Computing Uncertainty in Interval Based Sets

L.M. Rocha
Computer Research Group, MS P990
Los Alamos National Laboratory
Los Alamos, NM 87545

Vladik Kreinivich
Department of Computer Science
University of Texas at El Paso
El Paso, TX 79968


R. Baker Kearfott
Department of Mathematics
University of Southwestern Louisiana
U.S.L. Box 4-1010, Lafayette, LA 70504-1010

In: Applications of Interval Computation. R.B. Kearfott and V. Kreinovich (Eds.). Kluwer Academic Press. pp. 337-380.


The kinds of uncertainty present in different interval based techniques of representing uncertainty in knowledge based systems are discussed. Interval Valued Fuzzy Sets (IVFS) are shown to describe both fuzziness and nonspecificity in their membership degrees, while a structure referred to as evidence set further introduces conflict. A more realistic model of uncertainty is described by L-fuzzy sets, interval valued L-fuzzy sets, and L-evidence sets, where L is a finite set of possible degrees of belief. Measures of uncertainty of such structures are examined.


The majority of the existing knowledge-based system use real numbers to describe the experts' degrees of certainty in different statements. In many cases, algorithms for reasoning with uncertainty are based on the quantitative estimations of the current uncertainty of knowledge.

Recently, interval-based formalisms, such as interval valued fuzzy sets (IVFS), and evidence sets (a generalization of IVFS proposed in Rocha[1994a]) have been proposed for more adequate description of experts` uncertainty. To efficiently use these formalisms in reasoning, we thus need to design uncertainty measures for these interval-based generalizations of traditional uncertainty formalisms.

In the following, measures of uncertainty for interval based sets are derived from the information measures described by Ramer et al [1990, 1994]. It will be shown that the numerical characteristics of uncertainty present in interval based set structures may be better captured in a discrete formulation where the real unit interval of degrees of belief is replaced by a linearly ordered finite set. This discrete formulation is more coherent with human cognitive abilities.

We start with a review of relevant mathematical background in Section 2; the basic concepts of uncertainty representation, as well as evidence sets, will be explained and formally defined. In Section 3, measures of uncertainty will be developed, while in Section 4, the L-fuzzy set alternatives will be defined. Finally, in Section 5, the importance of evidence sets will be stressed as models capable of expressing all the recognized forms of uncertainty.