Realizability Semantics of Parametric Polymorphism, General References, and Recursive Types

Abstract

Abstract. We present a realizability model for a call-by-value, higherorder programming language with parametric polymorphism, general first-class references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include general reference types. The interpretation uses a new approach to modeling references.The universe of semantic types consists of world-indexed families of logical relations over a universal predomain. In order to model general reference types, worlds are finite maps from locations to semantic types: this introduces a circularity between semantic types and worlds that precludes a direct definition of either. Our solution is to solve a recursive equation in an appropriate category of metric spaces. In effect, types are interpreted using a Kripke logical relation over a recursively defined set of worlds.We illustrate how the model can be used to prove simple equivalences between different implementations of imperative abstract data types.

A Complete Characterization of Observational Equivalence in Polymorphic λ-Calculus with General References

“…(On the other hand, those examples have also shown that, with the help of Lemma 2, our "brute-force" method is often simple enough.) Logical relations are also better at giving a compositional model of universal types, as in [3] and [6,7].…”

“…A recent paper by Ahmed, Dreyer and Rossberg [3]discussed in Section 1, 7 and 8-follows this line of work. Another line of work has been carried out by Birkedal et al [6][7][8]. To the best of our knowledge, our method can prove strictly more examples of equivalences than all of the above approaches (though this is hard to prove generally, because completeness by itself does not always mean an automatic proof; recall that observational equivalence is undecidable in our language).…”

“…Indeed, some interesting equivalences cannot be proved by their method [3,Section 5.7 and 5.8]. Independently, Birkedal, Støvring and Thamsborg [6,7] have also developed logical relations in a language with polymorphic (and recursive) types and references. However, they are also incomplete and not yet useful enough for reasoning about observational equivalence involving local state in general [7, Section 1 and 6][6, Section 1].…”

A typed store-passing translation for general references

“…To reflect this, several semantic models of weak references in the literature follow the "possible worlds" approach [2,10,16,34]. There, the interpretation of a type is parameterized over a world, which represents the current state of the store.…”

“…. [10] or work with approximate solutions of it [2,14]. An alternative is to define a world as a map of memory addresses to syntactic types [16].…”