E. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings, Ninth Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, July 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... in [31, 24] Such atomic sheaves also become the theoretical basis of the approach for handling binders in abstract syntax in [4] The parametricity theory itself arose from an effort to get a model of polymorphic lambda calculus and to formalize logical relations in a categorical setting [29, 10, 30]. A structure known as reflexive graph has been used in the theory. The category of sheaves has been used to give a semantics of a type theory or a first order logic in [9, 11, 22] Goguen proposed a framework based on sheaves for defining semantics of concurrent object oriented languages [5] ....

....squares in category theory and form the basis of the uniformity concept captured by relational parametricity. O Hearn and Tennent [19] used reflexive graphs of categories to lift this picture to an abstract level, so that it could be applied to the semantics of Idealized Algol. See also [8, 23, 30] for other applications. Looking at the diagram (1) above, we notice two kinds of objects and morphisms: vertices and vertex morphisms: A i ; B i and A i f i B i edges and edge morphisms: R; S and A 0 f 0 B 0 A 1 R 6 f 1 B 1 S 6 These can be organized into two categories ....

E. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings, Ninth Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, July 1994.

....3. 3 Parametricity and Algebra Let us say that a parametric model of polymorphism is one where all elements p 2 8ff : T (ff) satisfy the relational parametricity condition For all relations R : D E, p[D] p[E] 2 T (R) This condition is stated informally here: Precise treatments include [36, 14, 25, 1, 35, 9]. Parametric models are not easy to find; the first was presented in [2] We are interested not so much in the details of a formalization of parametricity as its consequences. As a specific example, consider the type 8ff : ff ff. We can argue that there is only one parametric element p of this ....

E. P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings, 9th Annual IEEE Symposium on Logic in Computer Science, Paris, 1994. IEEE Computer Society Press, Los Alamitos, California.

....of as a parametric polymorphic function, for polymorphism over data types, while, e.g. a fixed point operator should be parametrically polymorphic over phrase types. Addendum, January, 1995. Since the work described in the body of this paper was done, there have been several developments. First, Robinson and Rosolini (1994) have given an elegant construction of parametric models of second order calculus utilizing reflexive graphs in the setting of internal category models of polymorphism. Second, Sieber (1994) has used his methods to give a full abstraction result for closed terms of up to second order type, in a ....

Robinson, E. P. and Rosolini, G. (1994). Reflexive graphs and parametric polymorphism. In (LICS, 1994).

....as a parametric polymorphic function, for polymorphism over data types, while, e.g. a fixed point operator should be parametrically polymorphic over phrase types. Addendum, January, 1995. Since the work described in the body of this paper was done, there have been several developments. First, Robinson and Rosolini (1994) have given an elegant construction of parametric models of second order calculus utilizing reflexive graphs in the setting of internal category models of polymorphism. Second, Sieber (1994) has used his methods to give a full abstraction result for closed terms of up to second order type, in a ....

Robinson, E. P. and Rosolini, G. (1994). Reflexive graphs and parametric polymorphism. In (LICS, 1994).

....R is a complete relation on cpo s. This ensures that the least fixedpoint operator exists in the model. By parametric model we mean what in [13] was referred to as the Identity Extension Lemma: A] I D = I [ A] D This is usually taken as the defining characteristic of Reynolds Parametricity [3, 14, 9], and the definition of 8 is arranged precisely to ensure the identity property (which is included as part of the Isomorphism Lemma) Lemma 1 (Isomorphism Functoriality Lemma) For any type A, the relational action of [ A] is functorial on isomorphisms. That is, i) if Rff is the graph of an ....

E. P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings, 9th Annual IEEE Symposium on Logic in Computer Science, Paris, 1994. IEEE Computer Society Press, Los Alamitos, California.

