Bifibrational Functorial Semantics of Parametric Polymorphism

Ghani, Neil; Johann, Patricia; Forsberg, Fredrik Nordvall; Orsanigo, Federico; Revell, Tim

doi:10.1016/j.entcs.2015.12.011

Cited by 12 publications

(58 citation statements)

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“…Additionally, the bifibration condition of Vol. 16:1 CALL-BY-NAME GRADUAL TYPE THEORY 7:41 Φ t 1 t 2 : Figure 8: [33] Cast Rules Derived [13] is essentially the same as the definition of an equipment, but with a twist: in gradual typing every contract induces an adjoint pair of terms, but there every term induces an adjoint pair of relations: the graph and "cograph". Hopefully the similarity with parametric logics will be useful in studying the combination of graduality with parametricity.…”

Section: Cast-lmentioning

confidence: 99%

Gradual type theory

New

Licata

Ahmed

2019

Proc. ACM Program. Lang.

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We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism. These dynamism judgments build on prior work in blame calculi and on the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality (η) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs. This construction is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing. CC Creative Commons 7:2 M.S. New and D.R. Licata Vol. 16:1"strictly less than", and similarly for other terms such as "greater than", "more dynamic", etc. 7:4 M.S. New and D.R. Licata Vol. 16:1 7:6 M.S. New and D.R. Licata Vol. 16:1 7:8 M.S. New and D.R. Licata Vol. 16:1 2.2. PTT Signatures. While gradual type theory proves that most operational rules of gradual typing are equivalences, some must be added as axioms. Compare Moggi's monadic metalanguage [22]: since it is a general theory of monads, it is not provable that an effect is 7:12 M.S. New and D.R. Licata Vol. 16:1

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Section: Cast-lmentioning

confidence: 99%

Gradual type theory

New

Licata

Ahmed

2019

Proc. ACM Program. Lang.

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“…With this adjustment, we have two canonical relationally parametric models of System F: i) the PER model of Longo and Moggi [9], internal to the theory of ω-sets and realizable functions, and ii) Reynolds' original model 1 , internal to CIC. After Reynolds' original paper, more abstract treatments of his ideas were given by, e.g., Robinson and Rosolini [15], O'Hearn and Tennent [11], Dunphy and Reddy [2], and Ghani et al [5]. The approach is to use a categorical structure -reflexive graph categories for [2,11,15] and fibrations for [5] -to represent sets and relations, and to interpret types as appropriate functors and terms as natural transformations.…”

Section: Introductionmentioning

confidence: 99%

“…After Reynolds' original paper, more abstract treatments of his ideas were given by, e.g., Robinson and Rosolini [15], O'Hearn and Tennent [11], Dunphy and Reddy [2], and Ghani et al [5]. The approach is to use a categorical structure -reflexive graph categories for [2,11,15] and fibrations for [5] -to represent sets and relations, and to interpret types as appropriate functors and terms as natural transformations. In particular, [2] aims to "[address] parametricity in all its incarnations", and similarly for [5].…”

Section: Introductionmentioning

confidence: 99%

“…The approach is to use a categorical structure -reflexive graph categories for [2,11,15] and fibrations for [5] -to represent sets and relations, and to interpret types as appropriate functors and terms as natural transformations. In particular, [2] aims to "[address] parametricity in all its incarnations", and similarly for [5]. Surprisingly and significantly, however, Reynolds' original model does not arise as a direct instance of either framework.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, unlike the frameworks [2,4,5,7,10,15], our definition of a parametric model recognizes the above proof-relevant model:…”

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confidence: 99%

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A General Framework for Relational Parametricity

Sojakova

Johann

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Reynolds' original theory of relational parametricity was intended to capture the idea that polymorphically typed System F programs preserve all relations between inputs. But as Reynolds himself later showed, his theory can only be formalized in a meta-theory with an impredicative universe, such as the Calculus of Inductive Constructions. Abstracting from Reynolds' ideas, Dunphy and Reddy developed their well-known framework for parametricity that uses parametric limits in reflexive graph categories and aims to subsume a variety of parametric models. As we observe, however, their theory is not sufficiently general to subsume the very model that inspired parametricity, namely Reynolds' original model, expressed inside type theory.To correct this, we develop an abstract framework for relational parametricity that generalizes the notion of a reflexive graph categories and delivers Reynolds' model as a direct instance in a natural way. This framework is uniform with respect to a choice of meta-theory, which allows us to obtain the well-known PER model of Longo and Moggi as a direct instance in a natural way as well. In addition, we offer two novel relationally parametric models of System F: i) a categorical version of Reynolds' model, where types are functorial on isomorphisms and all polymorphic functions respect the functorial action, and ii) a proof-relevant categorical version of Reynolds' model (after Orsanigo), where, additionally, witnesses of relatedness are themselves suitably related. We show that, unlike previously existing frameworks for parametricity, ours recognizes both of these new models in a natural way. Our framework is thus descriptive, in that it accounts for well-known models, as well as prescriptive, in that it identifies abstract properties that good models of relational parametricity should satisfy and suggests new constructions of such models.

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Parametricity for Primitive Nested Types

Johann

Ghiorzi

Jeffries

2021

Lecture Notes in Computer Science

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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.

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Bifibrational Functorial Semantics of Parametric Polymorphism

Cited by 12 publications

References 12 publications

Gradual type theory

Gradual type theory

A General Framework for Relational Parametricity

Parametricity for Primitive Nested Types

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