Institute for Res. and Appl. of Fuzzy Modeling, University of Ostrava Bráfova 7, CZ-701 03 Ostrav... more Institute for Res. and Appl. of Fuzzy Modeling, University of Ostrava Bráfova 7, CZ-701 03 Ostrava, Czech Republic e-mail: belohlav@osu.cz and Dept. Computer Science, Technical University of Ostrava, tr. 17. listopadu, CZ-708 33 Ostrava, Czech Republic
Page 1. Similarity Relations in Concept Lattices RADIM B ˇELOHL ´AVEK, Institute for Research and... more Page 1. Similarity Relations in Concept Lattices RADIM B ˇELOHL ´AVEK, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Bráfova 7, 701 03 Ostrava, Czech Republic. and Department of Computer Science, Technical University of Ostrava, tr. 17. ...
This article presents a concept interpretation of patterns for bidirectional associative memory (... more This article presents a concept interpretation of patterns for bidirectional associative memory (BAM) and a representation of hierarchical structures of concepts (concept lattices) by BAMs. The constructive representation theorem provides a storing rule for a training set that allows a concept interpretation. Examples demonstrating the theorems are presented.
The present paper deals with Codd's relational model of data. In particular, we d... more The present paper deals with Codd's relational model of data. In particular, we deal with fuzzy logic extensions of the relational model. Our main purpose is to examine relationships between some of the models which have been proposed in the literature. We concentrate on functional dependencies which is the most studied part of the relational model within fuzzy logic extensions.
An important topic in formal concept analysis is to cope with a possibly large number of formal c... more An important topic in formal concept analysis is to cope with a possibly large number of formal concepts extracted from formal context (input data). We propose a method to reduce the number of extracted formal concepts by means of constraints expressed by particular formulas (attribute-dependency formulas, ADF). ADF represent a form of dependencies specified by a user expressing relative importance of attributes. ADF are considered as additional input accompanying the formal context 〈X, Y, I〉. The reduction consists in considering formal concepts which are compatible with a given set of ADF and leaving out noncompatible concepts. We present basic properties related to ADF, an algorithm for generating the reduced set of formal concepts, and demonstrating examples.
We study rules $A \Longrightarrow B$ describing attribute dependencies in tables over domains wit... more We study rules $A \Longrightarrow B$ describing attribute dependencies in tables over domains with similarity relations. $A \Longrightarrow B$ reads “for any two table rows: similar values of attributes from A imply similar values of attributes from B”. The rules generalize ordinary functional dependencies in that they allow for processing of similarity of attribute values. Similarity is modeled by reflexive and symmetric fuzzy relations. We show a system of Armstrong-like derivation rules and prove its completeness (two versions). Furthermore, we describe a non-redundant basis of all rules which are true in a data table and present an algorithm to compute bases.
Formal concept analysis is a method of exploratory data analysis that aims at the extraction of n... more Formal concept analysis is a method of exploratory data analysis that aims at the extraction of natural clusters from object-attribute data tables. The clusters, called formal concepts, are naturally interpreted as human-perceived concepts in a traditional sense and can be partially ordered by a subconcept-superconcept hierarchy. The hierarchical structure of formal concepts (so-called concept lattice) represents a structured information obtained automatically from the input data table. The present paper focuses on the analysis of input data with a predefined hierarchy on attributes thus extending the basic approach of formal concept analysis. The motivation of the present approach derives from the fact that very often, people (consciously or unconsciously) attach various importance to attributes which is then reflected in the conceptual classification based on these attributes. We define the notion of a formal concept respecting the attribute hierarchy. Formal concepts which do not respect the hierarchy are considered not relevant. Elimination of the non-relevant concepts leads to a reduced set of extracted concepts making the discovered structure of hidden concepts more comprehensible. We present basic formal results on our approach as well as illustrating examples.
... theory of closure operators which operate with fuzzy sets (so-called fuzzy closure operators)... more ... theory of closure operators which operate with fuzzy sets (so-called fuzzy closure operators) are studied (Mashour and Ghanim, 1985; Bandler ... Introduced originally in the study of ideal systems of rings (Ward and Dilworth, 1939), residuated lattices have been introduced into the ...
