Semantics of Higher-Order Recursion Schemes

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 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. Whereas Fiore et al proved that the presheaf F-lambda of lambda-terms is an initial H-lambda-monoid, we work with the presheaf R-lambda 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 R-lambda.