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Topology and the Semantics of Logic Programs
 Fundamenta Informaticae
, 1995
"... We introduce topologies on spaces of interpretations which extend and generalise the query and positive query topologies of Batarekh and Subrahmanian. We study continuity in these topologies of the immediate consequence map associated with any normal logic program and provide necessary and sufficien ..."
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Cited by 41 (28 self)
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We introduce topologies on spaces of interpretations which extend and generalise the query and positive query topologies of Batarekh and Subrahmanian. We study continuity in these topologies of the immediate consequence map associated with any normal logic program and provide necessary and sufficient conditions for continuity to hold. We relate these ideas to (i) computational power of programs; (ii) canonical programs; (iii) the declarative semantics of definite programs; (iv) Maher's down continuous programs; (v) the decency thesis of Jaffar, Lassez and Maher, and (vi) compactness of sets of fixed points and models. 1 Introduction The classical approach to the study of fixed point semantics of definite logic programs P utilises two main facts: (1) The set I J L of all interpretations of the underlying language L based on a given preinterpretation J forms a complete lattice under the partial order of set inclusion. (2) The immediate consequence map T P defined on I J L is latti...
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
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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.
Uniformities on Free Semigroups
 Intern. J. Algebra and Computation
, 1999
"... . It is known that the profinite completions of a free semigroup which are associated with a pseudovariety of semigroups or of ordered semigroups, can be defined by an 'ecart or a quasi'ecart. We characterize those quasi'ecarts and those quasiuniformities which arise in this fashion. We also prov ..."
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Cited by 6 (5 self)
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. It is known that the profinite completions of a free semigroup which are associated with a pseudovariety of semigroups or of ordered semigroups, can be defined by an 'ecart or a quasi'ecart. We characterize those quasi'ecarts and those quasiuniformities which arise in this fashion. We also prove an Eilenberglike onetoone correspondence between pseudovarieties of ordered semigroups and socalled varieties of quasiuniformities. 1 Introduction Let A + be a free semigroup. If V is a pseudovariety of finite semigroups, the profinite topology on A + associated with V (see [1, 2]) is defined by a metric d V , as soon as it is Hausdorff. Now the question arises to know which metrics are obtained in this way. We show that a metric on A + is uniformly equivalent with a metric of the form d V if and only if it satisfies the following properties (Theorem 5.10): (1) it is ultrametric, (2) A + is precompact, (3) concatenation is uniformly continuous, (4) morphisms from A + into A...
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].
Fuzzy Orders in Approximate Reasoning
 Zollo G. (Ed), New Logics for the New Economy, Edizioni Scientifiche Italiane
"... An approach to fixed point theory based on the notion of fuzzy order is proposed. Such an approach extends both the fixed point theory in ordered sets and the fixed point theory in metric spaces. This since the fuzzy orders are strictly related with the quasimetrics as defined by A. K. Seda in [10] ..."
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Cited by 1 (0 self)
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An approach to fixed point theory based on the notion of fuzzy order is proposed. Such an approach extends both the fixed point theory in ordered sets and the fixed point theory in metric spaces. This since the fuzzy orders are strictly related with the quasimetrics as defined by A. K. Seda in [10]. The aim is to give new tools for logic programming and for approximate reasoning.
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...
The essence of ideal completion in quantitative form (Extended Abstract)
, 1995
"... 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 comple ..."
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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 Scottcontinuous 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...
Elements of generalized . . .
, 1996
"... Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enrichedcategorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and ..."
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Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enrichedcategorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations that combines and extends the standard ordertheoretic (Smyth and Plotkin, 1982) and metric (America and Rutten, 1989) approaches.