We study concept lattices with hedges. The principal aim is to control, in a parametrical way, th... more We study concept lattices with hedges. The principal aim is to control, in a parametrical way, the size of a concept lattice. The paper presents theoretical insight, comments, and examples. We show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the so-called main theorem of concept lattices. Furthermore, the isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. Further insight is provided in case one uses hedges only for attributes. We demonstrate by experiments that the size reduction using hedges as a parameter is smooth
Presented is a completeness theorem for fuzzy equational logic with truth values in a complete re... more Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established.
In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a for... more In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a formal (fuzzy) concept may be fuzzy sets. In this paper we focus on so-called crisply generated formal concepts. A concept $\langle{A,B}\rangle \in \mathcal{B}(X, Y, I)$ is crisply generated if A = D ↓ (and so B = D ↓↑) for some crisp (i.e., ordinary) set D ⊆ Y of attributes (generator). Considering only crisply generated concepts has two practical consequences. First, the number of crisply generated formal concepts is considerably less than the number of all formal fuzzy concepts. Second, since crisply generated concepts may be identified with a (ordinary, not fuzzy) set of attributes (the largest generator), they might be considered “the important ones” among all formal fuzzy concepts. We present basic properties of the set of all crisply generated concepts, an algorithm for listing all crisply generated concepts, a version of the main theorem of concept lattices for crisply generated concepts, and show that crisply generated concepts are just the fixed points of pairs of mappings resembling Galois connections. Furthermore, we show connections to other papers on formal concept analysis of data with fuzzy attributes. Also, we present examples demonstrating the reduction of the number of formal concepts and the speed-up of our algorithm (compared to listing of all formal concepts and testing whether a concept is crisply generated).
This paper is a follow up to “Belohlavek, Vychodil: What is a fuzzy concept lattice?, Proc. CLA 2... more This paper is a follow up to “Belohlavek, Vychodil: What is a fuzzy concept lattice?, Proc. CLA 2005, 34–45”, in which we provided a then up-to-date overview of various approaches to fuzzy concept lattices and relationships among them. The main goal of the present paper is different, namely to provide an overview of conceptual issues in fuzzy concept lattices. Emphasized are the issues in which fuzzy concept lattices differ from ordinary concept lattices. In a sense, this paper is written for people familiar with ordinary concept lattices who would like to learn about fuzzy concept lattices. Due to the page limit, the paper is brief but we provide an extensive list of references with comments.
Institute for Res. and Appl. of Fuzzy Modeling, University of Ostrava Bráfova 7, CZ-701 03 Ostrav... more Institute for Res. and Appl. of Fuzzy Modeling, University of Ostrava Bráfova 7, CZ-701 03 Ostrava, Czech Republic e-mail: belohlav@osu.cz and Dept. Computer Science, Technical University of Ostrava, tr. 17. listopadu, CZ-708 33 Ostrava, Czech Republic
Page 1. Similarity Relations in Concept Lattices RADIM B ˇELOHL ´AVEK, Institute for Research and... more Page 1. Similarity Relations in Concept Lattices RADIM B ˇELOHL ´AVEK, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Bráfova 7, 701 03 Ostrava, Czech Republic. and Department of Computer Science, Technical University of Ostrava, tr. 17. ...
This article presents a concept interpretation of patterns for bidirectional associative memory (... more This article presents a concept interpretation of patterns for bidirectional associative memory (BAM) and a representation of hierarchical structures of concepts (concept lattices) by BAMs. The constructive representation theorem provides a storing rule for a training set that allows a concept interpretation. Examples demonstrating the theorems are presented.
The present paper deals with Codd's relational model of data. In particular, we d... more The present paper deals with Codd's relational model of data. In particular, we deal with fuzzy logic extensions of the relational model. Our main purpose is to examine relationships between some of the models which have been proposed in the literature. We concentrate on functional dependencies which is the most studied part of the relational model within fuzzy logic extensions.
An important topic in formal concept analysis is to cope with a possibly large number of formal c... more An important topic in formal concept analysis is to cope with a possibly large number of formal concepts extracted from formal context (input data). We propose a method to reduce the number of extracted formal concepts by means of constraints expressed by particular formulas (attribute-dependency formulas, ADF). ADF represent a form of dependencies specified by a user expressing relative importance of attributes. ADF are considered as additional input accompanying the formal context 〈X, Y, I〉. The reduction consists in considering formal concepts which are compatible with a given set of ADF and leaving out noncompatible concepts. We present basic properties related to ADF, an algorithm for generating the reduced set of formal concepts, and demonstrating examples.
