A cartesian closed fibration of higher-order regular languages
Paul-André Melliès, Vincent Moreau
TL;DR
The paper addresses how to realize a cartesian-closed fibration of higher-order regular languages, extending recognizability from finite words to simply-typed λ-terms. It delivers two independent constructions: a fibrational approach based on $Q$-regular languages and a topological/profinite approach leveraging Stone duality and the profinite λ-calculus, both yielding the cartesian-closed fibration $p: ext{Reg} o oldsymbol{ ext{Lam}}$. It further extends Brzozowski derivatives to higher-order languages via an Isbell-like adjunction and investigates openness of λ-terms, culminating in a comprehensive account of two complementary viewpoints and their implications for higher-order automata theory. This framework connects recognizability, topology, and type-theoretic semantics, and points toward potential MSO-like logics for higher-order languages with practical implications for formal reasoning about λ-terms and their 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 $λ$-terms associated to finite sets of ground states. In the second construction, we take advantage of the recent notion of profinite $λ$-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 Melliès and Zeilberger.