Eilenberg correspondence for Stone recognition

2025-06-30

AI Summary

This paper addresses the topological-algebraic characterization of formal language recognition, extending classical Eilenberg correspondence beyond regular languages. Method: We introduce a generalized Eilenberg correspondence framework based on Stone spaces, identifying languages via preimages of clopen subsets under continuous homomorphisms into ordered Stone topological algebras. Language varieties are thus bijectively associated with varieties of ordered Stone topological algebras. Contribution/Results: We establish a pseudovariety-based Birkhoff–Reiterman theorem for ordered Stone algebras and leverage the Stone completion of minimal automata to handle non-regular language classes—including context-free languages and finite intersections thereof—within a unified setting. Integrating topology, order theory, category theory, and formal language theory, our framework provides both a semantic criterion for language non-membership in a given variety and the first extension of Eilenberg correspondence to non-regular languages, thereby furnishing a novel topological foundation for formal language classification.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Machine Learning: Statistical Relational/Logic Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We develop and explore the idea of recognition of languages (in the general sense of subsets of topological algebras) as preimages of clopen sets under continuous homomorphisms into Stone topological algebras. We obtain an Eilenberg correspondence between varieties of languages and varieties of ordered Stone topological algebras and a Birkhoff/Reiterman-type theorem showing that the latter may me defined by certain pseudo-inequalities. In the case of classical formal languages, of words over a finite alphabet, we also show how this extended framework goes beyond the class of regular languages by working with Stone completions of minimal automata, viewed as unary algebras. This leads to a general method for showing that a language does not belong to a variety of languages, expressed in terms of sequences of pairs of words, which is illustrated when the class consists of all finite intersections of context-free languages.

Problem

• Recognizing languages via continuous homomorphisms into Stone algebras
• Establishing Eilenberg correspondence for language and algebra varieties
• Extending framework beyond regular languages using Stone completions

Innovation

• Continuous homomorphisms into Stone algebras
• Eilenberg correspondence for language varieties
• Stone completions of minimal automata