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Non-parametric parametricity


Type abstraction and intensional type analysis are features seemingly at odds—type abstraction is intended to guarantee parametricity and representation independence, while type analysis is inherently non-parametric. Recently, however, several researchers have proposed and implemented “dynamic type generation” as a way to reconcile these features. The idea is that, when one defines an abstract type, one should also be able to generate at runtime a fresh type name, which may be used as a dynamic representative of the abstract type for purposes of type analysis. The question remains: in a language with non-parametric polymorphism, does dynamic type generation provide us with the same kinds of abstraction guarantees that we get from parametric polymorphism?

Our goal is to provide a rigorous answer to this question. We define a step-indexed Kripke logical relation for a language with both non-parametric polymorphism (in the form of type-safe cast) and dynamic type generation. Our logical relation enables us to establish parametricity and representation independence results, even in a non-parametric setting, by attaching arbitrary relational interpretations to dynamically generated type names. In addition, we explore how programs that are provably equivalent in a more traditional parametric logical relation may be “wrapped” systematically to produce terms that are related by our non-parametric relation, and vice versa. This leads us to develop a “polarized” variant of our logical relation, which enables us to distinguish formally between positive and negative notions of parametricity.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. W. Appel & D. McAllester (2001) An indexed model of recursive types for foundational proof-carrying code. ACM Trans. Program. Lang. Syst. 23 (5), 657683.

L. Birkedal & R. W. Harper (1999) Constructing interpretations of recursive types in an operational setting. Inf. Comput. 155, 363.

D. Grossman , G. Morrisett & S. Zdancewic (2000) Syntactic type abstraction. ACM Trans. Program. Lang. Syst. 22 (6), 10371080.

R. Harper & J. C. Mitchell (1999) Parametricity and variants of Girard's J operator. Inf. Process. Lett. 70 (1), 15.

J. C. Mitchell & G. D. Plotkin (1988) Abstract types have existential type. ACM Trans. Program. Lang. Syst. 10 (3), 470502.

E. Sumii & B. C. Pierce (2003) Logical relations for encryption. J. Comput. Secur. 11 (4), 521554.

E. Sumii & B. C. Pierce (2007a). A bisimulation for dynamic sealing. Theor. Comput. Sci. 375 (1–3), 161192.

E. Sumii & B. C. Pierce (2007b) A bisimulation for type abstraction and recursion. J. ACM 54 (5), 143.

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Journal of Functional Programming
  • ISSN: 0956-7968
  • EISSN: 1469-7653
  • URL: /core/journals/journal-of-functional-programming
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