....parametricity are the following 1. Reynolds and Ma [MR92] give a categorical formulation of parametricity, in the context of indexed category models of polymorphism [See87] There is also similar work of Mitchell and Scedrov [MS92] for essentially ML style polymorphism. 2. Robinson and Rosolini [RR94] give a related development, with the definitions reworked in the setting of internal category models of polymorphism. 3. Wadler [Wad89] does the same in the context of type frame models [BMM90] and provides many useful examples of reasoning with relational parametricity (see also [Has94] 4. ....

....over, and substitution of, relations as well as types, leading to a logical expression of the relational parametricity condition. In the first three cases we have a notion of what counts as a model of polymorphism, and additional requirements that parametric models must satisfy; MR92, RR94] contain useful discussions on relationships among the three approaches. One might expect that parametric models in the Ma Reynolds or Robinson Rosolini senses would provide sound models of the logic of PlotkinAbadi, but the details of this have not appeared as of yet. The question of ....

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E. P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings of the 9th Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, 1994.

....this is made precise by presenting a logic for parametricity. In particular it is shown show that parametricity entails that the 2nd order encodings of and types are strong, i.e. have the expected categorical properties. Several authors have presented categorical versions of parametricity [MR91,RR94,Has90] The system we present here can be encoded in System F. However, we believe that it is of independent interest since inductive and coinductive definitions should be considered as more primitive than 2nd order quantification. In particular it is very straightforward to identify a predicative ....

E.P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In S. Abramsky, editor, Proc. 9th Symposium in Logic in Computer Science, pages 364--371, Paris, 1994. I.E.E.E. Computer Society.

....we do not address is polymorphism. In particular, Reynolds has emphasised the importance of the identity extension lemma in his approach to parametricity [20, 10] one might hope to analyse this axiomatically along the lines of our work here, and perhaps relate it to the logic from [16] ideas of [22] should be relevant for this) We leave this as a problem for future work. We are grateful to Edmund Robinson for discussions and feedback throughout this work. In particular, his treatment of identities and soundness in [21] though slightly different, had an influence on us here. It would be ....

.... strictly generalise those in Hoare s approach to data refinement [4, 5, 7, 8] This allows us to extend Hoare s approach to higher types in a less restrictive way than the extension using embedding projection pairs proposed in [4] Reflexive graphs have also been used by Robinson and Rosolini [22], to give a construction of parametric models of polymorphism, and by Pitts [15] in his study of logical relations and recursive domains. So Ma and Reynolds used Cartesian closed categories, while O Hearn and Tennent used reflexive graphs of categories with no additional structure. For other kinds ....

E. P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In Proceedings, 9th Annual IEEE Symposium on Logic in Computer Science, Paris, 1994. IEEE Computer Society Press, Los Alamitos, California.

....constitutes a model of the polymorphic lambda calculus, as did Ma and Reynolds [38] in 1992. Ma and Reynolds also gave an abstract definition of what constitutes a relationally parametric model. Ma and Reynolds phrased their definitions using indexed category theory; in 1994 Robinson and Rosolini [60] gave another formulation based on internal categories. All this is also described in the language of fibred category theory in a forthcoming book by Jacobs [34] For the axiomatic theory of domains, the already mentioned work by Freyd [19, 18, 20] on algebraically compact categories has been ....

....a well complete object if all the oe(T ) are well complete and thus we have a notion of impredicative polymorphism for the wellcomplete objects. For this notion of polymorphism we shall be interested in the following questions: 1. Can we obtain a parametric model in the sense of Ma and Reynolds [38, 60] Making use of the fact that the well complete objects form an internal category in an ambient category with good closure properties we may be able to use the approach of Robinson and Rosolini [60] to get a parametric model. This will be very useful for programming language semantics, ....

[Article contains additional citation context not shown here]

E.P. Robinson and G. Rosolini. Reflexive graphs and parametric polymorphism. In S. Abramsky, editor, Proc. 9th Symposium in Logic in Computer Science, pages 364--371, Paris, 1994. I.E.E.E. Computer Society.

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