Algebraic Characterizations of Classes of Regular Languages in DynFO

TL;DR

This paper provides a detailed algebraic analysis of how different logical fragments and auxiliary relations can maintain classes of regular languages within the dynamic descriptive complexity framework, refining existing results.

Contribution

It refines Hesse's result by showing unary auxiliary data with one quantifier alternation can maintain all regular languages and offers precise algebraic characterizations of these classes.

Findings

Unary auxiliary data with one quantifier alternation maintains all regular languages.

Precise algebraic characterizations of classes maintainable with quantifier-free and positive existential formulas.

Extensions of previous results by Hesse and Gelade et al. in the context of dynamic descriptive complexity.

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.