# 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

and

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.

## Abstract

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.
## Introduction

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.