片段中可维持的正则语言类的细粒度结构。Hesse 的结果表明，即使仅允许使用一元辅助关系，正则语言类仍可由一阶公式维持。Gelade、Marquardt 与 Schwentick 的另一结果指出，正则语言类与使用二元辅助关系的无量词公式可维持的语言类重合。我们改进 Hesse 的结果，证明在仅使用一元辅助数据时，具有一次量词交替的公式即可维持所有正则语言。进而，我们获得了在一元辅助关系存在下，分别由无量词公式与正存在公式可维持的语言类的精确代数刻画。