Preface by the Editors of the Special Issue on Computational Intelligence and Mathematics

Mathematics 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.