A cartesian closed fibration of higher-order regular languages

2026-01-25

AI Summary

This work aims to construct a Cartesian closed fibration for higher-order regular languages in the sense of Salvati. To this end, two constructions are proposed: the first builds upon the category of regular languages of λ-terms with finite sets of base states, deriving the desired structure via fibration techniques; the second employs profinite λ-calculus, realizing the fibration through clopen subsets over Stone spaces and subsequently applying base change. This study is the first to integrate fibration theory, profinite λ-calculus, and Isbell-type adjunctions, thereby generalizing Brzozowski derivatives to higher-order regular languages. The resulting framework achieves a deep synthesis of categorical semantics and automata theory and demonstrates expressive power in capturing derivative operations on formal languages.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Natural Language Processing: Lexical Semantics and Morphology
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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 heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We explain how to construct in two different ways a cartesian closed fibration of higher-order regular languages in the sense of Salvati. In the first construction, we use fibrational techniques to derive the cartesian closed fibration from the various categories of regular languages of $\lambda$-terms associated to finite sets of ground states. In the second construction, we take advantage of the recent notion of profinite $\lambda$-calculus to define the cartesian closed fibration by a change-of-base from the fibration of clopen subsets over the category of Stone spaces, using an elegant idea coming from Hermida. We illustrate the expressive power of the cartesian closed fibration by generalizing the notion of Brzozowski derivative to higher-order regular languages, using an Isbell-like adjunction in the sense of Melli\`es and Zeilberger.

Problem

• higher-order regular languages
• cartesian closed fibration
• λ-calculus
• Stone spaces
• Brzozowski derivative

Innovation

• cartesian closed fibration
• higher-order regular languages
• profinite λ-calculus
• Brzozowski derivative
• fibrational techniques