M. Hyland, E. Robinson, and G. Rosolini. Algebraic types in per models. In Proc. Mathematical Foundations of Programming Semantics, volume 442 of LNCS. Springer-Verlag, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Lim( 1 ( 58 A rst attempt at producing the parametric limit of an indexed functor F : G H was to use the intersection R F (R) of saturated relations. There have been some results to suggest that this intersection R F (R) might be a PER rather than a more general relation [Has91, HRR90] Lacking a proof R F (R) is a PER, the parametric limit of F is given as the largest PER contained in R F (R) 8 edges R: n It is not hard to see that every PER contained in R F (R) is contained in Lim(F ) The relation Lim(F ) on IN is transitive, since there is the ....

M. Hyland, E. Robinson, and G. Rosolini. Algebraic types in per models. In Proc. Mathematical Foundations of Programming Semantics, volume 442 of LNCS. Springer-Verlag, 1990.

....between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91] to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89]; and to the completeness conjecture in [Gir91a] 3 We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear Logic. This refinement is not a complication; on the contrary, it makes the definitions smoother and more symmetric. Thus, we get a ....

J. M. E. Hyland, E. P. Robinson, and G. Rosolini. Algebraic types in per models. In Fifth Conference on Mathematical Foundations in Programming Semantics, pages 333--350. Springer-Verlag, 1989. Lecture Notes in Computer Science 442.

.... precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in programming language semantics [Mil75, Plo77] Plotkin s characterization of de nability in the calculus using logical relations [Plo80] studies of parametric polymorphism [BFSS90, HRR89], and the completeness conjecture in [Gir91] However, the contribution of [AJ92b] was to clearly identify and formulate this issue as a precise and interesting research programme, and to prove the rst in what has become quite a rich sequence of results. 28 Geometric proofs via proof nets. ....

M. Hyland, E. P. Robinson and G. Rosolini. Algebraic types in PER models. In Proceedings of the Fifth Conference on Mathematical Foundations of Programming Semantics, Springer LNCS Vol. 442, pp. 333-350, 1989.

.... precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in programming language semantics [Mil75, Plo77] Plotkin s characterization of de nability in the calculus using logical relations [Plo80] studies of parametric polymorphism [BFSS90, HRR89], and the completeness conjecture in [Gir91] However, the contribution of [AJ92b] was to clearly identify and formulate this issue as a precise and interesting research programme, and to prove the rst in what has become quite a rich sequence of results. 28 Geometric proofs via proof nets. ....

M. Hyland, E. P. Robinson and G. Rosolini. Algebraic types in PER models. In Proceedings of the Fifth Conference on Mathematical Foundations of Programming Semantics, Springer LNCS Vol. 442, pp. 333-350, 1989.

....between the two categories, are called adjoint models, following [Bie95, BW96] The problem of full completeness for second order (polymorphic) # calculus, i.e. Girard s system F ( Gir72] is a very important problem, which has been extensively studied. Here are some results in the literature. In [HRR90], the category of Partial Equivalence Relations (PER) over the open term model of the untyped # calculus has been proved to be fully (and faithfully) complete for algebraic types. These are ML types of rank less or equal than 2, like for instance the type #X. X # X) # X # X of Church s ....

....A logical relation technique relating the term algebra and the term subalgebra of partial involutions could be useful here. The interest of linear term algebras lies in the fact that the PER model generated by these is essentially the PER model shown to be fully complete at algebraic types in [HRR90]. We have presented a linear realizability technique for building PER categories over an LCA. These PER categories turn out to be linear categories. It would be interesting to carry on the investigation of the general properties of these categories, e.g. define coproducts, products, etc. 36 ....

