Parametric Polymorphism and models of storage

The term parametric polymorphism was introduced in by Strachey who distinguished it from “ad hoc” polymorphism. In the words of John Reynolds, “a parametric polymorphic function is one that behaves the same way for all types,” whereas an ad hoc polymorphic function may have unrelated meanings at different types. A very similar intuition arose in mathematics, some twenty years earlier, for which Eilenberg and MacLane proposed a “General Theory of Natural Equivalences”, what we now call category theory. Relating the two notions allows us to develop a broad view of “parametricity” (the idea of acting the same way for all types), which
is applicable not only to programming languages but also to mathematics and perhaps other disciplines. A particularly important application of this broad view is for Strachey’s “mathematical semantics” of programming languages.

For modelling imperative programming languages that operate by manipulating a store, Strachey presented a basic model of storage based on “locations”, while promising a further paper on an “abstract model of storage” to appear in future. Unfortunately the latter never appeared. In succeeding work, Reynolds proposed an abstract model of store as well as an “intuitionistic” functor category approach to model the stack discipline of the store. O'Hearn and Tennent made the crucial observation that the model needs to be embellished with parametricity to capture the data abstraction aspects of the store.

In this talk, Uday reviews these developments as well as his own recent work on integrating the Reynolds model with parametricity to capture the fact that state changes of the store are irreversible. One way to understand these developments is to view them as a “parametric mathematical semantics” of programming languages.