We develop a geometric view of Ralescu’s concept of a many-valued cardinality of fuzzy sets. The view facilitates an easy understanding of this concept and helps elucidate its nature. We demonstrate that the view enables to obtain properties of the cardinality concept by a straightforward geometric reasoning. We show that the geometric view may be utilized to analyze further approaches to cardinality of fuzzy sets. In particular, we observe that the view applies to a natural generalization of Ralescu’s concept in which the standard minimu is replaced by a general t-norm.

We describe an intuitive and practically significant empirical phenomenon that relates to the concept of cardinality of a fuzzy set, namely, an excessive accumulation of small degrees of membership. We argue and demonstrate by examples that the present notions of cardinality do not take this phenomenon into account properly and may thus prove insufficient in applications. We propose a new concept of cardinality, generalizing the wellknown Zadeh's sigma count, demonstrate using both intuitive and technical examples that it alleviates the insufficiency of the existing ones, and provide a theoretical analysis of this concept. We also propose topics for future theoretical and empirical research.

We present a novel algorithm for factorization of matrices with grades or, equivalently, for decomposition of fuzzy relations. The algorithm is inspired by a recent factorization method for Boolean matrices and develops two ideas in the setting of ordinal attributes. First, it uses formal concepts associated with the factorized matrix, or fuzzy relation, as essential components around which factors are built. Second, it steps back when computing new factors to check whether some computed factors may be eliminated or improved given the subsequently generated factors. The new algorithm thus uses convenient properties of formal concepts utilized by previous factorization algorithms. Still, unlike the previous algorithms, our algorithm allows for more general and therefore more precise factorizations due to a possible overcovering of the input data, which our new algorithm admits. We provide an experimental evaluation of the new algorithm and compare it to some existing algorithms for factorization of data with grades. The evaluation reveals that our new algorithm outperforms the current algorithms in terms of quality of factorization. We also present observations and improvements for factorization of Boolean matrices. We conclude with a discussion regarding open research topics.

We provide a detailed analysis and a first complete description of 8M—an old but virtually unknown algorithm for Boolean matrix factorization. Even though the algorithm uses a rather limited insight into the factorization problem from today’s perspective, we demonstrate that its performance is reasonably good compared to the currently available algorithms. Our analysis reveals that this is due to certain concepts employed by 8M that are not exploited by the current algorithms. We discuss the prospect of these concepts, utilize them to improve two well-known current factorization algorithms, and, furthermore, propose an improvement of 8M itself, which significantly enhances the performance of the original 8M. Our findings are illustrated by experimental evaluation.

We present an extension of the discrete basis problem, recently a profoundly studied problem, from the Boolean setting to the setting of fuzzy attributes, i.e., a setting of ordinal data. Our problem consists in finding for a given object-attribute matrix I containing truth degrees and a prescribed number k of factors the best approximate decomposition of I into an object-factor matrix A and a factor-attribute matrix B. Since such matrices represent fuzzy relations, the problem is related to but very different from that of decomposition of fuzzy relations as studied in fuzzy relational equations because neither A nor B are supposed to be known in our problem. We observe that our problem is NP-hard as an optimization problem. Consequently, we provide an approximation algorithm for solving this problem and provide its time complexity in the worst case. The algorithm is inspired by the Asso algorithm, which is known for Boolean attributes and is based on new considerations regarding associations among fuzzy attributes. We provide an experimental evaluation on various datasets and demonstrate that our algorithm is capable of extracting informative factors in data. We conclude with a discussion regarding future research issues.

We present a way to compute the set of fixpoints of a given fuzzy closure operator via algorithms for computing sets of fixpoints of ordinary closure operators. We assume that the fuzzy closure operator is given by a set of fuzzy sets generating this operator. The proposed way is based on certain reduction theorems which we provide and which relate fuzzy and ordinary closure operators and the sets of their fixpoints. We also present explicit description of selected algorithms which result using the presented approach.

We critically examine and point out weaknesses of the existing considerations in Boolean matrix factorization (BMF) regarding noise and the algorithms' ability to deal with noise. We argue that the current understanding is underdeveloped and that the current approaches are missing an important aspect. We provide a new, quantitative way to assess the ability of an algorithm to handle noise. Our approach is based on a common-sense definition of robustness requiring that the computed factorizations should not be affected much by varying the noise in data. We present an experimental evaluation of several existing algorithms and compare the results to the observations available in the literature. In addition to providing justification of some properties claimed in the literature without proper justification, our experiments reveal properties which were not reported as well as properties which counter certain claims made in the literature. Importantly, our approach reveals a line separating robust-to-noise from sensitive-to-noise algorithms, which has not been revealed by the previous approaches.

