Interval Valued R-Implications and Automorphisms

Interval fuzzy logic is firmly integrated with principles of fuzzy logic theory and interval mathematics. The former provides a complete and inclusive mathematical model of uncertainty from which the foundations of fuzzy control have widened the scope of control theory. The latter models the uncertainty and the errors in numerical computation, leading to self-validated methods. Both areas were independently developed in the mid 1960s improving the quantitative analysis of approximations to mathematically exact values, which may not be observable, representable or computable. Interval fuzzy connectives have been described in the terms of the combination of those theories. In this work, the best interval representation is considered for the study of R-implication in fuzzy logic. Based on the best interval representation, an interval fuzzy R-implication is obtained as a canonical extension satisfying the optimality property and preserving the same properties satisfied by the fuzzy R-implication. In addition, commutative diagrams relate fuzzy R-implications to interval fuzzy R-implications. This leads to the understanding of how interval automorphisms act on interval R-implications and generate other interval fuzzy R-implications

Properties of fuzzy implications obtained via the interval constructor

ABSTRACT The extension of classical logic connective to the unit interval [0

The best interval representation of fuzzy Simplications and automorphisms

Abstract — The aim of this work is to analyze interval fuzzy S-implications and interval automorphisms. Starting from any fuzzy S-implication, it is shown how to obtain an interval fuzzy S-implication canonically. We proved that the such interval fuzzy S-implications meet the optimality property and preserve the same properties satisfied by the fuzzy S-implication. In addition, commutative diagrams are used in order to relate fuzzy S-implications to interval fuzzy S-implications, and to understand how interval automorphisms act on interval Simplications, generating other interval fuzzy S-implications. I