Algebraic Characterizations of Classes of Regular Languages in DynFO
Corentin Barloy, Felix Tschirbs, Nils Vortmeier, Thomas Zeume
TL;DR
The paper connects dynamic descriptive complexity with algebraic language theory by establishing fine-grained characterizations of regular languages under unary and quantifier-free dynamic updates. It shows that UDynΣ2 with unary relations suffices to maintain all regular languages, and that UDynProp with unary updates exactly captures regular languages whose syntactic monoids are groups, while UDynΣ1+ corresponds to languages with ordered syntactic monoids in the wreath product $J^{+} \ast \mathbf{G}$. The work leverages Green's relations and substructure arguments to derive both upper bounds and lower bounds, highlighting the role of algebraic structure in dynamic maintenance and revealing open questions for UDynΣ1. Overall, it advances the synthesis of dynamic complexity and semigroup theory, suggesting lp-varieties as a productive path for further characterization.
Abstract
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.