We study rules $A \Longrightarrow B$ describing attribute dependencies in tables over domains wit... more We study rules $A \Longrightarrow B$ describing attribute dependencies in tables over domains with similarity relations. $A \Longrightarrow B$ reads “for any two table rows: similar values of attributes from A imply similar values of attributes from B”. The rules generalize ordinary functional dependencies in that they allow for processing of similarity of attribute values. Similarity is modeled by reflexive and symmetric fuzzy relations. We show a system of Armstrong-like derivation rules and prove its completeness (two versions). Furthermore, we describe a non-redundant basis of all rules which are true in a data table and present an algorithm to compute bases.
Formal concept analysis is a method of exploratory data analysis that aims at the extraction of n... more Formal concept analysis is a method of exploratory data analysis that aims at the extraction of natural clusters from object-attribute data tables. The clusters, called formal concepts, are naturally interpreted as human-perceived concepts in a traditional sense and can be partially ordered by a subconcept-superconcept hierarchy. The hierarchical structure of formal concepts (so-called concept lattice) represents a structured information obtained automatically from the input data table. The present paper focuses on the analysis of input data with a predefined hierarchy on attributes thus extending the basic approach of formal concept analysis. The motivation of the present approach derives from the fact that very often, people (consciously or unconsciously) attach various importance to attributes which is then reflected in the conceptual classification based on these attributes. We define the notion of a formal concept respecting the attribute hierarchy. Formal concepts which do not respect the hierarchy are considered not relevant. Elimination of the non-relevant concepts leads to a reduced set of extracted concepts making the discovered structure of hidden concepts more comprehensible. We present basic formal results on our approach as well as illustrating examples.
... theory of closure operators which operate with fuzzy sets (so-called fuzzy closure operators)... more ... theory of closure operators which operate with fuzzy sets (so-called fuzzy closure operators) are studied (Mashour and Ghanim, 1985; Bandler ... Introduced originally in the study of ideal systems of rings (Ward and Dilworth, 1939), residuated lattices have been introduced into the ...
We study concept lattices with hedges. The principal aim is to control, in a parametrical way, th... more We study concept lattices with hedges. The principal aim is to control, in a parametrical way, the size of a concept lattice. The paper presents theoretical insight, comments, and examples. We show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the so-called main theorem of concept lattices. Furthermore, the isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. Further insight is provided in case one uses hedges only for attributes. We demonstrate by experiments that the size reduction using hedges as a parameter is smooth
Presented is a completeness theorem for fuzzy equational logic with truth values in a complete re... more Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established.
In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a for... more In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a formal (fuzzy) concept may be fuzzy sets. In this paper we focus on so-called crisply generated formal concepts. A concept $\langle{A,B}\rangle \in \mathcal{B}(X, Y, I)$ is crisply generated if A = D ↓ (and so B = D ↓↑) for some crisp (i.e., ordinary) set D ⊆ Y of attributes (generator). Considering only crisply generated concepts has two practical consequences. First, the number of crisply generated formal concepts is considerably less than the number of all formal fuzzy concepts. Second, since crisply generated concepts may be identified with a (ordinary, not fuzzy) set of attributes (the largest generator), they might be considered “the important ones” among all formal fuzzy concepts. We present basic properties of the set of all crisply generated concepts, an algorithm for listing all crisply generated concepts, a version of the main theorem of concept lattices for crisply generated concepts, and show that crisply generated concepts are just the fixed points of pairs of mappings resembling Galois connections. Furthermore, we show connections to other papers on formal concept analysis of data with fuzzy attributes. Also, we present examples demonstrating the reduction of the number of formal concepts and the speed-up of our algorithm (compared to listing of all formal concepts and testing whether a concept is crisply generated).
This paper is a follow up to “Belohlavek, Vychodil: What is a fuzzy concept lattice?, Proc. CLA 2... more This paper is a follow up to “Belohlavek, Vychodil: What is a fuzzy concept lattice?, Proc. CLA 2005, 34–45”, in which we provided a then up-to-date overview of various approaches to fuzzy concept lattices and relationships among them. The main goal of the present paper is different, namely to provide an overview of conceptual issues in fuzzy concept lattices. Emphasized are the issues in which fuzzy concept lattices differ from ordinary concept lattices. In a sense, this paper is written for people familiar with ordinary concept lattices who would like to learn about fuzzy concept lattices. Due to the page limit, the paper is brief but we provide an extensive list of references with comments.
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