Inductive and Coinductive types with Iteration and Recursion in a Polymorphic Framework

We study (extensions of) polymorphic typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in terms of iteration. However, in the syntax we often have only weak initiality, which makes the definition of recursion in terms of iteration inefficient or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. (The same notion has, independently, been proposed by [Mendler 1991].) We then look at how these syntactic notions work out in the framework of K-models for polymorphic lambda calculus. It will turn out that with some quite weak extra assumptions, recursion can be defined in terms of corecursion and vice versa using polymorphism. This also works syntactically: We shall look at some slight extensions of polymorphic lambda calculus for which a scheme for either recursion or corecursion suffices to be able to define the other. As an application of this we look at the Calculus of Inductive Definitions ([Coquand and Mohring 1990] and [Dowek e.a. 1991]), which reflects our categorical notion of recursion and we show how to define coinductive types with corecursion in it.