On Typability for Rank-2 Intersection Types with Polymorphic Recursion

We show that typability for a natural form of polymorphic recursive typing for rank-2 intersection types is undecidable. Our proof involves characterizing typability as a context free language (CFL) graph problem, which may be of independent interest, and reduction from the boundedness problem for Turing machines. We also show a property of the type system which, in conjunction with the undecidability result, disproves a misconception about the Milner- Mycroft type system. We also show undecidability of a related program analysis problem.