International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

In this paper, we address the problem of identification of fuzzy measures through different representations, namely the Möbius, the Shapley and the Banzhaf interaction representations. In the first part of the paper, we recall the main results concerning these representations, and give a simple algorithm to compute them. Then we determine the bounds of the Möbius and the interaction representations for fuzzy measures. Lastly, the identification of fuzzy measures by minimizing a quadratic error criterion is addressed. We give expressions of the quadratic program for all the considered representations, and study the uniqueness of the solution.

A generalization of k-order additive discrete fuzzy measures recently introduced by Grabisch is shown. k-order additive fuzzy measures on general spaces are introduced. Connection of the proposed generalization with the general Möbius transform of Shafer is shown. General evaluation formula for the Choquet integral is given. Further generalizations concerning the type of applied arithmetics are proposed.

The k-order maxitive fuzzy measures on general spaces are discussed. The sum of k_i-order maxitive measures is shown to be a ∑ k_i-order maxitive fuzzy measure. The sum of two maxitive measures (possibility measures) is discussed and completely analyzed.

Aggregation of object or action attribute values in the presence of interaction among attributes, is of great practical importance in many fields. A discrete fuzzy Choquet integral is a very promising tool for the interaction modeling, but it allows the modeling of positive interaction and redundancy and their combinations. In this paper, the modeling of logically more complex interaction by Choquet integral is performed using a general discrete measure.

Modal logic interpretations of plausibility and belief measures are developed based on the observation that the accessibility relation in a model of modal logic, regarded as a multivalued mapping, induces a plausibility measure and a belief measure on the set of possible worlds.

Operation with functions known as pseudo-convolution and its generalization as well as theirs basic properties has been presented. Then, it has been proved that pseudo-convolution which core is pseudo-integral based on max or min decomposable measure can be obtained as limit of g-convolutions, i.e., pseudo-convolutions with pseudo-integrals based on ⊕-decomposable measures where ⊕ is generated pseudo-addition, as their cores.

This paper improves a previously proposed axiomatic setting for qualitative decision under uncertainty in the von Neumann and Morgenstern' style, where only ordinal linear scales are required for assessing uncertainty and utility. Two qualitative criteria are axiomatized in a finite setting: a pessimistic one and an optimistic one, respectively obeying an uncertainty aversion axiom and an uncertainty-attraction axiom. These criteria generalize the well-known maximin and maximax criteria, making them more realistic. They are suited to one-shot decisions and they are not based on the notion of mean value, but take the form of medians. Elements for a qualitative case-based decision methodology are also proposed, with pessimistic and optimistic evaluations formally similar to the expressions which cope with uncertainty, up to modifying factors which cope with the lack of normalization of similarity evaluations. Finally two extensions of the model are analysed: (i) the case of generalized possibilistic mixtures, using a t-norm instead of min, and (ii) the case of evaluating either preferences or uncertainty on Cartesian products of ordinal scales.

As an alternative to quantum logics MV algebras have been suggested^2,20. In a quite general class of MV algebras there have been constructed sum of any two observables^10,14,20. Recently the notion of product MV algebra has been introduced^4,5,12,17. In any weakly σ-distributive product MV algebra the joint observable of any observables can be constructed and on the base also the sum of any observables. In this paper we show that the two concepts of constructing of the sum are equivalent.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to vladik@cs.utep.edu, or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.