Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

On triangular norms and uninorms definable in ŁΠ12

AbstractIn this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ12. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0,1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation–annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms

Information Aggregation in Intelligent Systems Using Generalized Operators

Aggregation of information represented by membership functions is a central matter in intelligent systems where fuzzy rule base and reasoning mechanism are applied. Typical examples of such systems consist of, but not limited to, fuzzy control, decision support and expert systems. Since the advent of fuzzy sets a great number of fuzzy connectives, aggregation operators have been introduced. Some families of such operators (like t-norms) have become standard in the field. Nevertheless, it also became clear that these operators do not always follow the real phenomena. Therefore, there is a natural need for finding new operators to develop more sophisticated intelligent systems. This paper summarizes the research results of the authors that have been carried out in recent years on generalization of conventional operators

NEW APPROACH TO INFORMATION AGGREGATION

In this paper new types of aggregation operators, namely absorbing-norms and parametric type of operator families called distance-based or evolutionary operators are introduced. Absorbing- norms are commutative, associative binary operators having an absorbing element from the uni! interval. A detailed discussion of properties and structure of these operators is given in the paper. Two types of distance-based operators are defined. The maximum and minimum distance operators with respect to e have the value of the element, which is farther, or nearer to e, respectively, where e is an arbitrary element of the unit interval [0,1]. The special cases e = O and e = 1lead to the max and min operators. The new operators are evolutionary types in the sense that if e is increasing starting from zero till e = 1 the min operator is developing into the max operator, while on the other side the max is transformed into the min operator. It is shown that the evolutionary operators can be constructed by means of min and max operators, which are also special cases of the operators. The maximum distance operators are special operators called uninorms and the minimum distance ones are absorbing-norms

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

TOWARDS THE GENERALIZATION OF T-OPERATORS: A DISTANCE BASED APPROACH

Aggregation is one of the key issues in the development of intelligent systems, just like with neural networks, fuzzy knowledge based systems, vision systems, and decision-making systems. From the point of view of a particular application the choice of the most appropriate operator is an important part of system design. This paper gives a brief summary of the best known operatorst- such as t-norms, t-conorms, uninorms, averaging and compensative operators, and outlines their most important properties. Two new pairs of distances, based on binary operations and their generalizations, are introduced, based on the fuzzy entropy approach, and their properties are outlined

Aggregated fuzzy answer set programming

Fuzzy Answer Set programming (FASP) is an extension of answer set programming (ASP), based on fuzzy logic. It allows to encode continuous optimization problems in the same concise manner as ASP allows to model combinatorial problems. As a result of its inherent continuity, rules in FASP may be satisfied or violated to certain degrees. Rather than insisting that all rules are fully satisfied, we may only require that they are satisfied partially, to the best extent possible. However, most approaches that feature partial rule satisfaction limit themselves to attaching predefined weights to rules, which is not sufficiently flexible for most real-life applications. In this paper, we develop an alternative, based on aggregator functions that specify which (combination of) rules are most important to satisfy. We extend upon previous work by allowing aggregator expressions to define partially ordered preferences, and by the use of a fixpoint semantics