Variations on Algebra: monadicity and generalisations of equational theories

Dedicated to Rod Burstal

Bifibrational Functorial Semantics For Parametric Polymorphism

Reynolds’ theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Semantically, reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective by showing the relevance of bifibrations. We develop a bifibrational framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds’ Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality

Bifibrational functorial semantics of parametric polymorphism

Reynolds' theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Semantically, reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective by showing the relevance of bifibrations. We develop a bifibrational framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds' Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality

A Relationally Parametric Model of Dependent Type Theory

Reynolds’ theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds’ original work exploited the typing discipline of the polymorphically typed -calculus System F, but there is now considerable interest in extending relational parametricity to type systems that are richer and more expressive than that of System F.This paper constructs parametric models of predicative and impredicative dependent type theory. The significance of our models is twofold. Firstly, in the impredicative variant we are able to deduce the existence of initial algebras for all indexed functors. To our knowledge, ours is the first account of parametricity for dependent types that is able to lift the useful deduction of the existence of initial algebras in parametric models of System F to the dependently typed setting. Secondly, our models offer conceptual clarity by uniformly expressing relational parametricity for dependent types in terms of reflexive graphs, which allows us to unify the interpretations of types and kinds, instead of taking the relational interpretation of types as a primitive notion. Expressing our model in terms of reflexive graphs ensures that it has canonical choices for the interpretations of the standard type constructors of dependent type theory, except for the interpretation of the universe of small types, where we formulate a refined interpretation tailored for relational parametricity. Moreover, our reflexive graph model opens the door to generalizations of relational parametricity, for example to higher-dimensional relational parametricity

The Marriage of Univalence and Parametricity

International audienceReasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, which are frequently used to mechanize mathematical results and carry out program verification efforts, equality is appallingly syntactic and, as a result, exploiting equivalences is cumbersome at best. Parametricity and univalence are two major concepts that have been explored in the literature to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport. This work first clarifies the limitations of these two concepts when considered in isolation, and then devises a fruitful marriage between both. The resulting concept, called univalent parametricity, is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences. Our approach handles both type and term dependency, as well as type-level computation. In addition to the theory of univalent parametricity, we present a lightweight framework implemented in the Coq proof assistant that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally-efficient representation, as soon as they are proven equivalent. The combination of parametricity and univalence supports transport à la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to transport more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq. This work paves the way to easier-to-use proof assistants by supporting seamless programming and proving modulo equivalences

Parametricity as isomorphism

SIGLEAvailable from British Library Document Supply Centre- DSC:7623.6171(SU-CS-R--5/92) / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Parametricity as isomorphism

SIGLEAvailable from British Library Document Supply Centre- DSC:7623.6171(SU-CS-R--5/92) / BLDSC - British Library Document Supply CentreGBUnited Kingdo