@TECHREPORT{Wells95typabilityis, author = {J. B. Wells}, title = {Typability is undecidable for F+eta}, institution = {}, year = {1995} }

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Abstract

System F is the well-known polymorphically-typed-calculus with universal quanti ers (\8"). F+ is System F extended with the eta rule, which says that if term M can be given type and M-reduces to N, then N can also be given the type. Adding the eta rule to System F is equivalent to adding the subsumption rule using the subtyping (\containment") relation that Mitchell de ned and axiomatized [Mit88]. The subsumption rule says that if M can be given type and is a subtype of type,thenMcan be given type. Mitchell's subtyping relation involves no extensions to the syntaxoftypes, i.e., no bounded polymorphism and no supertype of all types, and is thus unrelated to the system F (\F-sub"). Typability for F+ is the problem of determining for any termMwhether there is any type that can be given to it using the type inference rules of F+. Typability has been proven undecidable for System F [Wel94] (without the eta rule), but the decidability oftypability has been an open problem for F+. Mitchell's subtyping relation has recently been proven undecidable [TU95, Wel95b], implying the undecidability of\type checking " for F+. This paper reduces the problem of subtyping to the problem of typability for F+,thus proving the undecidability oftypability. The proof methods are similar in outline to those used to prove the undecidability oftypability for System F, but the ne details di er greatly. 1