There have been several pioneering approaches to the extraction of attribute implications from a formal context, dating from the 1980's: the one of Guigues and Duquenne based on so-called non-redundancy nodes, another one proposed by Ganter highlighting the concept of pseudo-closed set, and the best-known one relying on the recursive computation of so-called pseudo-intents in the book by Ganter and Wille. The Guigues and Duquenne approach has never been compared in detail in the literature with the other two, although they turn out to be equivalent. This paper tries to fill this gap, proposing a unified view, hopefully more easy to grasp.

Fuzzy relation equations (FRE) are a useful formalism with a broad number of applications in different computer science areas. Testing if a solution exists and, if so, computing the unique greatest solution is straightforward. In contrast, the computation of minimal solutions is more complex. In particular, even in FRE with a very simple structure, the number of minimal solutions can increase exponentially. However, minimal solutions are immensely useful since, under mild conditions, they (together with the greatest solution) allow one to describe the entire space of solutions to an FRE. The main result of this work is a new method for enumerating the set of minimal solutions. It works by establishing a relationship between coverings of FRE and maximal independent elements of (hyper-)boxes. We can thus make efficient enumeration methods for maximal independent elements of (hyper-)boxes applicable also to our setting of FRE, where the operator considered in the composition of fuzzy relations only needs to preserve suprema of arbitrary subsets and infima of non-empty subsets. More specifically, we thus show that the enumeration of the minimal solutions of an FRE can be done with incremental quasi-polynomial delay.

A method to solve a latticed optimization problem constrained by a bipolar fuzzy relation equation is presented in this paper, under the hypothesis of a partially monotonic objective function. The solving strategy consists of transforming the problem into optimizing an order-preserving function in all arguments subject to another bipolar fuzzy relation equation. As a result, all the solutions of the original optimization problem can be deduced from the extremal elements of the feasible domain of the transformed problem. The presented approach embraces the particular case of linear optimization constrained by bipolar fuzzy relation equations.

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.

Preface by the Editors of the Special Issue on Computational Intelligence and MathematicsMathematics is the foundation of almost all areas of sciences, especially it is so with Engineering, Computer Science, Physics, Chemistry, and Business, and new mathematical models and approaches continuously improve the efficiency of current methodologies and provide solutions for new challenges.An important and emerging field of Computer Science is Computational Intelligence (CI), whose aim is to provide methods to be able to deal with complex real-world problems for which traditional approaches are not feasible.Some of the methods that CI encompasses are, among others, fuzzy logic, evolutionary computation, neural networks, as well as probabilistic and statistical approaches, such as Bayesian networks or kernel methods.Approaches based on CI often provide a compromise between resource intensity and accuracy or precision of the solution.So, for example, NP-hard problems, which are known to be intractable (at least as far no polynomial complexity algorithm has been ever found for any of the-mathematically equivalent-NP-hard problems), may be rather well solved for limited size and limited problem structure by various heuristics.The classical Traveling Salesman Problem (TSP) may be rather well tackled by the Lin-Kernighan heuristics for smaller size graphs (up to a few 100), while the CONCORDE approach delivers good results up to sizes 1000-1500.However, for problems with sizes over 2000 often there is no known solution as far.This is an excellent training field for CI approaches, and reference data sets are available in abundance.It is clear that both areas, CI and Mathematics, are closely related since the latter is the fundamental base of the former, and continuous interactions between them will bring more and more robust and efficient CI models and approaches-while sometimes the new CI methods deliver unexpected results useful even for getting closer to the solution of mathematically unsolved problems.This special issue includes a small selection of papers written by scientists and engineers working in the field of CI and applied mathematics and proposes some new results that might interest fellow scientists in both fields.The first paper by Rodriguez-Lorenzo et al. defines a sound and complete inference system for triadic conditional attribute implication (CAI) generated from a formal triadic context and expressed as a set of axioms "a la Armstrong."Moreover, it proposes a method to compute CAIs from Biedermann's implications and introduces an algorithm to compute the closure of an attribute set X with respect to a set of CAIs given a set of conditions.