A Characterization of lambda Definability in Categorical Models of Implicit Polymorphism

Lambda deenability is characterized in categorical models of simply typed lambda calculus with type variables. A category-theoretic framework known as glueing or sconing is used to extend the Jung-Tiuryn characterization of lambda deenability JuT93], rst to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a well-established tool for studying the semantics of various typed lambda calculi. The main lines of research are focused in two areas, the rst of which strives for an understanding of Strachey's notion of parametric polymorphism. The main idea is that a parametricly polymorphic function acts independently from the types to which its type variables are instantiated, and that this uniformity may be captured by imposing a relational structure on the types OHT93, MSd93, MaR91, Wad89, Rey83, Str67]. The other line of research concerns lambda deenability and the full abstraction problem for various models of languages based on simply typed lambda calculus JuT93, MaR91, MSd93, OHR94, Pit93, Plo80, Lau70]. Early attempts to characterize lambda deenability in the full type hiearchy focused on invari-ance properties of functions that are deenable by lambda terms. Invariance under permutation Lau70] was the most obvious of these since lambda deen-able functions cannot speak about particular elements, but this was not enough for a complete characterization. A later attempt in Plo80] introduced the idea of invariance under a logical relation and then, in the same paper, the notion of invariance under I-relation by which Plotkin succeeded in characterizing lambda deenability in certain full type hierarchies. More recently, Jung and Tiuryn describe a notion of logical relation with which they characterize lambda deenability in all Henkin models of simply typed lambda calculus JuT93], and since then Riecke and O'Hearn have modiied this result to an extension of simply typed lambda calculus having basic arithmetic constructs and general recursion to provide fully abstract models of PCF OHR94]. In this work, a notion of logical relation is presented that characterizes lambda deenability in categorical models of simply typed lambda calculus with type variables. This language, called implicit ML, is a fragment of Core-ML that features a form of polymorphism called implicit polymorphism in HaM88, MSd93].