Characterization and Decidability of FC-Definable Regular Languages

2025-05-14

AI Summary

This paper investigates the expressive power of first-order logic FC—featuring only the successor relation—over finite words, focusing on characterizing the class of definable regular languages and settling its decidability. Using a synthesis of syntactic monoid theory, automata-theoretic equivalences, and logical semantics, we establish, for the first time, that certain regular languages are not FC-definable. We provide three equivalent, decidable characterizations of the FC-definable languages: algebraically, as those whose syntactic monoids satisfy a specific identity; automata-theoretically, as those recognized by deterministic finite automata without nontrivial cycles; and via regular expressions, as star-free generalized regular expressions extended with Kleene stars over terminal words. Furthermore, we devise effective bidirectional translations between FC formulas and succinct regular expressions. These results fully resolve the characterization and decidability problems for FC-definable regular languages.

Technology Category

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

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

FC is a first-order logic that reasons over all factors of a finite word using concatenation, and can define non-regular languages like that of all squares (ww). In this paper, we establish that there are regular languages that are not FC-definable. Moreover, we give a decidable characterization of the FC-definable regular languages in terms of algebra, automata, and regular expressions. The latter of which is natural and concise: Star-free generalized regular expressions extended with the Kleene star of terminal words.

Problem

• Determine which regular languages are FC-definable
• Provide algebraic and automata-based characterizations of FC-definable regular languages
• Develop concise regular expressions for FC-definable languages

Innovation

• Decidable characterization of FC-definable regular languages
• Algebra, automata, and regular expressions analysis
• Star-free generalized regular expressions with Kleene star