Parametricity for Nested Types and GADTs

TL;DR

This paper develops a parametricity framework for nested data types and GADTs within a Hindley-Milner-style calculus, enabling derivation of free theorems and handling truly nested types directly.

Contribution

It introduces a novel calculus for nested types with a parametric model that supports primitive pattern matching, map, and fold, and extends parametricity to GADTs with limitations.

Findings

Constructed a parametric model for nested types

Derived free theorems and a shortcut fusion transformation

Identified limitations in interpreting GADTs functorially

Abstract

This paper considers parametricity and its consequent free theorems for nested data types. Rather than representing nested types via their Church encodings in a higher-kinded or dependently typed extension of System F, we adopt a functional programming perspective and design a Hindley-Milner-style calculus with primitives for constructing nested types directly as fixpoints. Our calculus can express all nested types appearing in the literature, including truly nested types. At the level of terms, it supports primitive pattern matching, map functions, and fold combinators for nested types. Our main contribution is the construction of a parametric model for our calculus. This is both delicate and challenging. In particular, to ensure the existence of semantic fixpoints interpreting nested types, and thus to establish a suitable Identity Extension Lemma for our calculus, our type system must explicitly track functoriality of types, and cocontinuity conditions on the functors interpreting them must be appropriately threaded throughout the model construction. We also prove that our model satisfies an appropriate Abstraction Theorem, as well as that it verifies all standard consequences of parametricity in the presence of primitive nested types. We give several concrete examples illustrating how our model can be used to derive useful free theorems, including a short cut fusion transformation, for programs over nested types. Finally, we consider generalizing our results to GADTs, and argue that no extension of our parametric model for nested types can give a functorial interpretation of GADTs in terms of left Kan extensions and still be parametric.