J.Hyland, E.Robinson, G.Rosolini. Algebraic types in PER models, MFPS Conf. Proc., M.Main et al. eds, LNCS 442, 1990, 333--350. 38

....between the two categories, are called adjoint models, following [Bie95, BW96] The problem of full completeness for second order (polymorphic) calculus, i.e. Girard s system F ( Gir72] is a very important problem, which has been extensively studied. Here are some results in the literature. In [HRR90], the category of Partial Equivalence Relations (PER) over the open term model of the untyped calculus has been proved to be fully (and faithfully) complete for algebraic types. These are ML types of rank less or equal than 2, like for instance the type 8X: X X) X X of Church s numerals. ....

....A logical relation technique relating the term algebra and the term subalgebra of partial involutions could be useful here. The interest of linear term algebras lies in the fact that the PER model generated by these is essentially the PER model shown to be fully complete at algebraic types in [HRR90]. ffl We have presented a linear realizability technique for building PER categories over an LCA. These PER categories turn out to be linear categories. It would be interesting to carry on the investigation of the general properties of these categories, e.g. define coproducts, products, etc. ....

J.Hyland, E.Robinson, G.Rosolini. Algebraic types in PER models, MFPS Conf. Proc., M.Main et al. eds, LNCS 442, 1990, 333--350. 38

....relationship between polymorphism on one hand and functional languages on the other. Perhaps even more fascinating is the question of parametricity where one s goal is to find models in which the interpretations of the encodings are initial among the models of the corresponding structures. See [5, 12, 18, 6] for some results on the topic. There has also been attempt to carry out the similar construction in dependent calculi [3, 17] But so far there has been no good result. We are not sure to what extent this is due to our ignorance of the free structures of generalized algebras [2] From another ....

M. Hyland, E. Robinson, and G. Rosolini. Algebraic types in per models. In M. Main et al, editor, Mathematical Foundations of Programming Semantics, LNCS 442, pages 333--350. Springer-Verlag, 1990.

....together with a monoidal adjunction between the two categories, are called adjoint models, following [Bie95,BW96] The problem of full completeness for second order (polymorphic) calculus, i.e. Girard s system F ( Gir72] is a very important problem, which has been extensively studied. In [HRR90], the category of Partial Equivalence Relations (PER) over the open term model of the untyped calculus has been proved to be fully (and faithfully) complete for algebraic types, a small subclass of MLtypes. ML types are universal closures of simple types, i.e. types of the form 8X 1 : Xn :T ....

J.Hyland, E.Robinson, G.Rosolini. Algebraic types in PER models, MFPS Conf. Proc., M.Main et al. eds, LNCS 442, 1990, 333--350.

.... Lemma, along the lines of [AJM96] The full completeness result obtained in this paper is interesting, since, until now, the research on full completeness for System F has produced fully complete denotational models only for a small subclass of ML types, i.e. the algebraic types (see [HRR90]) In the literature, there are two fully complete models for the whole system F: i.e. that of [BC88] and that of [Hug99] The first model is based on a quotient of a term model, the latter is a game model. But both these models still have a somewhat syntactical flavor, and their constructions ....

....A logical relation technique relating the term algebra and the term subalgebra of partial involutions could be useful here. The interest of linear term algebras lies in the fact that the PER model generated by these is essentially the PER model shown to be fully complete at algebraic types in [HRR90]. ffl We have presented a linear realizability technique for building PER categories over an LCA. These PER categories turn out to be linear categories. It would be interesting to carry on the investigation of the general properties of these categories, e.g. define coproducts, products, etc. ....

J.Hyland, E.Robinson, G.Rosolini. Algebraic types in PER models, MFPS, M.Main et al. eds, LNCS 442, 1990, 333--350.

....between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91] to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89]; and to the completeness conjecture in [Gir91a] We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear Logic. This refinement is not a complication; on the contrary, it makes the definitions smoother and more symmetric. Thus, we get a ....

J. M. E. Hyland, E. P. Robinson, and G. Rosolini. Algebraic types in per models. In Fifth Conference on Mathematical Foundations in Programming Semantics, pages 333-- 350. Springer-Verlag, 1989. Lecture Notes in Computer Science 442.

