The distributive, graded lattice of EL concept descriptions and its neighborhood relation: Extended Version

For the description logic EL, we consider the neighborhood relation which is induced by the subsumption order, and we show that the corresponding lattice of EL concept descriptions is distributive, modular, graded, and metric. In particular, this implies the existence of a rank function as well as the existence of a distance function

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data

Most specific consequences in the description logic EL

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic EL and its variants are provided, and means for its computation are developed. Algebraic properties of most specific consequences are explored. Furthermore, several applications that make use of this new notion are proposed and, in particular, it is shown how given terminological knowledge can be incorporated in existing approaches for the axiomatization of observations. For instance, a procedure for an incremental learning of concept inclusions from sequences of interpretations is developed

An algebraic theory of normal forms

AbstractIn this paper we present a general theory of normal forms, based on a categorial result (Dubuc, 1974) for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in (Fine, 1975) and our approach will extend it to other cases and make it more direct: operations of a purely geometric and combinatorial nature (cuts of leaves and roots, renaming labels and more generally segment-by-label replacements) will be introduced in order to give a mathematical description of the basic logical/algebraic constructions (free algebras, morphisms among them, canonical models, the lattice of varieties).We begin (Section 1) by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later (Sections 2 and 3) we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible (we shall give examples in Section 5).The central part of the paper (Sections 4 and 5) is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 (that can be read independently on Section 5) deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F (the parallel results for intuitionistic logic are in Pitts, 1992; Ghilardi, 1992; Ghilardi and Zawadowski, 1993).As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic (a possible reference is Rasiowa (1974)). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane (1971) or the appendix

Many-Body Invariants for Super Spin Chains with Antiunitary Symmetries

Proposals for many-body invariants for super-spin chains with anti-unitary symmetries are evaluated. Symmetry protected phases are modeled as homotopy classes of gapped ground states. The formalism of matrix product states is systematically extended to fermionic systems with anti-unitary symmetries. A basis-independent diagrammatic approach capable of handling anti-unitary symmetries is developed. Suggestions from the literature for observables of a twisted entanglement entropy type are calculated and proven to be topological invariants of fermionic matrix product states. The viability of classifications via these invariants is discussed as well as the connection to the cohomology classification of one-dimensional fermionic symmetry protected phases. Taking the limit of diverging bond dimension while controlling the correlation length, the homotopy invariance is proved to persist

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

Framing university small group talk : knowledge construction through lexical concepts

PhD ThesisKnowledge construction in educational discourse continues to interest practitioners and researchers due to the conceptually “natural” connection between knowledge and learning for professional development. Frames have conceptual and practical advantages over other units of inquiry concerning meaning negotiation for knowledge construction. They are relatively stable data-structures representing prototypical situations retrieved from real world experiences, cover larger units of meaning beyond the immediate sequential mechanism at interaction, and have been inherently placed at the semantic-pragmatic interface for empirical observation. Framing in a particular context – university small group talk has been an under-researched field, while the relationship between talk and knowledge through collaborative work has been identified below/at the Higher Educational level. Involving higher level cognitive activities and distinct interactional patterns, university small group talk is worth close examination and systematic investigation. This study applies Corpus Linguistics and Interactional Linguistics approaches to examine a subset of a one-million-word corpus of university small group talk at a UK university. Specifically, it provides a detailed examination of the participants’ framing behaviours for knowledge construction through their talk of disciplinary lexical concepts. Analysis reveals how the participants draw upon schematized knowledge structures evoked by particular lexical choices and how they invoke expanded scenarios via pragmatic mappings in the ongoing interaction. Additionally, it is demonstrated how the framing moves are related to the structural uniqueness of university small group talk, the contextualized speaker roles and the institutional procedures and routines. This study deepens the understanding of the relationship between linguistically constructed knowledge and the way interlocutors conceptualize the world through institutionalized collaboration, building upon the existing research on human reliance upon structures to interpret reality at both the conceptual and the action levels. The study also addresses interaction research in Higher Educational settings, by discussing how the cognitive-communicative duality of framing is sensitive to various contextual resources, distinct discourse structures and task procedures through the group dynamics