Structural machine learning with galois lattice and graphs (original) (raw)

Galois-lattices: A possible representation of knowledge structures

Evaluation in Education, 1985

The Galois-lattice is a graphic method of representing knowledge structures. The nodes of Glattices represent all the possible concepts in a given body of knowledge in the sense that a notion defines a set of individuals or properties with no exceptions or idiosyncrasies. A G-lattice provides a tool to represent all the possible developmental phases to reach a given total knowledge via different partial knowledge structures. Glattices are pure algebraic structures and may contain contingent features caused by irregular and random data. In this chapter some algorithms are proposed to develop the Glattice method so that statistical analyses can be incorporated. These algorithms are based on the possible omissions of individuals or properties from the data so that the resulting G-lattice be reduced as much as possible. The greatly simplified structure represented by the resulting G-lattice is still valid for a large percentage of individuals or properties. This is an alternative approach to statistical significance as this approach refers to strict logical relations. A comparison with the Rasch model is given. On Binary Variables Binary variables abound in educational research. Right or wrong answers, presence or absence of certain properties, choices and rejections in tests can be considered as sets of binary variables. On the other hand, there are only a few mathematical procedures that are developed for binary variables exclusively. Statisticians usually propose building up indices from binary variables and then apply common statistical procedures to analyse relationships between the indices.

Extracting formal concepts out of relational data

2003

Relational datasets, i.e., datasets in which individuals are described both by their own features and by their relations to other individuals, arise from various sources such as databases, both relational and object-oriented, or software models, e.g., UML class diagrams. When processing such complex datasets, it is of prime importance for an analysis tool to hold as much as possible to the initial format so that the semantics is preserved and the interpretation of the final results eased. There have been several attempts to introduce relations into the Galois lattice and formal concept analysis fields. We propose a novel approach to this problem which relies on an enhanced version of the classical binary data descriptions based on the distinction of several mutually related formal contexts.

A partition-based approach towards constructing Galois (concept) lattices

Discrete Mathematics, 2002

Galois lattices and formal concept analysis of binary relations have proved useful in the resolution of many problems of theoretical or practical interest. Recent studies of practical applications in data mining and software engineering have put the emphasis on the need for both e cient and exible algorithms to construct the lattice. Our paper presents a novel approach for lattice construction based on the apposition of binary relation fragments. We extend the existing theory to a complete characterization of the global Galois (concept) lattice as a substructure of the direct product of the lattices related to fragments. The structural properties underlie a procedure for extracting the global lattice from the direct product, which is the basis for a full-scale lattice construction algorithm implementing a divide-and-conquer strategy. The paper provides a complexity analysis of the algorithm together with some results about its practical performance and describes a class of binary relations for which the algorithm outperforms the most e cient lattice-constructing methods.

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.

Graph based concept learning

2000

We introduce the graph-based relational concept learner SubdueCL. We start with a brief description of other graph-based learning systems: the Galois lattice, Conceptual Graphs, and the Subdue system. We then prcsent our new system SubdueCL and finally we show some preliminary results of a comparison of SubducCL with the two Inductive Logic Programming (ILP) systems Foil and Progol.

Towards a theory of formal classification

2005

Classifications have been used for centuries with the goal of cataloguing and searching large sets of objects. In the early days it was mainly books; lately it has become Web pages, pictures and any kind of electronic information items. Classifications describe their contents using natural language labels, an approach which has proved very effective in manual classification. However natural language labels show their limitations when one tries to automate the process, as they make it almost impossible to reason about classifications and their contents. In this paper we introduce the novel notion of Formal Classification, as a graph structure where labels are written in a logical concept language. The main property of Formal Classifications is that each node can be associated a normal form formula which univocally describes its contents. This in turn allows us to reduce document classification and query answering to fully automatic propositional reasoning.

Symbolic galois lattices with pattern structures

Rough Sets, Fuzzy Sets …, 2011

Concept lattices are mathematical structures useful for many tasks in knowledge discovery and management. A concept lattice is basically obtained from binary data encoding the membership of some attributes to some objects. Dealing with complex data brings the important problem of discretization and the associated loss of information. To avoid discretization, (i) pattern structures and (ii) symbolic data analysis provide means to analyze such complex data directly. We compare both these approaches and show how they are mutually beneficial.

Knowledge Organisation and Information Retrieval with Galois Lattices

Lecture Notes in Computer Science, 2004

In this paper we investigate the application of Galois (or concept) lattices on different data sources (e.g. web documents or bibliographical items) in order to organise knowledge that can be extracted from the data. This knowledge organisation can then be used for a number of purposes (e.g. knowledge management in an organisation, document retrieval on the Web, etc.). Galois lattices can be considered as classification tools for knowledge units in concept hierarchies that can be used within a knowledge-based system. Moreover, Galois lattices can be used in parallel with domain ontologies for building more precise and more concise concept ontologies, and for guiding the knowledge discovery process.