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The offered methods of Conceptual Knowledge Processing are procedures which are well-planed to mean and purpose and therewith lead to skills for solving practical tasks. The used means and skills have been mainly created as translations of mathematical means and skills of Formal Concept Analysis. Those transdisciplinary translations may be understood as transformations from mathematical thinking,...
Motivated by the desire to estimate the number of order-preserving self maps of an ordered set, we compare three algorithms (Simple Sampling [4], Partial Backtracking [10] and Heuristic Sampling [1]) which predict how many nodes of a search tree are visited. The comparison is for the original algorithms that apply to backtracking and for modifications that apply to forward checking. We identify generic...
The paper is an overview of recent developments concerning attribute implications in a fuzzy setting. Attribute implications are formulas of the form A B, where A and B are collections of attributes, which describe dependencies between attributes. Attribute implications are studied in several areas of computer science and mathematics. We focus on two of them, namely, formal concept...
This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a theory for the assessment of a student’s knowledge in a topic and gives details on a practical implementation in the form of a software. The basic concept of the theory is the ‘knowledge state,’ which is the complete set of problems that an individual is capable of solving in a particular topic, such as Arithmetic...
Preconcept Lattices are identified to be (up to isomorphism) the completely distributive complete lattices in which the supremum of all atoms is equal or greater than the infimum of all coatoms. This is a consequence of the Basic Theorem on Preconcept Lattices, which also offers means for checking line diagrams of preconcept lattices.
Galois connections between concept lattices can be represented as binary relations on the context level, known as dual bonds. The latter also appear as the elements of the tensor product of concept lattices, but it is known that not all dual bonds between two lattices can be represented in this way. In this work, we define regular Galois connections as those that are represented by a dual bond in...
A main goal of Formal Concept Analysis (FCA) from its very beginning has been the support of rational communication by formalizing and visualizing concepts. In the last years, this approach has been extended to traditional logic based on the doctrines of concepts, judgements and conclusions, leading to a framework called Contextual Logic. Much of the work on Contextual Logic has been inspired by the...
Computing the maximal bicliques of a bipartite graph is equivalent to generating the concepts of the binary relation defined by the matrix of this graph. We study this problem for special classes of input relations for which concepts can be generated much more efficiently than in the general case; in some special cases, we can even say that the number of concepts is polynomially bounded, and all concepts...
We introduce the epistemic extension, a logic transformation based on the modal logic AIK (All I Know) for use in the framework of Logical Concept Analysis (LCA). The aim is to allow for the distinction between negation, opposition, and possibility in a unique formalism. The difference between negation and opposition is examplified by the difference between “young/not young” and “young/old”. The difference...
In the paper “Which concept lattices are pseudocomplemented?” ([GK05]) we gave a contextual characterization of pseudocomplemented concept lattices by means of the arrow relations. In this contribution we use this description to embed finite semilattices into pseudocomplemented semilattices. This process can be used to define a negation on concepts. AMS Subject Classification: 06D15,...
In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed...
An interactive discovery method for finding association rules is presented. It consists in a user-guided search using reduction operators on a rule. Rules are generated on-demand according to the navigation made by the user. Main interest of this approach is that, at each step, the user has only a linear number of new rules to analyze and that all computations are done in polynomial time. Several...
The paper provides evidence that spatial indexing structures offer faster resolution of Formal Concept Analysis queries than B-Tree/Hash methods. We show that many Formal Concept Analysis operations, computing the contingent and extent sizes as well as listing the matching objects, enjoy improved performance with the use of spatial indexing structures such as the RD-Tree. Speed improvements can vary...
Homographs are words with identical spellings but different origins and meanings. Natural language processing must deal with the disambiguation of homographs and the attribution of senses to them. Advances have been made using context to discriminate homographs, but the problem is still open. Disambiguating homographs is possible using formal concept analysis. This paper discusses the issues, illustrated...
Suppose that whenever event x occurs, a second event y must subsequently occur. We say that x “causes” y, or y is causally dependent on x. Deterministic causality abounds in software where execution of one routine can necessarily force execution of a subsequent sub-routine. Discovery of such causal dependencies can be an important step to understanding the structure of undocumented, legacy code. ...
This paper discusses an interpretation of relation algebra and fork algebra with respect to FCA contexts. In this case, “relation algebra” refers to the DeMorgan-Peirce-Schroeder-Tarski algebra and not to the “relational algebra” as described by Codd. The goal of this interpretation is to provide an algebraic formalisation of object-relational databases that is based on binary relations and thus closer...
This paper presents a spring-based lattice drawing method that uses natural spring lengths determined by assigning a dissimilarity value, the size of symmetric difference, to every pair of concept extents. This extends previous work on incorporating support structure in a concept lattice diagram, in which the support weight function was applied to modify any layout. That work was a partial advance...
With the help of the left-relation on lattices [11] we give two characterizations for planar lattices. They can be used to decide already in a context, whether the associated concept lattice is planar. With the help of these results we hope to find a quick algorithm to recognize planar lattices and draw them in the plane in the near future.
Good quality concept lattice drawings are required to effectively communicate logical structure in Formal Concept Analysis. Data analysis frameworks such as the Toscana System use manually arranged concept lattices to avoid the problem of automatically producing high quality lattices. This limits Toscana systems to a finite number of concept lattices that have been prepared a priori. To extend the...
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