# Semantics of Higher-Order Recursion Schemes

• Stefan Milius
• Jiří Velebil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

## Abstract

Higher-order recursion schemes are equations defining recursively 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 λ-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite λ-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. Whereas Fiore et al proved that the presheaf F λ of λ-terms is an initial H λ -monoid, we work with the presheaf R λ of rational infinite λ-terms and prove that this is an initial iterative H λ -monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in R λ .

## Keywords

Higher-order recursion schemes infinite λ-terms sets in context rational tree

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