Minimal solutions play a crucial role in describing all solutions of relational equations. For this reason, the problem of computing minimal solutions has for long been examined. The literature contains several algorithms for computing minimal solutions. Recently, contributions regarding computational complexity of the problem itself appeared. The complexity aspect is clearly of fundamental importance. However, the existing results contain serious flaws. In this paper, we inspect the existing contributions, clarify the flaws, examine the problem of complexity of computing minimal solutions, prove that there is no efficient algorithm computing all minimal solutions, and discuss further ramifications of our observations.

Boolean matrix factorization (BMF), or decomposition, received a considerable attention in data mining research. In this paper, we argue that research should extend beyond the Boolean case toward more general type of data such as ordinal data. Technically, such extension amounts to replacement of the two-element Boolean algebra utilized in BMF by more general structures, which brings non-trivial challenges. We first present the problem formulation, survey the existing literature, and provide an illustrative example. Second, we present new theorems regarding decompositions of matrices with ordinal data. Third, we propose a new algorithm based on these results along with an experimental evaluation.

The paper presents results on representation of the basic structures related to ternary fuzzy relations by the structures related to ordinary ternary relations, such as Galois connections, closure operators, and trilattices (structures of maximal Cartesian subrelations). These structures appear as the fundamental structures in relational data analysis such as formal concept analysis or association rules. We prove several representation theorems that allow us to automatically transfer some of the known results from the ordinary case to fuzzy case. The transfer is demonstrated by examples.

We present results regarding row and column spaces of matrices whose entries are elements of residuated lattices. In particular, we define the notions of a row and column space for matrices over residuated lattices, provide connections to concept lat

We describe optimal decompositions of matrices whose entries are elements of a residuated lattice L, such as L=[0, 1]. Such matrices represent relationships between objects and attributes with the entries representing degrees to which attributes represented by columns apply to objects represented by rows. Given such an n × m object-attribute matrix I, we look for a decomposition of I into a product A ○ B of an n × k object-factor matrix A and a k × m factor-attribute matrix B with entries from L with the number k of factors as small as possible. We show that formal concepts of I, which play a central role in the Port-Royal approach to logic and which are the fixpoints of particular Galois connections associated to I, are optimal factors for decomposition of I in that they provide us with decompositions with the smallest number of factors. Moreover, we describe transformations between the space of original attributes and the space of factors induced by a decomposition I = A ○ B. The article contains illustrative examples demonstrating the significance of the presented results for factor analysis of relational data. In addition, we present a general framework for a calculus of matrices with entries from residuated lattices in which both the matrix products and decompositions discussed in this article as well as triangular products and decompositions discussed elsewhere can be regarded as two particular cases of a general type of product and decomposition. We present the results for matrices, i.e. for relations between finite sets in terms of relations, but the arguments behind are valid for relations between infinite sets as well.

The article deals with Codd's relational model of data and its fuzzy logic extensions. Our main purpose is to examine, from the point of view of fuzzy logic in the narrow sense, some of the extensions proposed in the literature and the relationships between them. We argue that fuzzy logic in the narrow sense is important for the fuzzy logic extensions because it provides conceptual and methodological foundations, clarity and simplicity. We present several comparative observations as well as new technical results.

We present a fast bottom-up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees, along with the partial order on the set of all fixpoints. Fuzzy closure operators appear in several areas of fuzzy logic and its applications, including formal concept analysis (FCA) that we use as a reference area of application in this paper. Several problems in FCA, such as computing all formal concepts from data with graded attributes or computing non-redundant bases of all attribute dependencies, can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data. The development of a general algorithm that is applicable, in particular, to these problems is the ultimate purpose of this paper. We present the algorithm, its theoretical foundations, and experimental evaluation.

Confluence and termination are essential properties of relations connected to the idea of rewriting and substituting which appear in abstract rewriting systems. This paper presents a preliminary study of confluence, termination, and related properties from the point of view of fuzzy logic leaving the ordinary notions a particular case when the underlying structure of truth degrees is the two-valued Boolean algebra. The main motivation of this study is the fact that there are natural examples where the notion of substitutability is inherently fuzzy rather than crisp.

