Concept-Relation Algebra (original) (raw)

Conceptual graphs and formal concept analysis

Conceptual structures: Fulfilling Peirce's dream, 1997

Analysis may be combined to obtain a formalization of Elementary Logic which is useful for knowledge representation and processing. For this, a translation of conceptual graphs to formal contexts and concept lattices is described through an example. Using a suitable mathematization of conceptual graphs, basics of a uni ed mathematical theory for Elementary Logic are proposed.

A Parallel between Extended Formal Concept Analysis and Bipartite Graphs Analysis

Lecture Notes in Computer Science, 2010

The paper offers a parallel between two approaches to conceptual clustering, namely formal concept analysis (augmented with the introduction of new operators) and bipartite graph analysis. It is shown that a formal concept (as defined in formal concept analysis) corresponds to the idea of a maximal bi-clique, while a "conceptual world" (defined through a Galois connection associated of the new operators) is a disconnected sub-graph in a bipartite graph. The parallel between formal concept analysis and bipartite graph analysis is further exploited by considering "approximation" methods on both sides. It leads to suggests new ideas for providing simplified views of datasets.

Interval-valued fuzzy graph representation of concept lattice

2012

Formal Concept Analysis (FCA) with fuzzy setting has been successfully applied by researchers for data analysis and representation. Reducing the number of fuzzy formal concepts and their lattice structure are addressed as a major issues. In this study, we try to link between interval-valued fuzzy graph and fuzzy concept lattice to overcome from the issue. We show that proposed method reduces the number of fuzzy formal concepts and their lattice structure while preserving specialized and generalized concepts. Proposed ...

Graph partitions and concept lattices

DIMACS series in discrete mathematics and theoretical computer science, 2007

We apply the graph decomposition method known as rooted level aware breadth first search to partition graph-connected formal contexts and examine some of the consequences for the corresponding concept lattices. In graph-theoretic terms, this lattice can be viewed as the lattice of maximal bicliques of the bipartite graph obtained by symmetrizing the object-attribute pairs of the input formal context. We find that a rooted breadth-first search decomposition of a graph-connected formal context leads to a closely related partition of the concept lattice, and we provide some details of this relationship. The main result is used to describe how the concept lattice can be unfolded, according to the information gathered during the breadth first search. We discuss potential uses of the results in data mining applications that employ concept lattices, specifically those involving association rules.

Lattices, closures systems and implication bases: A survey of structural aspects and algorithms

Theoretical Computer Science, 2016

Concept lattices and closed set lattices are graphs with the lattice property. They have been increasingly used this last decade in various domains of computer science, such as data mining, knowledge representation, databases or information retrieval. A fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table. A consequence is the existence of a bijective link between lattices, contexts (via the table) and a set of implicational rules (via the canonical (direct) basis). The possible transformations between these objects give rise to relevant tools for data analysis. In this paper, we present a survey of lattice theory, from the algebraic definition of a lattice, to that of a concept lattice, through closure systems and implicational rules; including the exploration of fundamental bijective links between lattices, reduced contexts and bases of implicational rules; and concluding with the presentation of the main generation algorithms of these objects.

Formal concept analysis over graphs and hypergraphs

Formal Concept Analysis (FCA) provides an account of classification based on a binary relation between two sets. These two sets contain the objects and attributes (or properties) under consideration. In this paper I propose a generalization of formal concept analysis based on binary relations between hypergraphs, and more generally between pre-orders. A binary relation between any two sets already provides a bipartite graph, and this is a well-known perspective in FCA. However the use of graphs here is quite different as it corresponds to imposing extra structure on the sets of objects and of attributes. In the case of objects the resulting theory should provide a knowledge representation technique for structured collections of objects. The generalization is achieved by an application of work on mathematical morphology for hypergraphs.

Simple concept graphs: A logic approach

Lecture Notes in Computer Science, 1998

Conceptual Graphs and Formal Concept Analysis are combined by developing a logical theory for concept graphs of relational contexts. Therefore, concept graphs are introduced as syntactical constructs, and their semantics is de ned based on relational contexts. For this contextual logic, a sound and complete system of inference rules is presented and a standard graph is introduced that entails all concept graphs being valid in a given relational context. A possible use for conceptual knowledge representation and processing is suggested.

Bipolar fuzzy graph representation of concept lattice

Information Sciences, Elsevier, 2014

Formal Concept Analysis (FCA) is a mathematical framework for knowledge processing tasks. FCA has been successfully incorporated into fuzzy setting and its extension (interval-valued fuzzy set) for handling vagueness and impreciseness in data. However, the analysis in such settings is restricted to unipolar space. Recently, some applications of bipolar information are shown in bipolar fuzzy graph, lattice theory as well as in FCA. The adequate analysis of bipolar information using FCA requires incorporation of bipolar fuzzy set and an appropriate lattice structure. For this purpose, we propose an algorithm for generating the bipolar fuzzy formal concepts, a method for (α,β)-cut of bipolar fuzzy formal context and its implications with illustrative examples.

Conceptual Graphs Are Also Graphs

Lecture Notes in Computer Science, 2014

The main objective of this paper is to add one more brick in building the CG model as a knowledge representation model autonomous from logic. The CG model is not only a graphical representation of logic, it is much more: it is a declarative model encoding knowledge in a mathematical theory, namely labelled graph theory, which has e cient computable forms, with a fundamental graph operation on the encodings to do reasoning, projection, which is a labelled graph morphism. Main topics of this paper are: a generalized formalism for simple CGs; a strong equivalence between CSP (Constraint Satisfaction Problem) and labelled graph morphism. This correspondence allows the transportation of e cient algorithms from one domain to the other, and con rms that projection |or more generally labelled graph morphism| rmly moors CGs to combinatorial algorithmics, which is a cornerstone of computer science. The usual sound and complete rst order logic semantics for CGs is still valid for our generalized model. This, plus the ease of doing important reasonings with CGs |for instance plausible reasonings by using some maximal join operations| without, at least for the moment, logical semantics, strengthens our belief that CGs must also be studied and developed independently from logic.