Parametric Polymorphism — Universally

Abstract

Abstract. In the 1980s, John Reynolds postulated that a parametrically polymorphic function is an ad-hoc polymorphic function satisfying a uniformity principle. This allowed him to prove that his set-theoretic semantics has a relational lifting which satisfies the Identity Extension Lemma and the Abstraction Theorem. However, his definition (and subsequent variants) have only been given for specific models. In contrast, we give a model-independent axiomatic treatment by characterising Reynolds' definition via a universal property, and show that the above results follow from this universal property in the axiomatic setting.

Comprehensive Parametric Polymorphism: Categorical Models and Type Theory

“…The results of the present paper should be contrasted with those of other recent work by first two authors and colleagues [9,10]. In this paper, we have axiomatised category-theoretic structure modelling relational parametricity for the specific type theory λ2, where the resulting structure encompasses both 'syntactic' and 'semantic' models.…”

“…In this paper, we have axiomatised category-theoretic structure modelling relational parametricity for the specific type theory λ2, where the resulting structure encompasses both 'syntactic' and 'semantic' models. In contrast, [9,10] axiomatise the category-theoretic structure required on a 'semantic' model for Reynolds' original set-theoretic definition of relational parametricity [23] to generalise to the model. Interestingly, the category-theoretic notion of bifibration occurs both as a central ingredient in the axiomatisation of [9,10], and, in the guise of direct-image relations, as a vital tool in the present paper.…”

“…In contrast, [9,10] axiomatise the category-theoretic structure required on a 'semantic' model for Reynolds' original set-theoretic definition of relational parametricity [23] to generalise to the model. Interestingly, the category-theoretic notion of bifibration occurs both as a central ingredient in the axiomatisation of [9,10], and, in the guise of direct-image relations, as a vital tool in the present paper. A novelty in the present paper is that the bifibrational structure is derived rather than assumed.…”

Universal properties for universal types in bifibrational parametricity

“…This paper is an extended version of a paper that appeared in the proceedings of WoLLIC 2015 [Ghani et al, 2015b]. This version includes the completely new Sections 3.3 and 4.3.2, which generalise the results from the conference version.…”

On completeness and parametricity in the realizability semantics of System F

“…In other words, IEL expresses the fact that the maps over S C corresponding to types must be "equality-preserving". In more generality, in [GFO15] and [GJF + 15] it is explained in detail how the interpretation of the quantifier in Reynolds' parametric models can be defined in terms of equality-preserving functors.…”