Abstract. This paper unifies and/or generalizes several approaches to information, including the information flow of Barwise and Seligman, the formal conceptual analysis of Wille, the lattice of theories of Sowa, the categorical general systems theory of Goguen, and the cognitive semantic theories of Fauconnier, Turner, Gärdenfors, and others. Its rigorous approach uses category theory to achieve independence from any particular choice of representation, and institutions to achieve independence from any particular choice of logic. Corelations and colimits provide a general formalization of information integration, and Grothendieck constructions extend this to several kinds of heterogeneity. Applications include modular programming, Curry-Howard isomorphism, database semantics, ontology alignment, cognitive semantics, and more. 1

Algebraic work [9] shows that the deep theory of possible world semantics is available in the more general setting of substructural logics, at least in an algebraic guise. The question is whether it is also available in a relational form. This article seeks to set the stage for answering this question. Guided by the algebraic theory, but purely relationally we introduce a new type of frames. These structures generalize Kripke structures but are two-sorted, containing both worlds and co-worlds. These latter points may be viewed as modelling irreducible increases in information where worlds model irreducible decreases in information. Based on these structures, a purely model theoretic and uniform account of completeness for the implication-fusion fragment of various substructural logics is given. Completeness is obtained via a generalization of the standard canonical model construction in combination with correspondence results. 1

Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may o#er connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in [25].

Because botanical taxonomies are prototypical classifications it would seem that it should be easy to formalize them as concept lattices or type hierarchies.

Song and Bruza [6] introduce a framework for Information Retrieval(IR) based on Gardenfor’s three tiered cognitive model; Conceptual Spaces[4]. They instantiate a conceptual space using Hyperspace Analogue to Language (HAL)[3] to generate higher order concepts which are later used for adhoc retrieval. In this poster, we propose an alternative implementation of the conceptual space by using a probabilistic HAL space (pHAL). To evaluate whether converting to such an implementation is beneficial we have performed an initial investigation comparing the concept combination of HAL against pHAL for the task of query expansion. Our experiments indicate that pHAL outperforms the original HAL method and that better query term selection methods can improve performance on both HAL and pHAL.

this paper we propose a semiotic conceptual framework which is compatible with Peirce's definition of signs and uses formal concept analysis for its conceptual structures. The goal of our research is to improve the use of formal languages such as ontology languages and programming languages. Even though there exist a myriad of theories, models and implementations of formal languages, in practice it is often not clear which strategies to use. AI ontology language research is in danger of repeating mistakes that have already been studied in other disciplines (such as linguistics and library science) years ago

Dealing with conceptual hierarchies is a fundamental task when working with conceptual structures. In this paper we present a projection of the implications underlying such a hierarchy into the n- dimensional space. This vector representation of the structure allows more complex mathematical treatment of the conceptual hierarchy.

The focus of research on representing and reasoning with knowledge traditionally has been on single specifications and appropriate inference paradigms to draw conclusions from such data. Accordingly, this is also an essential aspect of ontology research which has received much attention in recent years. But ontologies introduce another new challenge based on the distributed nature of most of their applications, which requires to relate heterogeneous ontological specifications and to integrate information from multiple sources. These problems have of course been recognized, but many current approaches still lack the deep formal backgrounds on which todays reasoning paradigms are already founded. Here we propose category theory as a well-explored and very extensive mathematical foundation for modelling distributed knowledge. A particular prospect is to derive conclusions from the structure of those distributed knowledge bases, as it is for example needed when

Abstract: A system of a number of relatively stable units that can combine more or less freely to form somewhat less stable structures has a capacity to carry information in a more or less arbitrary way. I call such a system a physical information system if its properties are dynamically specified. All physical information systems have certain general dynamical properties. DNA can form such a system, but so can, to a lesser degree, RNA, proteins, cells and cellular subsystems, various immune system elements, organisms in populations and in ecosystems, as well as other higher-level phenomena. These systems are hierarchical structures with respect to the expression of lower level information at higher levels. This allows a distinction between macro and microstates within the system, with resulting statistical (entropy driven) dynamics, including the possibility of self-organization, system bifurcation, and the formation of higher levels of information expression. Although lowerlevel information is expressed in an information hierarchy, this in itself is not sufficient for reference, function, or meaning. Nonetheless, the expression of information is central to the realization of all of these. ‘Biological information ’ is thus ambiguous between syntactic information in a hierarchical modular system, and functional information. However, the

Abstract. This paper is a study into some properties and applications of Moss ’ coalgebraic or ‘cover ’ modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇-algebra and of a positive modal ∇-algebra. We establish a categorical isomorphism between the category of modal ∇algebra and that of modal algebras, and similarly for positive modal ∇-algebras and positive modal algebras. We then turn to a presentation, in terms of relation lifting, of the Vietoris hyperspace in topology. The key ingredient is an F-lifting construction, for an arbitrary set functor F, on the category Chu of two-valued Chu spaces and Chu transforms, based on relation lifting. As a case study, we show how to realize the Vietoris construction on Stone spaces as a special instance of this Chu construction for the (finite) power set functor. Finally, we establish a tight connection with the axiomatization of the modal ∇-algebras.