In this sequel to our previous article (Corderoet al., 2020) on general inference systems for reasoning with if–then dependencies, we study transformations of if–then rules to semantically equivalent collections of if–then rules suitable to solve several problems related to reasoning with data dependencies. We work in a framework of general lattice-based if–then rules whose semantics is parameterized by systems of isotone Galois connections. This framework allows us to obtain theoretical insight as well as algorithms on a general level and observe their special cases by choosing various types of parameterizations. This way, we study methods for automated reasoning with different types of if–then rules in a single framework that covers existing as well as novel types of rules. Our approach supports a large family of if–then rules, including fuzzy if–then rules with various types of semantics. The main results in this article include new observations on the syntactic inference of if–then rules, complete collections of rules, reduced normal forms of collections of rules, and automated reasoning methods. We demonstrate the generality of the framework and the results by examples of their particular cases focusing on fuzzy if–then rules.

We study the semantics of fuzzy if-then rules called fuzzy attribute implications parameterized by systems of isotone Galois connections. The rules express dependences between fuzzy attributes in object-attribute incidence data. The proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches. We formalize the general parameterizations, propose bivalent and graded notions of semantic entailment of fuzzy attribute implications, show their characterization in terms of least models and complete axiomatization, and provide characterization of bases of fuzzy attribute implications derived from data.

We deal with the expressive power of if-then rules called fuzzy attribute implications (FAIs) which can be seen as formulas A ⇒ B where both A and B are conjunctions of subformulas containing propositional variables and constants for truth degrees and whose interpretation is parameterized by linguistic hedges. The formulas admit the following model-theoretical property: a system of fuzzy sets [Formula: see text] is a fuzzy closure system satisfying an additional condition of being closed under a*-shifts (so-called L*-closure system) if and only if [Formula: see text] is a system of models of a set of FAIs. In this paper, we point out the importance of constants of truth degrees in the antecedents of formulas by showing that simpler formulas are not sufficient to describe all L*-closure systems. As a practical consequence, the simpler formulas cannot be used as an alternative description of concept lattices with linguistic hedges.

We present a novel approach to compute formal concepts of formal context. In terms of operations with Boolean matrices, the presented algorithm computes all maximal rectangles of the input Boolean matrix which are full of 1s. The algorithm combines b

We address important issues of dimensionality reduction of transactional data sets where the input data consists of lists of transactions, each of them being a finite set of items. The reduction consists in finding a small set of new items, so-called factor-items, which is considerably smaller than the original set of items while comprising full or nearly full information about the original items. Using this type of reduction, the original data set can be represented by a smaller transactional data set using factor-items instead of the original items, thus reducing its dimensionality. The procedure utilized in this paper is based on approximate Boolean matrix decomposition. In this paper, we focus on the role of frequent closed item sets that can be used to determine factor-items. We present the factorization problem, its reduction to Boolean matrix decompositions, experiments with publicly available data sets, and an algorithm for computing decompositions.

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.

Our paper deals with the expressive power of fuzzy attribute implications which are if-then rules describing dependencies among graded attributes. In our previous work, we have shown that fuzzy attribute implications are important in data mining because they can be used as a concise description of all if-then dependencies which are hidden in object-attribute fuzzy relational data. Fuzzy attribute implications can be seen as formulas (in the narrow sense) of the form A rArr B where both A and B are conjunctions of subformulas containing constants for truth degrees acting as (constants for) threshold truth degrees. This paper investigates possibility to replace sets of fuzzy attribute implications with fuzzy sets of ordinary if-then formulas in the sense of Pavelkapsilas abstract logic. We reveal the impossibility to replace fuzzy attribute implications by the ordinary formulas without losing their expressive power. From the technical point of view, we present counterexamples demonstrating that fuzzy sets of the ordinary attribute implications cannot be used to describe sets of fixed points of arbitrary fuzzy closure operators.

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.

Presented is indirect method for generating non-redundant bases of data tables with fuzzy attributes. Fuzzy attribute implications (FAIs) are formulas describing particular dependencies of attributes in data. Non-redundant bases are minimal sets of FAIs describing all FAIs which are true (valid) in given data. Our method is based on reducing sets of FAIs describing all dependencies which can contain redundant FAIs. By removing the redundant FAIs we obtain this way non-redundant bases. We show that the procedure can generate smaller bases than previous methods. We present new theoretical results, the algorithm, its complexity analysis, and statistics demonstrating efficiency of the method.

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