Programming Examples Needing Polymorphic Recursion

Inferring types for polymorphic recursive function definitions (abbreviated to polymorphic recursion) is a recurring topic on the mailing lists of popular typed programming languages. This is despite the fact that type inference for polymorphic recursion using @?-types has been proved undecidable. This report presents several programming examples involving polymorphic recursion and determines their typability under various type systems, including the Hindley-Milner system, an intersection-type system, and extensions of these two. The goal of this report is to show that many of these examples are typable using a system of intersection types as an alternative form of polymorphism. By accomplishing this, we hope to lay the foundation for future research into a decidable intersection-type inference algorithm. We do not provide a comprehensive survey of type systems appropriate for polymorphic recursion, with or without type annotations inserted in the source language. Rather, we focus on examples for which types may be inferred without type annotations, with an emphasis on systems of intersection- types.