Comprehensive Parametric Polymorphism: Categorical Models and Type Theory

Ghani, Neil; Forsberg, Fredrik Nordvall; Simpson, Alex

doi:10.1007/978-3-662-49630-5_1

Cited by 5 publications

(10 citation statements)

References 23 publications

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Section: Discussionmentioning

confidence: 99%

“…We can now be more specific about how our approach compares to the external approaches in [4,7,10,15], all of which are based on a reflexive graph of λ2-fibrations. The definition in [4] appears to be too restrictive: it requires a comprehension structure that, e.g., the λ2-fibration corresponding to Reynolds' model does not admit. In addition, none of these frameworks seem to recognize the λ-fibration corresponding to the proof-relevant model as parametric, for the following reason: it is unclear how to define the family of adjoints for the second fibration (called r in [7]) of "heterogeneous" reflexive graph functors in a way that is compatible with the adjoint structure on the original λ2-fibration.…”

Section: Discussionmentioning

confidence: 99%

“…The identity reflexive graph natural transformation serves as the identity for the composition of reflexive graph natural transformations 4…”

mentioning

confidence: 99%

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

mentioning

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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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

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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“…Several type systems have been designed to support formal reasoning about relational properties for functional programs. Some of the earlier works in this direction have focused on the semantics foundations of parametricity, like the work by on System R, a relational version of System F. e recent work by Ghani et al (2016a) has further extended this approach to give be er foundations to a combination of relational parametricity and impredicative polymorphism. Interestingly, similarly to RHOL, System R also supports relations between expressions at di erent types, although, since System R does not support re nement types, the only relations that System R can support are the parametric ones on polymorphic terms.…”

Section: Introductionmentioning

confidence: 99%

A relational logic for higher-order programs

Aguirre

Barthe

Gaboardi

et al. 2017

Proc. ACM Program. Lang.

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Relational program veri cation is a variant of program veri cation where one can reason about two programs and as a special case about two executions of a single program on di erent inputs. Relational program veri cation can be used for reasoning about a broad range of properties, including equivalence and re nement, and specialized notions such as continuity, information ow security or relative cost. In a higher-order se ing, relational program veri cation can be achieved using relational re nement type systems, a form of re nement types where assertions have a relational interpretation. Relational re nement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very di erent structures.We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive de nitions. RHOL retains the type-directed avour of relational re nement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we de ne sound embeddings for several existing relational type systems such as relational re nement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work. ACM Reference format:Contributions. We present a new logic, called Relational Higher Order Logic (RHOL, § 5), for reasoning about relational properties of higher-order programs wri en in a variant of Plotkin's PCF ( § 2). e logic manipulates judgments of the form:where Γ is a simply typed context, σ 1 and σ 2 are (possibly di erent) simple types, t 1 and t 2 are terms, Ψ is a set of assertions, and ϕ is an assertion. Our logic retains the type-directed nature of (relational) re nement type systems, and features typing rules for reasoning about structurally similar terms. However, disentangling types from assertions also makes it possible to de ne type-directed rules operating on a single term (le or right) of the judgment. is confers great expressivity to the logic, without signi cantly a ecting its type-directed nature, and opens the possibility to alternate freely between two-sided and one-sided reasoning, as done in rst-order imperative languages. e validity of judgments is expressed relative to a set-theoretical semantics-our variant of PCF is restricted to terms which admit a set-theoretical semantics, including strongly normalizing terms. More precisely, a judgment Γ | Ψ ⊢ t 1 : σ 1 ∼ t 2 : σ 2 | ϕ is valid if for every valuation ρ (mapping variables in the context Γ to elements in the interpretation of their types), the interpretation of ϕ is true whenever the interpretation of (a...

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A relational logic for higher-order programs

Aguirre¹,

Barthe²,

Gaboardi³

et al. 2019

J. Funct. Prog.

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Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security, or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called relational higher-order logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavor of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic, and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule, and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

show abstract

Comprehensive Parametric Polymorphism: Categorical Models and Type Theory

Cited by 5 publications

References 23 publications

A General Framework for Relational Parametricity

A General Framework for Relational Parametricity

A relational logic for higher-order programs

A relational logic for higher-order programs

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