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Continuous Domains in Formal Concept Analysis

Wang Longchun, Guo Lankun, Li Qingguo

Fundamenta Informaticae

2021

Vol. 179, nr 3

295--319

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Hitzler P., Krötzsch M., Zhang G-Q.

Fundamenta Informaticae

2006

Vol. 74, nr 2,3

301-328

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 category-theoretical perspective. To this end, we build on the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.

Multi lingual sequent calculus and coherent spaces

Jung A., Kegelmann M.

Fundamenta Informaticae

1999

Vol. 37, Nr 4

369-412

We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalized in this fashion. The usual cut rule does not make sense for sequents which connect different logical systems because it mixes formulas from antecedent and succedent. We propose a different cut rule which addresses this problem. The new cut rule can be used as a basis for composition in a suitable category of logical sys-tems. As it turns out, this category is equivalent to coherent spaces with certain relations be-tween them. Finally, cut elimination in this set-up can be employed to provide a new explanation of the domain constructions in Samson Abramsky's Domain Theory in Logical Form.