Semantics of Higher-Order Recursion Schemes

TL;DR

This paper investigates the semantics of higher-order recursion schemes, establishing their unique solutions in a category-theoretic framework using presheaves of rational infinite b5-terms.

Contribution

It extends the semantics of higher-order recursion schemes to rational infinite b5-terms, proving the existence of unique solutions in an initial iterative Hb5-monoid.

Findings

Every guarded higher-order recursion scheme has a unique uninterpreted solution.

The presheaf of rational infinite b5-terms forms an initial iterative Hb5-monoid.

The semantics are established in a category of sets in context, capturing variable binding and substitution.

Abstract

Higher-order recursion schemes are recursive equations defining new operations from given ones called "terminals". Every such recursion scheme is proved to have a least interpreted semantics in every Scott's model of \lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite \lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in \lambda-calculus by an endofunctor H\lambda and they explained simultaneous substitution of \lambda-terms by proving that the presheaf of \lambda-terms is an initial H\lambda-monoid. Here we work with the presheaf of rational infinite \lambda-terms and prove that this is an initial iterative H\lambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid.