David Harmanec

MEASURING TOTAL UNCERTAINTY IN DEMPSTER-SHAFER THEORY: A NOVEL APPROACH

A novel approach to measuring uncertainty and uncertainty-based information in Dempster-Shafer theory is proposed (independently also proposed by Maeda et al. [1993]). It is shown that the proposed measure of total uncertainty in Dempster-Shafer theory is both additive and subadditive, has a desired range, and collapses correctly to either the Shannon entropy or the Hartley measure of uncertainty for special probability assignment functions. The paper is restricted, for the sake of simplicity, to finite sets.

Constructing fuzzy measures in expert systems

Abstract This paper is an overview of results regarding various representations of fuzzy measures and methods for constructing fuzzy measures in the context of expert systems, which were obtained by the authors and their associates during the last three years. Included are methods for constructing fuzzy measures by various transformations, by extension, by statistical inference, and by various data-driven methods based either on the Sugeno-integral or the Choquet-integral and using neural networks, genetic algorithms, or fuzzy relation equations.

ON THE INTEGRATION OF UNCERTAINTY THEORIES

Investigations pursued in this paper contribute to a research project introduced by Resconi, Klir, and St. Clair [1] whose purpose is to employ syntactic and semantic structures of modal logic as a unifying framework within which various uncertainty theories can be formalized, compared, and organized hierarchically. This paper focuses on the explicit use of modal logic semantics to formalize fuzzy sets, belief measures, plausibility measures, and Sugeno λ-measures.

ON THE COMPUTATION OF UNCERTAINTY MEASURE IN DEMPSTER-SHAFER THEORY

An algorithm for computing the recently proposed measure of uncertainly AU for Dempster-Shafer theory is presented. The correctness of the algorithm is proven. The algorithm is illustrated by simple examples. Some implementation issues are also discussed.

Interpretations of various uncertainty theories using models of modal logic: a summary

Abstract This paper summarizes our efforts to establish the usual semantics of propositional modal logic as a unifying framework for various uncertainty theories. Interpretations for fuzzy set theory, Dempster-Shafer theory, probability theory, and possibility theory are discussed. Some properties of these interpretations are also presented, as well as directions for future research.

ON MODAL LOGIC INTERPRETATION OF POSSIBILITY THEORY

This paper is a continuation in a series of papers started by Resconi et. al. [8] in which we try to develop interpretations of various uncertainty theories within the framework of standard modal logic. In this paper, we deal with possibility theory. We suggest an interpretation of possibility measures and necessity measures and show that this interpretation is complete for rational-valued measures.

On modal logic interpretation of Dempster–Shafer theory of evidence

This article further develops one branch of research initiated in an article by Resconi, Klir, and St. Clair (G. Resconi, G. J. Klir, and U. St. Clair, Int. J. Gen. Syst., 21(1), 23‐50 (1992) and continued in another article by Resconi et al. (Int. J. Uncertainty, Fuzziness and Knowledge‐Based Systems, 1(1), 1993). It fully formulates an interpretation of the Dempster‐Shafer theory in terms of the standard semantics of modal logic. It is shown how to represent the basic probability assignment function as well as the commonality function of the Dempster‐Shafer theory by modal logic and that this representation is complete for rational‐valued functions (basic assignment, belief, or plausibility functions).

Types and Measures of Uncertainty

This paper is an overview of the current state of affairs in the area of measuring uncertainty. Three basic types of uncertainty are introduced: nonspecificity and conflict, which result from information deficiency, and fuzziness, which results from linguistic imprecision. Well-justified measures of these types of uncertainty in fuzzy set theory, possibility theory, Dempster-Shafer, theory as well as in classical set theory and probability theory are overviewed.

Modal logic interpretation of Dempster-Shafer theory: An infinite case

Abstract A modal logic interpretation of belief and plausibility measures defined on infinite sets is established. As a special case, a modal logic interpretation of necessity and possibility measures defined on infinite sets is also established. It is proven in both cases that the interpretation is complete. These results establish, in effect, modal logic interpretations of the Dempster-Shafer theory and possibility theory.

ON INFORMATION-PRESERVING TRANSFORMATIONS

In this paper, we investigate uncertainty-invariant transformations between the Dempster-Shafer theory, possibility theory and probability theory. These transformations are based on a well-justified measure of uncertainty in the Dempster-Shafer theory. The measure is defined as the maximum of the Shannon entropy for all probability distributions that conform to the given belief function.