Parametricity, type equality, and higher-order polymorphism

Vytiniotis, Dimitrios; Weirich, Stephanie

doi:10.1017/s0956796810000079

Cited by 26 publications

(23 citation statements)

References 40 publications

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“…• The result is already exposed by Vytiniotis & Weirich (2010), but it is remarkable that its proof is one line long given our framework.…”

Section: Type Equalitymentioning

confidence: 65%

“…One direction of research is concerned with parametricity in extensions of System F. Our work is directly inspired by Vytiniotis & Weirich (2010), which extend parametricity to (an extension of) Fω: indeed, Fω can be seen as a PTS with one more product rule than System F.…”

Section: Related Workmentioning

confidence: 99%

“…Recent works (Takeuti 2004, personal communication;Johann & Voigtländer 2006;Neis et al 2009;Vytiniotis & Weirich 2010) have developed parametricity results for several such calculi, but not in a common framework. In this paper, we apply and extend Reynolds' (1983) idea to a large class of PTSs and provide a framework that unifies previous descriptions of parametricity and forms a basis for future studies of parametricity in specific type systems.…”

Section: Introductionmentioning

confidence: 99%

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Bernardy¹,

Jansson²,

Paterson³

2012

J. Funct. Prog.

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This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link: R O S S P A T E R S O NCity University, London, UK (e-mail:)ross@soi.city.ac.uk) AbstractReynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process. 83(1), 513-523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.

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“…• The result is already exposed by Vytiniotis & Weirich (2010), but it is remarkable that its proof is one line long given our framework.…”

Section: Type Equalitymentioning

confidence: 65%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bernardy¹,

Jansson²,

Paterson³

2012

J. Funct. Prog.

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“…Johann and Voigtländer [2005] have applied it to a system with explicit strictness; Vytiniotis and Weirich [2010] to Fω extended with representation types; Takeuti [2004] sketches how it can be applied to the λ-cube, Neis et al [2009] to a system with dynamic casting. In this paper, we apply Reynolds' idea to dependently-typed systems.…”

Section: Introductionmentioning

confidence: 99%

Parametricity and dependent types

2010

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones. Permanent

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“…Throughout the section, we fix the function h. Following the original work [19], we make use of free theorems [2,20,22,23] in the proofs. We assume that a polymorphic function h that we bidirectionalize is total, and sources and views do not contain any undefined values.…”

Section: Correctnessmentioning

confidence: 99%

Bidirectionalization for free with runtime recording

Matsuda

Wang

2013

Proceedings of the 15th Symposium on Principles and Practice of Declarative Programming

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A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. Over the years, a lot of effort has been made to offer better language support for programming such transformations. In particular, a technique known as bidirectionalization is able to analyze and transform unidirectional programs written in general purpose languages, and "bidirectionalize" them.Among others, a technique termed as semantic bidirectionalization proposed by Voigtländer stands out in term of user-friendliness. The unidirectional program can be written using arbitrary language constructs, as long as the function is polymorphic and the language constructs respect parametricity. The free theorems that follow from the polymorphic type of the program allow a kind of forensic examination of the transformation, determining its effect without examining its implementation. This is convenient, in the sense that the programmer is not restricted to using a particular syntax; but it does require the transformation to be polymorphic.In this paper, we lift this polymorphism requirement to improve the applicability of semantic bidirectionalization. Concretely, we provide a type class PackM γ α µ, which intuitively reads "a concrete datatype γ is abstracted to a type α, and the 'observations' made by a transformation on values of type γ are recorded by a monad µ". With PackM , we turn monomorphic transformations into polymorphic ones, that are ready to be bidirectionalized. We demonstrate our technique with a case study of standard XML queries, which were considered beyond semantic bidirectionalization because of their monomorphic nature.

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Parametricity, type equality, and higher-order polymorphism

Cited by 26 publications

References 40 publications

Proofs for free

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Parametricity and dependent types

Bidirectionalization for free with runtime recording

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