The undecidability of type related problems in the type-free style System F with finitely stratified polymorphic types

We consider here a number of variations on System F that are predicative second-order systems whose terms are intermediate between the Curry style and the Church style. As in the Church style, the terms we deal with here contain the information on where universal quantifier elimination and introduction in the type inference process must take place. However, they omit the information on what types are involved in the rules, which is similar to Curry forms. This can be viewed as a version of the partial type reconstruction problem considered by Boehm and Pfenning in which type erasure is done in a systematic way. In this paper we prove the undecidability of the type checking, type inference, and typability problems for the system. This demonstrates that the reason for undecidability is not the absence of the information where the second-order rules should be applied but the actual shape of the polymorphic types to be used in the derivation. Moreover, the proof works for the predicative version of the system with finitely stratified polymorphic types. The result includes bounds on the Leivant levels of types used in the instances leading to undecidability.