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In this paper, we survey the use of order-theoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of order-theoretic 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...

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 ffl-adjoint 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 pseudo-metric spaces, and it is shown how to solve domain equations in a non-categorical framework.

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 net-version of the Yoneda completion, complementing the net-version 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 quasi-metric spaces idempotent under the Yoneda completion is precisely the class of Smyth-completable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]

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 ffl-ball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the ffl-ball topology for metric spaces. AMS subject classification (1991): 68Q10, 68Q55 Keywords: generalized metric, preorder, metric, Alexandroff topology, Scott topology, ffl-ball 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...

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].

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 ffl-ball topology; 2. a topology for algebraic complete generalized ultrametric spaces generalizing both the Scott topology for arbitrary algebraic complete partial orders and the ffl-ball 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...

It is well known that one can use an adaptation of the inverse-limit 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 metric-space 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 domain-theoretic work by Smyth and Plotkin. Our primary motivation for solving generalized recursive metric-space equations comes from recent and ongoing work on Kripke-style 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

this article is twofold. From a mathematical perspective we present a notion of convergence which is suitably general such as to include the convergence of chains to their least upper bounds in preordered sets, and the convergence of Cauchy sequences to their metric limits in metric spaces. Rather than presenting this theory from a purely mathematical perspective however, we will use it to introduce a simple-minded domain theory based on a generic notion of approximation. We might hope that this is not the only use of the concepts we present, although it is the one that motivated us in the first place. One possible kind of approximation one uses in domain theory as it is used in the study of denotational semantics of programming languages is the binary one that we have in preorders: either an element is below another in the preordering, or it is not. Another kind of approximation is the metric one, where we do not just say whether one element approximates another, but to which degree it does so, with a non-negative real number. It turns out that we can separate out from a large part of domain theory considerations about a particular notion of approximation, and just state a few axioms that should hold about a notion of approximation. In this general theory, which encompasses preorders and metric spaces among many other kinds of structures, we can then do general domain theory. The requirements for our notion of approximation turns out to be that of a quite well-known mathematical structure, viz. that of a commutative, unital quantale. One such quantale is the two point lattice, which gives rise to the theory of preorders, and another is that of the non-negative real numbers, turned up-side down, giving rise to generalized metric spaces. The advantage of this separation ...

Robert C. Flagg and Philipp Sunderhauf y University of Southern Maine fflagg,psunderg@usm.maine.edu December 12, 1995 Abstract If a posets lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with Scott-continuous functions as morphisms. The same applies for the quantitative version of ideal completion suggested in the literature. As in the case of posets, it seems advantageous to consider a different topology with the completed spaces. We introduce Smyth completion as tool to automatically end up with the right topology after completing. 1 Introduction This paper is part of the ongoing foundational work on quantitative domain theory [Smy88, BBR95, Rut95, FW95, Wag94], which refines ordinary do- Supported by the Deutsche Forschungsgemeinschaf...