This paper deals with background knowledge in knowledge extraction from binary data. A background knowledge represents an additional piece of information a user may have along with the input data. Such information can be considered as specifying the type of knowledge a user is looking for in the data. In particular, we emphasize the need for taking into account background knowledge in formal concept analysis. We present an approach to modeling background knowledge that represents user's priorities regarding attributes and their relative importance. Such priorities serve as a constraint-only those formal concepts that are compatible with user's priorities are considered relevant, extracted from data, and presented to the user. Our approach has two main practical features. First, the number of formal concepts presented to the user may get significantly reduced. As a result, the user is supplied with relevant formal concepts only and is not overloaded by a large number of possibly nonrelevant formal concepts. Second, different priorities lead to different pieces of knowledge extracted from data. This way, the input data may be repeatedly used in knowledge extraction for different purposes corresponding to different priorities. We concentrate on foundational aspects such as mathematical feasibility, reasoning with background knowledge, removing redundancy from background knowledge, and computational tractability, and present several illustrative examples. In addition, we discuss directions for future research.

We present extensions of ordinary compositions of fuzzy relations. The extensions consist in parameterizing the ordinary compositions by means of particular unary functions on the scales of truth degrees. The approach is inspired by our previous work on formal concept analysis of data with fuzzy attributes where such parameterization of one particular type of fuzzy relational composition was used to control the number of clusters extracted from data. We present definitions and basic properties of the parameterized compositions, examples, and implications for several domains of application of fuzzy relational compositions.

Concept lattices are systems of conceptual clusters, called formal concepts, which are partially ordered by the subconcept/superconcept relationship. Concept lattices are basic structures used in formal concept analysis. In general, a concept lattice may contain overlapping clusters and need not be a tree. On the other hand, tree-like classification schemes are appealing and are produced by several clustering methods. In this paper, we present necessary and sufficient conditions on input data for the output concept lattice to form a tree after one removes its least element. We present these conditions for input data with yes/no attributes as well as for input data with fuzzy attributes. In addition, we show how Lindig's algorithm for computing concept lattices gets simplified when applied to input data for which the associated concept lattice is a tree after removing the least element. The paper also contains illustrative examples.

The present paper deals with dependencies developed within the theory of knowledge spaces. Knowledge spaces represent a new paradigm in psychological approaches to assessment of knowledge. A distinguishing feature of knowledge spaces is their non-numerical character. The aim of the present paper is twofold. First, we bring up several remarks on data dependencies studied within knowledge spaces. Second, we consider the dependencies in a framework which is more general than that of classical knowledge spaces. Namely, we abandon the assumption that a knowledge state is a set of problems/questions which an individual is able to solve. Instead, we assume that a knowledge state is a graded set (fuzzy set) of problems. Our assumption accounts for situations where it is possible that an individual can solve a particular problem partially, rather than just "can solve" or "cannot solve". We propose a definition of dependencies and validity of dependencies in knowledge spaces with graded knowledge states, provide selected properties of the dependencies, and a lemma which serves as a bridge to existing results on so-called fuzzy attribute implications.

The present paper deals with scaling within the framework of formal concept analysis (FCA) of data with fuzzy attributes. In ordinary FCA, the input is a data table with yes/no attributes. Scaling is a process of transformation of data tables with general attributes, e.g. nominal, ordinal, etc., to data tables with yes/no attributes. This way, data tables with general attributes can be analyzed by means of FCA. We propose a new way of scaling, namely, scaling of general attributes to fuzzy attributes. After such a scaling, the data can be analyzed by means of FCA developed for data with fuzzy attributes. Compared to ordinary scaling to yes/no attributes, our scaling procedure is less sensitive to how a user defines a scale which eliminates the arbitrariness of user's definition of a scale. This is the main advantage of our approach. In addition, scaling to fuzzy attributes is appealing from the point of view of knowledge representation and is connected to Zadeh's concept of linguistic variable. We present a general definition of scaling, examples comparing our approach to ordinary scaling, and theorems which answer some naturally arising questions regarding sensitivity of FCA to the definition of a scale.

