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15
Realizability semantics of parametric polymorphism, general references, and recursive types
, 2010
"... Abstract. We present a realizability model for a callbyvalue, higherorder programming language with parametric polymorphism, general firstclass references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include gener ..."
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Cited by 19 (12 self)
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Abstract. We present a realizability model for a callbyvalue, higherorder programming language with parametric polymorphism, general firstclass references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include general reference types. The interpretation uses a new approach to modeling references. The universe of semantic types consists of worldindexed families of logical relations over a universal predomain. In order to model general reference types, worlds are finite maps from locations to semantic types: this introduces a circularity between semantic types and worlds that precludes a direct definition of either. Our solution is to solve a recursive equation in an appropriate category of metric spaces. In effect, types are interpreted using a Kripke logical relation over a recursively defined set of worlds. We illustrate how the model can be used to prove simple equivalences between different implementations of imperative abstract data types. 1
Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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Cited by 10 (2 self)
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
On the Yoneda completion of a quasimetric space
 Theoretical Computer Science
, 2002
"... Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR ..."
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Cited by 8 (4 self)
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Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98], which finds its roots in work by Lawvere ([Law73], cf. also [Wag94]) and which is related to early work by Stoltenberg (e.g. [Sto67], [Sto67a] and [FG84]), and the Smyth completion ([Smy89],[Smy91],[Smy94],[Sun93] and [Sun95]). A netversion of the Yoneda completion, complementing the netversion of the Smyth completion ([Sun95]), is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98]: "An interesting question is to characterize the family of generalized metric spaces for which [the Yoneda] completion is idempotent (it contains at least all ordinary metric spaces)." We show that the largest class of quasimetric spaces idempotent under the Yoneda completion is precisely the class of Smythcompletable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]
Solutions of Functorial and NonFunctorial Metric Domain Equations
, 1995
"... A new method for solving domain equations in categories of metric spaces is studied. The categories CMS ß and KMS ß are introduced, having complete and compact metric spaces as objects and ffladjoint pairs as arrows. The existence and uniqueness of fixed points for certain endofunctors on these ..."
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Cited by 7 (3 self)
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A new method for solving domain equations in categories of metric spaces is studied. The categories CMS ß and KMS ß are introduced, having complete and compact metric spaces as objects and ffladjoint pairs as arrows. The existence and uniqueness of fixed points for certain endofunctors on these categories is established. The classes of complete and compact metric spaces are considered as pseudometric spaces, and it is shown how to solve domain equations in a noncategorical framework.
Alexandroff and Scott Topologies for Generalized Metric Spaces
 Proceedings of the 11th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences
"... Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a to ..."
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Cited by 4 (1 self)
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Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a topology for generalized metric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary metric spaces reduces to the fflball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the fflball topology for metric spaces. AMS subject classification (1991): 68Q10, 68Q55 Keywords: generalized metric, preorder, metric, Alexandroff topology, Scott topology, fflball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been develop...
The Essence of Ideal Completion in Quantitative Form
 GHK
, 1996
"... This paper is part of the ongoing foundational work on quantitative domain theory [Smy88,BvBR95,Rut96,FWS96,Sun94,Wag94], which refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of degree of approximation (cf. the introduction of [FWS96]). ..."
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Cited by 3 (0 self)
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This paper is part of the ongoing foundational work on quantitative domain theory [Smy88,BvBR95,Rut96,FWS96,Sun94,Wag94], which refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of degree of approximation (cf. the introduction of [FWS96]). We investigate the generalization of ideal completion of posets for quantitative domains suggested in [BvBR95] and [FWS96].
Alexandroff and Scott Topologies for Generalized Ultrametric Spaces
, 1995
"... Both preorders and ordinary ultrametric spaces are instances of generalized ultrametric spaces. Every generalized ultrametric space can be isometrically embedded in a (complete) function space by means of an ultrametric version of the categorical Yoneda Lemma. This simple fact gives naturally ris ..."
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Cited by 1 (0 self)
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Both preorders and ordinary ultrametric spaces are instances of generalized ultrametric spaces. Every generalized ultrametric space can be isometrically embedded in a (complete) function space by means of an ultrametric version of the categorical Yoneda Lemma. This simple fact gives naturally rise to: 1. a topology for generalized ultrametric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary ultrametric spaces reduces to the fflball topology; 2. a topology for algebraic complete generalized ultrametric spaces generalizing both the Scott topology for arbitrary algebraic complete partial orders and the fflball topology for complete ultrametric spaces. 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (cf., e.g., the recent textbooks [Win93] and [BV95]). Parts of their theory have been developed because of semantic necessity (see, e.g., [SP82] and [AR89]). Generalized ultram...
CATEGORIES ENRICHED OVER A QUANTALOID: ISBELL ADJUNCTIONS AND KAN ADJUNCTIONS
"... Abstract. Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in ..."
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Cited by 1 (0 self)
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Abstract. Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Qcategories factors through both of these functors. 1.
Solutions of Generalized Recursive MetricSpace Equations
"... It is well known that one can use an adaptation of the inverselimit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metricspace structure on each set of morphisms: the exact nat ..."
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Cited by 1 (1 self)
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It is well known that one can use an adaptation of the inverselimit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metricspace structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domaintheoretic work by Smyth and Plotkin. Our primary motivation for solving generalized recursive metricspace equations comes from recent and ongoing work on Kripkestyle models in which the sets of worlds must be recursively defined. For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category. As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. 1