A Predicative Type-Theoretic Interpretation of Objects

Predicative type theories are powerful tools for giving foundational interpretations of programming languages. Due to their explicit inductive construction, predicative type theories have multiple mathematical models that provide precise de nitions of programming language features. However, not all features have predicative interpretations, and current interpretations of objects rely on impredicative type theories, such as Girard's System F, because of the di culty in specifying a type for objects in the presence of self-application. In this paper we show that objects have a predicative interpretation. We show that predicativity is associated with method monotonicity, and that binary methods prevent the inductive type construction. Our interpretation di ers from impredicative accounts by replacing the use of recursive types for objects with conditions for method polymorphism over the self type. We further give a propositional meaning to objects in the type theory, providing a calculus for formal veri cation. Our interpretation has been veri ed in the Nuprl predicative type theory.