Two Parametricities Versus Three Universal Types

The formal calculus System F models the essence of polymorphism and abstract data types, features that exist in many programming languages. calculus’ core property is parametricity: a theorem expressing language’s abstractions validating important principles like information hiding modularity. When combined with recursive mutable state, continuations or exceptions, formulation parametricity needs to be adapted follow suit, for example using techniques step-indexing, Kripke world-indexing biorthogonality. However, it less clear how this should change when untyped languages, gradual dynamic sealing runtime type analysis (typecase) alongside generation. Extensions these have been proven satisfy forms (with worlds carrying semantic interpretations types). relative power modified formulations respect others expressiveness without extensions are unknown. In paper, we explain aforementioned different settings common characteristic: they do not enforce preserve lexical scope F’s variables. Formally, results existence universal (note: same as universally-quantified type). We why standard incompatible such resolve this. Building on insights, answer two open conjectures from literature, negatively, point out deficiency current proposals combining types.