Linear Abadi & Plotkin Logic

We present a formalization of a version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including existential types, inductive types, coinductive types and general recursive types. We show that the recursive types satisfy a universal property called dinaturality, and we develop reasoning principles for the constructed types. In the case of recursive types, the reasoning principle is a mixed induction / coinduction principle, with the curious property that coinduction holds for general relations, but induction only for a limited collection of “admissible” relations. A similar property was observed in Pitts analysis of recursive types in domain theory [18]. In a future paper we will develop a category theoretic notion of models of the logic presented here, and show how the results developed in the logic can be transfered to the models. ∗Corresponding author. Address: L. Birkedal, IT University of Copenhagen, Rued Langgaardsvej 7, DK–2300 Copenhagen S, DENMARK, [email protected]