A cartesian closed fibration of higher-order regular languages

TL;DR

This paper constructs a cartesian closed fibration of higher-order regular languages using fibrational techniques and profinite λ-calculus, enabling advanced operations like a generalized Brzozowski derivative.

Contribution

It introduces two novel constructions of the cartesian closed fibration for higher-order regular languages, expanding the theoretical framework in this area.

Findings

Constructed the fibration via fibrational techniques from regular languages of λ-terms.

Defined the fibration using profinite λ-calculus and change-of-base from Stone spaces.

Generalized Brzozowski derivative to higher-order regular languages.

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.