Algebraic Characterizations of Classes of Regular Languages in DynFO

2026-01-26

AI Summary

This study investigates the classes of regular languages maintainable in Patnaik and Immerman’s dynamic descriptive complexity framework DynFO when restricted to unary auxiliary relations, focusing on fragments of first-order logic—specifically, quantifier-free, positive existential, and those allowing a single alternation of quantifiers. By integrating algebraic theory of formal languages with fine-grained logical analysis, we provide the first precise algebraic characterizations of the language classes maintainable in the quantifier-free and positive existential fragments. Moreover, we strengthen Hesse’s result by showing that first-order formulas with a single quantifier alternation, together with unary auxiliary relations, suffice to maintain all regular languages. This work systematically elucidates the intricate correspondence among logical expressiveness, constraints on auxiliary relations, and the algebraic structure of regular languages.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Machine Learning: Statistical Relational/Logic Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• regular languages
• dynamic descriptive complexity
• algebraic characterizations
• first-order logic
• auxiliary relations

Innovation

• dynamic descriptive complexity
• regular languages
• algebraic characterization
• quantifier alternation
• unary auxiliary relations