This article deals with the claims that “a standard version of fuzzy logic collapses mathematically to two-valued logic” made by Charles Elkan in two papers [Proc 11th National Conf on AI, Menlo Park, CA: AAAI Press, 1993, pp 698–703; IEEE Expert 1994;9:3–8]. Although Elkan's effort to trivialize fuzzy logic has been questioned by numerous authors, our aim is to examine in detail his formal arguments and make some new observations. We present alternative, considerably simpler proofs of Elkan's theorems and use these proofs to argue that Elkan's claims are unwarranted. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 203–207, 2007.

We present a logic, called fuzzy attribute logic (FAL), for reasoning about formulas describing particular attribute dependencies. The formulas are of a form A ⇒ B where A and B are collections of attributes. Our formulas can be interpreted in two ways. First, in data tables with entries containing degrees to which objects (table rows) have attributes (table columns). Second, in database tables where each domain is equipped with a similarity relation assigning a degree of similarity to any pair of domain elements. We assume that the scale of degrees is equipped with fuzzy logical connectives and forms an arbitrary complete residuated lattice. This covers many structures used in fuzzy logic applications as well as structures used in formal systems of fuzzy logic. If the scale contains only two degrees, 0 (falsity) and 1 (truth), two well-known calculi become particular cases of our logic. Namely, with the first interpretation, our logic coincides with attribute logic used in formal concept analysis; with the second interpretation, our logic coincides with Armstrong system for reasoning about functional dependencies. We prove completeness of FAL over arbitrary complete residuated lattices in two versions. First, in the ordinary style, completeness asserts that a formula A ⇒ B is entailed by a collection T of formulas if A ⇒ B is provable from T. Second, in the graded style, completeness asserts that a degree to which A ⇒ B is entailed by a collection T of formulas equals a degree to which A ⇒ B is provable from T.

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

The paper studies if and to what extent it is possible to reduce notions related to attribute implications from data with fuzzy attributes to corresponding notions related to attribute implications from data with crisp attributes. We provide a reduction (transformation) theorem. Still, working directly in fuzzy setting is beneficial. Namely, we prove that we can compute a complete and non-redundant basis of fuzzy attribute implications of data with fuzzy attributes such that the basis is at most as large as (and can be smaller than) any basis of implications of the corresponding data table with crisp attributes

Studied in the paper are systems of equations which naturally arise in the formalization of the Port-Royal theory of concepts. The unknown quantity is a relation between objects and attributes. We study the case where the relation is fuzzy with truth values in a complete residuated lattice, covering therefore the special cases of complete Boolean algebras, Heyting algebras, MV-algebras etc. We answer the question of solvability, structure of solutions, and show how solvability of non-solvable systems may be attained by so-called decrease of logical precision.

The paper studies tolerance relations from the point of view of fuzzy logic and fuzzy set theory. Particularly, we focus on the study of natural groupings induced by a fuzzy tolerance, relations induced by fuzzy tolerances, representational issues, and an algorithm for generating all fuzzy tolerance classes.

New results on fuzzy concepts and concept lattices are presented. The issue of logical precision, a phenomenon that—although completely degenerate in two‐valued logic—naturally arises in the context of fuzzy logic is studied. We present basic results regarding logical precision and concept lattices. Since, as far as we know, the phenomenon has not been discussed in literature so far, we briefly discuss its role in the general context of relational systems. We argue that logical precision should be considered as a possible way to treat the problem of information granularity from the logical point of view.

We argue that model theory for fuzzy logic, if developed closely to the motivations of the fuzzy approach, can be considered a firm base for soft computing (esp. for fuzzy relational modeling). We present some basic concepts and results concerning similarity issues and morphisms. The results are applied to selected models of fuzzy relational modeling.

This paper studies the issue of similarity relations in fuzzy concept lattices. Fuzzy concepts and fuzzy concept lattices represent a formal approach to the modelling of non-sharp (fuzzy) concepts and conceptual structures in the sense of traditional (Port-Royal) logic. Applications of concept lattices are in representation of conceptual knowledge and in conceptual analysis of (fuzzy) data. Similarity relations are defined and considered on three levels: similarity of objects (and similarity of attributes), similarity of concepts, and similarity of concept lattices. We show a way to factorize (simplify) concept lattices by the similarity of concepts. Also shown is how to reduce the computation of the similarity relations.

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.