An extension lemma and homogeneous programming

1985 ◽  
Vol 47 (3) ◽  
pp. 321-336 ◽  
Author(s):  
J. Gwinner
Keyword(s):  
Homogeneous Programming ◽  
Extension Lemma

scholarly journals Universal properties for universal types in bifibrational parametricity

2019 ◽  
Vol 29 (06) ◽  
pp. 810-827
Author(s):  
Neil Ghani ◽  
Fredrik Nordvall Forsberg ◽  
Federico Orsanigo
Keyword(s):  
Ad Hoc ◽  
Universal Property ◽  
Axiomatic Treatment ◽  
Model Independent ◽  
Universal Properties ◽  
Extension Lemma ◽  
Uniformity Principle

AbstractIn 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) has 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.

Homogeneous programming

1980 ◽  
Vol 31 (1) ◽  
pp. 101-105 ◽  
Author(s):  
T. Fujimoto
Keyword(s):  
Homogeneous Programming

scholarly journals Parametricity for Primitive Nested Types

2021 ◽  
pp. 324-343
Author(s):  
Patricia Johann ◽  
Enrico Ghiorzi ◽  
Daniel Jeffries
Keyword(s):  
Pattern Matching ◽  
Functional Programming ◽  
Type System ◽  
Parametric Model ◽  
Model Construction ◽  
Data Types ◽  
Nested Data ◽  
Extension Lemma ◽  
System F

AbstractThis 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.

Homogeneous programming and symmetrization

1983 ◽  
Vol 41 (2) ◽  
pp. 285-291
Author(s):  
L. Thorlund-Petersen
Keyword(s):  
Homogeneous Programming

scholarly journals Parametricity for Nested Types and GADTs

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Patricia Johann ◽  
Enrico Ghiorzi
Keyword(s):  
Pattern Matching ◽  
Functional Programming ◽  
Type System ◽  
Parametric Model ◽  
Model Construction ◽  
Data Types ◽  
Nested Data ◽  
Kan Extensions ◽  
Extension Lemma ◽  
System F

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.

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