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We briefly introduce the notion of an inductive database, explain its relation to constraint-based data mining, and illustrate it on an example. We then discuss constraints and constraint-based data mining in more detail. We further give an overview of recent developments in the area, focusing on those made within the IQ project and presented in a recent volume with the same title as this paper, edited...
The Semantic Web [27] is gaining momentum. Driven by over 10 years of focused project funding in the US and the EU, Semantic Web Technologies are now entering application areas in industry, academia, government, and the open Web. The Semantic Web is based on the idea of describing the meaning - or semantics - of data on the Web using metadata - data that describes other data - in the form of...
We investigate the relations between, on the one hand, Galois connections and the related types of maps and, on the other hand, the axiomatic Arrowian approach for the aggregation (or consensus) problem in lattices. In the latter one wants to ”aggregate” n-tuples (n ≥ 2) of elements of a lattice L into an element of this lattice representing their ”consensus”, subject to satisfying some desirable...
A Web service is a software functionality accessible through the network. Web services are intended to be composed into coarser-grained applications. Achieving a required composite functionality requires the discovery of a collection of Web services out of the enormous service space. Each service must be examined to verify its provided functionality, making the selection task neither efficient nor...
Any monotone Boolean function on a lattice can be described by the set of its minimal 1 values. If a lattice is given as a concept lattice, this set can be represented by the set of minimal hypotheses of a classification context. Enumeration of minimal hypotheses in output polynomial time is shown to be impossible unless P = NP, which shows that dualization of monotone functions on lattices with quasipolynomial...
The Border algorithm and the iPred algorithm find the Hasse diagrams of FCA lattices. We show that they can be generalized to arbitrary lattices. In the case of iPred, this requires the identification of a join-semilattice homomorphism into a distributive lattice.
We present an approach that enables one to select a reasonable small number of possibly important formal concepts from the set of all formal concepts of a given input data. The problem to select a small number of concepts appears in applications of formal concept analysis when the number of all formal concepts of the input data is large. Namely, a user often asks for a list of “important concepts”...
We examine the enumeration problem for essential closed sets of a formal context. Essential closed sets are sets that can be written as the closure of a pseudo-intent. The results for enumeration of essential closed sets are similar to existing results for pseudo-intents, albeit some differences exist. For example, while it is possible to compute the lectically first pseudo-intent in polynomial time,...
In universal algebra and in lattice theory the notion of varieties is very prominent, since varieties describe the classes of all algebras (or of all lattices) modeling a given set of equations. While a comprehensive translation of that notion to a similar notion of varieties of complete lattices – and thus to Formal Concept Analysis – has not yet been accomplished, some characterizations of the doubly...
We present a comparison between Hierarchical Classes Analysis and the formal concept analytical approach to Factor Analysis regarding the factorization problem of binary matrices. Both methods decompose a binary matrix into the Boolean matrix product of two binary matrices such that the number of factors is as small as possible. We show that the two approaches yield the same decomposition even though...
DNA micro-arrays are a mechanism for eliciting gene expression values, the concentration of the transcription products of a set of genes, under different chemical conditions. The phenomena of interest—up-regulation, down-regulation and co-regulation—are hypothesized to stem from the functional relationships among transcription products. In [1,2,3] a generalisation of Formal Concept Analysis...
A numerical dataset is usually represented by a table where each entry denotes the value taken by an object in line for an attribute in column. A bicluster in a numerical data table is a subtable with close values different from values outside the subtable. Traditionally, largest biclusters were found by means of methods based on linear algebra. We propose an alternative approach based on concept...
Conventional means of querying a database for relevant information require some degree of knowledge about the nature and the structural representation of such information. The paper addresses the case where sufficient knowledge is not readily available a priori. A method for data exploration based on the refinement of graphs, which represent summarized views of the underlying data, is proposed and...
It is a well-known fact that complete tolerance relations on concept lattices are in one-to-one correspondence with some superrelations (called block relations) of the incidence relation of the underlying formal context. However, sometimes it is useful to consider more general superrelations of the incidence relation, leading to tolerance relations, not compatible with the lattice structure of the...
While extending partial orders towards linear orders is a very well-researched topic, the combination of two ordered sets has not yet attracted too much attention. In the underlying article, however, we describe the possibilities to merge two given quasiordered sets in the sense that the restriction of the combined order towards the given ordered sets returns the initial orders again. It turns out...
Ternary and more generally n-ary relations are commonly found in real-life applications and data collections. In this paper, we define new notions and propose procedures to mine closed tri-sets (triadic concepts) and triadic association rules within the framework of triadic concept analysis. The input data is represented as a formal triadic context of the form $\mathbb{K}:=(K_1, K_2, K_3, Y)$...
In this paper we propose the characterization of two new structures, the Agree Concept Lattice and the Quotient Agree Lattice of a database relation. Both of them are of great interest for multidimensional database analysis. They provide a formal framework which makes it possible to improve computation time, reduce representation and easily navigate through the Hasse diagram. These structures are...
We present a view of abstraction based on a structure preserving reduction of the Galois connection between a language of terms and the powerset of a set of instances O. Such a relation is materialized as an extension-intension lattice, namely a concept lattice when L is the powerset of a set P of attributes. We define and characterize an abstractionA as some part of either...
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