Typability is Undecidable for F+Eta

• Authors: J. B. Wells
• Venue: Typability is Undecidable for F+Eta
• Year: 1996

Abstract System F is the well-known polymorphically-typed lambda calculus with universal quantifiers. F+eta is System F extended with the eta rule, which says that if term M can be given type tau and M eta-reduces to N , then N can also be given the type tau. Adding the eta rule to System F is equivalent to adding the subsumption rule using the subtyping (containment) relation that Mitchell defined and axiomatized [Mit88]. The subsumption rule says that if M can be given type tau and tau is a subtype of type sigma, then M can be given type sigma. Mitchell''s subtyping relation involves no extensions to the syntax of types, i.e., no bounded polymorphism and no supertype of all types, and is thus unrelated to the system "F-sub". Typability for F+eta is the problem of determining for any term M whether there is any type tau that can be given to it using the type inference rules of F+eta. Typability has been proven undecidable for System F [Wel94] (without the eta rule), but the decidability of typability has been an open problem for F+eta. Mitchell''s subtyping relation has recently been proven undecidable [TU95,Wel95b], implying the undecidability of "type checking" for F+eta. This paper reduces the problem of subtyping to the problem of typability for F+eta, thus proving the undecidability of typability. The proof methods are similar in outline to those used to prove the undecidability of typability for System F, but the fine details differ greatly.