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Sequents, Frames, and Completeness
"... . Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the ..."
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. Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms. Key words: sequent structures, lattices, frames, domain theory, resolution, category. Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [1315]. It can be seen as a generalisation of the ear...
Morphisms in Logic, Topology, and Formal Concept Analysis
, 2005
"... The general topic of this thesis is the investigation of various notions of morphisms between logical deductive systems, motivated by the intuition that additional (categorical) structure is needed to model the interrelations of formal specifications. This general task necessarily involves considera ..."
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The general topic of this thesis is the investigation of various notions of morphisms between logical deductive systems, motivated by the intuition that additional (categorical) structure is needed to model the interrelations of formal specifications. This general task necessarily involves considerations in various mathematical disciplines, some of which might be interesting in their own right and which can be read independently.
Information Systems Revisited: The General Continuous Case∗
"... In this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logicoriented style as the information systems first introduced by Scott in 1982: the ..."
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In this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logicoriented style as the information systems first introduced by Scott in 1982: they consist of a set of tokens, a consistency predicate and an entailment relation satisfying a set of natural axioms. It is shown that continuous information systems are closely related to abstract bases. bases and/or continuous domains are equivalent, it follows that the category of continuous information systems is also equivalent to that of continuous domains. In applications mostly subclasses of continuous domains are considered. The domains have e.g. to be pointed, algebraic, boundedcomplete or FS. Conditions are presented that when fulfilled by an continuous information system force the generated domain to belong to the required subclass. In most cases the requirements are not only sufficient but also necessary. 1
A Logical Approach to Stable Domains
, 2006
"... Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunctions in the ..."
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Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic Ldomains.
IOS Press A Categorical View on Algebraic Lattices in Formal Concept Analysis
"... Abstract. Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or ..."
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Abstract. Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a categorytheoretical perspective. To this
A Logical Approach to Stable Domains
, 2006
"... Abstract Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunction ..."
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Abstract Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic Ldomains. Keywords: Disjunctive propositional logic, domain theory, information system, Ldomain, domain theory in logical form.