Towards a Simpler Account of Modules and Generativity: Abstract Types

We present a variant of the explicitly-typed secondorder polymorphic λ-calculus with primitive open existential types, i.e. a collection of more atomic constructs for introduction and elimination of existential types. We equip the language with a call-by-value small-step reduction semantics that enjoys the subject reduction property. Traditional closed existential types can be defined as syntactic sugar. Conversely, programs with no free existential types can be rearranged to only use closed existential types, but the translation is not compositional. We argue that open existential types model abstract types and modules with generativity. We also introduce a new notion of paths at the level of types instead of terms that allows for writing type annotations more concisely and modularly.