M. Eugenia Cornejo

Attribute and size reduction mechanisms in multi-adjoint concept lattices

Attribute reduction and size reduction in concept lattices are key research topics in Formal Concept Analysis (FCA). This paper combines both strategies in the multi-adjoint concept lattice framework in order to simplify the information provided by the original context. Specifically, we present three procedures which merge the attribute reduction and the size reduction by means of an irreducible α -cut concept lattice, analyzing the obtained properties.

Adjoint negations, more than residuated negations

Negation operators are useful operators, which have extensively been studied and used in different fuzzy settings, such as in fuzzy logic. This paper introduces a new and flexible kind of negation operator, based on adjoint triples, and which is called adjoint negation relative to a fixed element.Besides proving several interesting properties, the comparison with weak negations, studied by Trillas, Esteva et al., and pairs of weak negations introduced by Georgescu and Popescu, is presented. Furthermore, adjoint negations are used in order to define the corresponding dual disjunctive to an adjoint conjunctor.

On the use of thresholds in multi-adjoint concept lattices

The size of the concept lattices increases exponentially from the number of objects and attributes. This situation is more complicated in the fuzzy case, in which the considered carriers to evaluate the objects and attributes, and for the relation, are also taken into account. Hence, it is very important to study mechanisms to reduce the size of fuzzy concept lattices maintaining the main information. One of the most important mechanisms to reduce the size of concept lattices is the use of thresholds in the concept-forming operators. This paper studies this mechanism in the general fuzzy framework of multi-adjoint concept lattices, obtaining interesting properties and consequences.

Selecting the Coherence Notion in Multi-adjoint Normal Logic Programming

This paper is focused on looking for an appropriate coherence notion which allows us to deal with inconsistent information included in multi-adjoint normal logic programs. Different definitions closely related to the inconsistency concept have been studied and an adaptation of them to our logic programming framework has been included. A detailed reasoning is presented in order to motivate and justify the suitability of the chosen coherence notion.

Characterizing reducts in multi-adjoint concept lattices

Abstract The construction of reducts, that is, minimal sets of attributes containing the main information of a database, is a fundamental task in different frameworks, such as in Formal Concept Analysis (FCA) and Rough Set Theory (RST). This paper will be focused on a general fuzzy extension of FCA, called multi-adjoint concept lattice, and we present a study about the attributes generating meet-irreducible elements and on the reducts in this framework. From this study, we introduce interesting results on the cardinality of reducts and the consequences in the classical case.

Bipolar fuzzy relation equations based on the product T-norm

Bipolar fuzzy relation equations are given from the fuzzy relation equations introduced by Sanchez in the 1980s considering a negation operator in the equations. Numerous applications require variables that show a bipolar character such as decision making and revenue management, hence the importance of studying bipolar fuzzy relation equations. According to the literature, bipolar max-min equations have already been studied and a characterization of their solutions, by means of a finite set of maximal and minimal solution pairs, has been provided. This paper will present a first study on bipolar max-product fuzzy relation equations with one equation containing different variables, which includes different interesting properties in order to guarantee both their solvability and the existence of the greatest (least) solution or maximal (minimal) solutions. Moreover, a characterization of the solvability of a particular system of two bipolar max-product fuzzy relation equations is given.

Measuring the Incoherent Information in Multi-adjoint Normal Logic Programs

Databases usually contain incoherent information due to, for instance, the presence of noise in the data. The detection of the incoherent information is an important challenge in different topics. In this paper, we will consider a formal notion for this kind of information and we will study different measures in order to detect incoherent information in a general fuzzy logic programming framework. As a consequence, we can highlight some irregular data in a multi-adjoint normal logic program and so, in other useful and more particular frameworks.

Multi-adjoint Relation Equations: A Decision Support System for Fuzzy Logic

Fuzzy relation equations (FRE) are an important decision support system (DSS), for example, in fuzzy logic. FRE have recently been extended to a more general framework, called multiadjoint relation equations (MARE). This paper shows MARE as a fundamental DSS in multi‐adjoint logic programming. For that purpose, multi‐adjoint logic programs will be interpreted as a MARE, and the solvability of them will be given in terms of concept lattice theory. Furthermore, two approximations (optimistic and pessimistic approximations) of unsolvable equations will be obtained from a multiadjoint object‐oriented concept lattice. Finally, a real‐life example will be studied.

Syntax and semantics of multi-adjoint normal logic programming

Abstract Multi-adjoint logic programming is a general framework with interesting features, which involves other positive logic programming frameworks such as monotonic and residuated logic programming, generalized annotated logic programs, fuzzy logic programming and possibilistic logic programming. One of the most interesting extensions of this framework is the possibility of considering a negation operator in the logic programs, which will improve its flexibility and the range of real applications. This paper introduces multi-adjoint normal logic programming, which is an extension of multi-adjoint logic programming including a negation operator in the underlying lattice. Beside the introduction of the syntax and semantics of this paradigm, we will provide sufficient conditions for the existence of stable models defined on a convex compact set of an euclidean space. Finally, we will consider a particular algebraic structure in which sufficient conditions can be given in order to ensure the unicity of stable models of multi-adjoint normal logic programs.

Cuts or thresholds, what is the best reduction method in fuzzy formal concept analysis?

Recently α-cut irreducible and δ1δ2-multi-adjoint concept lattices have been introduced as two different methodologies focus on reducing the size of a given fuzzy concept lattice. The philosophy of both methodologies is completely different and so, the obtained lattices too. This paper analyzes the differences and proposes that the best is to combine both methodologies in order to obtain new procedures to reduce the information retrieval, only considering the important information for the user and with the advantages of both philosophies.