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Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
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On the duality of compact vs. open
 Papers on General Topology and Applications: Eleventh Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences
, 1996
"... It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We presen ..."
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Cited by 12 (1 self)
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It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We present a similar approach to describe the Stoneduals of coherent spaces, thus dropping the requirement of having a base of compactopens (or, alternatively, replacing algebraicity of the lattices by continuity). The construction via strong proximity lattices is resembling the classical case, just replacing the order by an order of approximation. Our development enlightens the fact that “open ” and “compact ” are dual concepts which merely happen to coincide in the classical case.
Stably Compact Spaces and Closed Relations
, 2001
"... Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and stud ..."
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Cited by 11 (2 self)
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Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and study the resulting categorical properties. Apart from extending ordinary continuous maps, these morphisms have a number of pleasing properties, the most prominent, perhaps, being that they correspond to preframe homomorphisms on the localic side. We exploit this Stonetype duality to establish that the category of stably compact spaces and closed relations has bilimits.
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...
Types, Logic, and Semantics for Nested Databases
 In: Proc. of the Conference on Mathematical Foundations of Programming Semantics95, Electronic
, 1995
"... This work presents first steps towards a denotational semantics for relational databases. It is argued that such a semantics will increase the chances of successfully incorporating relational databases into typed programming languages. Database relations are seen as sets of data of a common structur ..."
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Cited by 3 (2 self)
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This work presents first steps towards a denotational semantics for relational databases. It is argued that such a semantics will increase the chances of successfully incorporating relational databases into typed programming languages. Database relations are seen as sets of data of a common structure. The main problem therefore is to model a type of sets. We propose the snack powerdomain for this purpose. Technically, the paper attempts to clarify two aspects of the domain theoretic background of this approach. We give a localic description of the snack powerdomain construction which reveals its logical simplicity. Second, we study a subdomain relation between Scottdomains on the denotational and the logical level. Again, the logical version is simple and intuitive. Such a relation is indispensable for introducing database operations such as `natural join'. 1 Introduction The theory of relational databases (see [16,11] for surveys) is highly developed and proves its usefulness in pra...
An InformationSystem Representation Of The Smyth Powerdomain
 International Symposium on Domain Theory
, 1999
"... . This paper provides a representation of the Smyth powerdomain as information systems. A new notion of ideal elements, called disjunctive states, is introduced. Disjunctive states are built from clauses over the token set of the underlying information system in order to represent disjunctive in ..."
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. This paper provides a representation of the Smyth powerdomain as information systems. A new notion of ideal elements, called disjunctive states, is introduced. Disjunctive states are built from clauses over the token set of the underlying information system in order to represent disjunctive information. At the heart of this representation is a hyperresolution rule, whose completeness hinges upon a crucial combinatorial lemma for the proper bookkeeping of intermediate clauses. Our main representation result uses the HoffmanMislove Theorem to establish an orderisomorphism between disjunctive states and compact, saturated sets. As an application, we provide a couple of specific Smyth powerdomain examples that are useful for disjunctive logic programming. Moreover, the notion of disjunctive state is immediately applicable to sequent structures, or nondeterministic information systems. We show that the hyperresolution rule is sound and complete for sequent structures as we...
JOINS IN THE FRAME OF NUCLEI
"... Abstract. Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixedpoint ..."
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Abstract. Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixedpoint theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the technique, we offer a simple (and also intuitionistic) proof of the localic Hofmann–Mislove Theorem. 1.
ReGrouping Information in a Domain Theoretic Data Model
, 1995
"... This paper shows how nested database relations can be represented in the domain theoretic database model and operations nest and unnest can be defined within that model such that unnest ffi nest = id and nest ffi unnest id. We thus get a formal semantics of nested databases as well as operations fo ..."
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This paper shows how nested database relations can be represented in the domain theoretic database model and operations nest and unnest can be defined within that model such that unnest ffi nest = id and nest ffi unnest id. We thus get a formal semantics of nested databases as well as operations for nesting and unnesting which keep as much information as possible and therefore build the ideal passage between relations of different nesting levels. The organisation of the paper is as follows: In Section 1 we give a short introduction to nested relational databases. Section 2 illustrates the domain theoretic approach. In particular, the snackpowerdomain which models settypes is defined, and the interpretation of database relations in the new setting is given. We begin Section 3 with giving an intuitive, pictureoriented account of nested snacks (these will model nested relations). After that we turn to defining nest and unnest and proving the result claimed above. Finally we show how the snack representation of a nested relation behaves under unnest and nest in Section 4. 2 Nested Relational Databases