Coimplicações Fuzzy Valoradas Intervalarmente

Abstract

Traditional digital logic deals with variables assuming only two possible states: false and true. But for a large number of real world modeling, we want intermediary values. The concept of duality, stating that something can and must coexist with its opposite, makes the fuzzy logic seem natural, even inevitable. Thus, the fuzzy logic introduces the ability to infer conclusions and generates responses based on vague information, which is also ambiguous and qualitatively incomplete and inaccurate. In this context, the way of thinking of fuzzy-based systems is similar to humans, representing the expressions of natural language in a very simple and intuitive way, leading to the construction of systems easy to understand and to maintain. Other important area of research based on mathematical models for the treatment of uncertainty considers interval mathematics, which has been applied in the representation of inaccurate data. In interval mathematics, the principle of correctness is the assurance that in the computation of an algorithm, the interval output contains all possible outcomes corresponding to punctual data for an interval input. In addition, the optimality principle, determines that the interval output is the smallest possible one satisfying accuracy. Thus, the correctness is the minimum condition while the optimality is the ideal condition to be satisfied by interval computations. Based on these statements, intervals can be used to represent unknown values and to represent continuous values in scientific computing algorithms. The aims of interval valued fuzzy logic are to consider the interval fuzzy constructions as fuzzy constructors which are correct and to analyze criteria to ensure optimality. The extension of the interval connectives of fuzzy logic in this work is based on the canonical interval representation of real functions and in this case, restricted to the unit interval [0; 1] of the real line. Such representation always returns the smallest interval containing the image of the function. Considering concepts and foundations of both approaches, fuzzy logic and interval mathematics, this work studies the operators defined as coimplications. They are characterized as dual structure of the fuzzy implications. Moreover, it seeks to introduce the extension of the interval fuzzy coimplications, and analyses the satisfaction of properties similar to the respective classes of fuzzy coimplications. In particular, we show that valued interval fuzzy coimplications are representations of fuzzy coimplications satisfying these two principles. This Work also analise the dual structure of interval conjugate fuzzy implications, which are obtained from interval automorphisms