Abstract

We consider the problems of typability 1 and type checking 2 in the Girard/Reynolds secondorder polymorphic typed -calculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pure -terms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lower-bound complexity results for System F have been achieved, but they have remained "embarrassing open problems" 3 for System F itself. We first prove that type checking in System F is undecidable by a reduction from semi-unification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructing -terms such that in all type derivations, specific bound variables must always be assigned a specific type. Usi...

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