Siegfried Gottwald

Set theory for fuzzy sets of higher level

Abstract We extend the usual notion of fuzzy set in such a way that the elements of fuzzy sets again can be fuzzy sets. For such fuzzy sets of higher level the fuzzy set theoretic operations are generalized up to the notion of a fuzzy mapping. In our presentation of the results we use a suitable many valued logic, indicating in this way the close formal connections between fuzzy and classical set theory.

Solvability of fuzzy relational equations and manipulation of fuzzy data

Abstract The problem of characterization of fuzzy relational equations with respect to their solvability property is studied. Some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered. Two stages of manipulation process are indicated: combining pieces of evidence and inferring their mutual correspondence. Both of them are formulated and solved by the use of fuzzy relational equations. The role of a solvability index is extensively studied. Numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers.

On the existence of solutions of systems of fuzzy equations

Abstract A necessary and sufficient condition for the existence of solutions of a fuzzy equation given by E. Sanchez is extended to systems of fuzzy equations.

Mathematical fuzzy logic as a tool for the treatment of vague information

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations.

Triangular norm-based mathematical fuzzy logics

Abstract In this chapter, we consider particular classes of infinite-valued propositional logics which are strongly related to t-norms as conjunction connectives and to the real unit interval as set of their truth degrees, and which have their implication connectives determined via an adjointness condition. Such systems have in the last ten years been of considerable interest, and the topic of important results. They generalize well-known systems of infinite-valued logic, and form a link to as different areas as e.g. linear logic and fuzzy set theory. We survey the most important ones of these systems, always explaining suitable algebraic semantics and adequate formal calculi, but also discussing complexity issues. Finally we mention a type of extension which allows for graded notions of provability and entailment.

A new axiomatization for involutive monoidal t-norm-based logic

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the involutive monoidal t-norm-based residuated logic.

Solvability and approximate solvability of fuzzy relation equations

We give here a discussion of approximate solvability of a system of fuzzy relation equations. We demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations. First we explain the general framework, and later on we prove some particular results related to the problem of the best approximation.

Approximately solving fuzzy relation equations: some mathematical results and some heuristic proposals

Abstract Fuzzy relation equations are intimately connected with the approach toward fuzzy control. Unfortunately, systems of such equations are solvable only under quite restrictive conditions. Therefore a notion of approximate solution is discussed which applied also in cases where such systems do not have solutions. For some types of fuzzy equations best possible approximate solutions are known, for other types some strategies are considered to suitably guess approximate solutions. Furthermore, that approach is applied to some types of “cascaded” fuzzy relation equations which are connected with ideas to join neural network structures with fuzzy control.

Many-Valued Logic And Fuzzy Set Theory

Rather early in the (short) history of fuzzy sets it became clear that there is an intimate relationship between fuzzy set theory and many-valued logic. In the early days of fuzzy sets the main connection was given by fuzzy logic — in the understanding of this notion in those days: and this was as switching logic within a multiple-valued setting.

Fuzzy uniqueness of fuzzy mappings

Abstract For fuzzy mappings between fuzzy sets of higher level we study a notion of fuzzy uniqueness. i.e. of fuzzy functions, which formalizes the intuition that ‘almost identical’ images under f . Characterizations are given for such fuzzy mappings which are really (i.e. with truth value 1) fuzzy unique or fuzzy biunique, and for those fuzzy sets which can be domains of really fuzzy unique fuzzy mappings. Finally, as an application fuzzy cardinals, their ordering, sum, and product are introduced, and some properties proved.