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26
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
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Generalized Metrics and Uniquely Determined Logic Programs
 Theoretical Computer Science
"... The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard ..."
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Cited by 27 (16 self)
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The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard approaches to denotational semantics then become inapplicable. In this paper, we show how generalized metric spaces can be used to obtain fixedpoint semantics for several classes of programs relative to the supported model semantics, and investigate relationships between the underlying spaces we employ. Our methods allow the analysis of classes of programs which include the acyclic, locally hierarchical, and acceptable programs, amongst others, and draw on fixedpoint theorems which apply to generalized ultrametric spaces and to partial metric spaces.
Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding
, 1996
"... Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdo ..."
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Cited by 23 (3 self)
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Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the fflball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.
Generalized Ultrametric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding
, 1995
"... Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion ..."
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Cited by 15 (5 self)
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Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special cases of preorders and ordinary ultrametric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the fflball topology; 3. lower, upper, and convex powerdomains, and the powerdomain of compact subsets. Interestingly, all constructions are formulated in terms of (an ultrametric version of) the Yoneda (1954) lemma.
Induction And Recursion On The Real Line
"... We characterize the real line by properties similar to the socalled Peano axioms for natural numbers. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, s ..."
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Cited by 11 (9 self)
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We characterize the real line by properties similar to the socalled Peano axioms for natural numbers. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. In order to obtain such a characterization, we introduce a notion of infinitely iterated composition of morphisms in categories, and we state a fixed point theorem and an infinite composition theorem for uniform spaces. 1 Introduction We characterize the real line by properties similar to the socalled Peano axioms for natural numbers [10, 11, 19]. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. 1.1 Programme We begin by characterizing the unit interval I = [0; 1]. F...
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]
Some Issues Concerning Fixed Points in Computational Logic: QuasiMetrics, Multivalued Mappings and the KnasterTarski Theorem
, 2000
"... Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of ..."
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Cited by 8 (7 self)
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Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of the Banach contraction mapping theorem, and formulate a version of the classical fixedpoint theorem (sometimes attributed to Kleene) which is new. All these results have applications to the semantics of disjunctive logic programs, and we will describe a class of programs to which the new theorem can be applied. We also show that a unification of the latter two theorems is possible, using quasimetrics, which parallels the wellknown unification of Rutten and Smyth in the case of conventional programming language semantics.
Multivalued Mappings, FixedPoint Theorems and Disjunctive Databases
 Eds.) Proc. 3rd Irish Workshop on Formal Methods (IWFM'99), Electronic Workshops in Computing (eWiC), British Computer Society
, 1999
"... In this paper, we discuss the semantics of disjunctive programs and databases and show how multivalued mappings and their fixed points arise naturally within this context. A number of fixedpoint theorems for multivalued mappings are considered, some of which are already known and some of which are ..."
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Cited by 7 (5 self)
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In this paper, we discuss the semantics of disjunctive programs and databases and show how multivalued mappings and their fixed points arise naturally within this context. A number of fixedpoint theorems for multivalued mappings are considered, some of which are already known and some of which are new. The notion of a normal derivative of a disjunctive program is introduced. Normal derivatives are normal logic programs which are determined by the disjunctive program. Thus, the wellknown singlestep operator associated with a normal derivative is singlevalued, and its fixed points can be found by wellestablished means. It is shown how fixed points of the multivalued mapping determined by a disjunctive program relate to the fixed points of the singlestep operators coming from its normal derivatives. This procedure has potential for simplifying the construction of models of disjunctive databases, and this point is discussed. Most of the results for multivalued mappings rest on corres...
On upper weightable spaces
 Proc. 11th Summer Conference on General Topology and
, 1996
"... The weightable quasipseudometric spaces have been introduced by Matthews as part of the study of the denotational semantics of dataflow networks (e.g. [Mat92] and [Mat92a]). The study of these spaces has been continued in the context of Nonsymmetric Topology by Kunzi and Vajner ([KV93] and [Kün93] ..."
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Cited by 5 (5 self)
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The weightable quasipseudometric spaces have been introduced by Matthews as part of the study of the denotational semantics of dataflow networks (e.g. [Mat92] and [Mat92a]). The study of these spaces has been continued in the context of Nonsymmetric Topology by Kunzi and Vajner ([KV93] and [Kün93]). We introduce and motivate the class of upper weightable quasipseudometric spaces. The relationship with the development of a topological foundation for the complexity analysis of programs ([Sch95]) is discussed, which leads to the study of the weightable optimal (quasipseudometric) join semilattices.
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...