Parametricity for Nested Types and GADTs

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.

Unnecessary reliance on multilevel modelling to analyse nested data

A simulation experiment to confirm the validity of summary-statistics approaches for analysing nested data

Parametricity for Primitive Nested Types

This paper considers parametricity and its resulting 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 term level, 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: 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 prove that our model satisfies an appropriate Abstraction Theorem and verifies all standard consequences of parametricity for primitive nested types.

Foundations of Fine-Grained Explainability

Explainability is the process of linking part of the inputs given to a calculation to its output, in such a way that the selected inputs somehow “cause” the result. We establish the formal foundations of a notion of explainability for arbitrary abstract functions manipulating nested data structures. We then establish explanation relationships for a set of elementary functions, and for compositions thereof. A fully functional implementation of these concepts is finally presented and experimentally evaluated.