....between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91] to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89]; and to the completeness conjecture in [Gir91a] We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear Logic. This refinement is not a complication; on the contrary, it makes the definitions smoother and more symmetric. Thus, we get a ....

J. M. E. Hyland, E. P. Robinson, and G. Rosolini. Algebraic types in per models. In Fifth Conference on Mathematical Foundations in Programming Semantics, pages 333--350. Springer-Verlag, 1989. Lecture Notes in Computer Science 442.

....ideas gives us good uniformity conditions for reasoning about local variables. We will be working with categories equipped with a notion of realizability. These structures can be viewed elegantly as internal categories in the effective topos, or in the category of sets (see (Hyland, 1988; Hyland et al. 1989; Longo and Moggi, 1991) To simplify the presentation we will keep internal category aspects of the model in the background (though this viewpoint certainly guides the definitions) We use m Delta n to denote Kleene application on , the natural numbers (i.e. the application of the m th ....

....m Delta hexpand ; h A Delta r Gamma1 i Delta a)i Delta Lemma 13 and Theorem 14 can then be used to show that j codes a well defined map and that it is an isomorphism whose inverse is realized by k where k Delta p Delta a = p Delta hexpand ; ai. Using known facts about PER models ((Hyland et al. 1989; Freyd et al. 1992a) we immediately obtain that, for example, comm comm] 1 is isomorphic to the PER N that relates each natural number to itself. We do not know if this theorem goes through for the Moggi Hyland interpretation of polymorphism with PERs. It does whenever A is a product of Algol ....

Hyland, J., Robinson, E. P., and Rosolini, G. (1989). Algebraic types in PER models. In Main, M. et al., editors, Mathematical Foundations of Programming Semantics, volume 442 of Lecture Notes in Computer Science, pages 333--350, Berlin. Springer-Verlag. Proceedings of the 1989 Conference.

.... settheoretic models exist to satisfy a completeness theorem (in stark contrast to the classical case) and ffl the study by Hyland, Edmund Robinson and Pino Rosolini elucidating subtle completeness properties of internal categories in the effective topos and other categories related to it [HRR90] also, Hyl82, Rob89, FRR92b] Work on realizability models has flourished. Specific directions include algebraic characterizations of low order types, work on recursion, and on subtypes; just a few examples are [HRR88, BFSS90, Ama89, FMRS90, BL90, Mit88] Other pointers to work on ....

....alternative formalizations. For example: ffl Is the PER model relationally parametric Is the trimmed down model of [BFSS90] any different than the standard PER model Answering this would be a first step to relating relational parametricity and Strachey parametricity. Some partial results are [HRR90, FRR92a] ffl For what n and m are n ary and m ary relational parametricity different Similarly for Kripke relations [Plo80, JT93] Most of the discussion so far has avoided fixed points. Certainly relational parametricity and a polymorphic fixed point operator Y : 8ff : ff ff) ff can ....

J. M. E. Hyland, E. P. Robinson, and G. Rosolini. Algebraic types in PER models. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Mathematical Foundations of Programming Semantics, volume 442 of Lecture Notes in Computer Science, pages 333--350. Springer Verlag, 1990.

....investigate forms of parametricity. More is known in the context of these models than in any other, but that is still not very much. We have known for some time that the interpretations of some of the algebraic types in the classical Per model of the natural numbers were the initial algebras (see [HRR90a, FRR92a]) This has been proposed as a test of the degree of parametricity of a model. However, the proofs at least seem to depend on the particular realizability algebra on which the model is based (natural numbers with Kleene application) This raises the unpleasant possibility that the degree of ....

J.M.E. Hyland, E.P. Robinson, and G. Rosolini. Algebraic types in PER models. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science 442, pages 333--350. Springer-Verlag, 1990.

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J. M. E. Hyland, E. Robinson, G. Rosolini. Algebraic types in PER models, MFPS Conf. Proc., M.Main et al. eds, LNCS 442, 1990, 333-350.

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