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\title{Interval Fuzzy Data Processing: Case When Degree of Belief Is
Decribed by an Interval}
\author{Hung T. Nguyen and Elbert Walker}

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\begin{abstract}
The necessity to use intervals to describe degrees of belief in expert
systems stems from the following
three sources:
\begin{itemize}

\item Experts {\it are} often uncertain about their degrees of belief.

\item Operations with degrees of belief 
are not uniquely defined. For example, if we know the degrees of
belief (subjective probabilities) $a=d(A)$ and $b=t(B)$ 
of the statements $A$ and $B$, then the only thing that we know about
the degree of belief of $A\& B$ is that it lies between
$\min(a+b-1,0)$ and $\min(a,b)$. Therefore, even if we start with precise
values, e.g., $a=0.3$ and $b=0.4$, we get an {\it interval}
$[0,0.3]$ of possible values of $d(A\& B)$. This idea was proposed and
used by L. Kohout.

\item Even if for basic logical 
operations $*$ ($\&$, $\vee$, $\neg$), we fix a method of describing
$d(A* B)$ as a function
of $d(A)$ and $d(B)$, there is a third source of interval uncertainty:
For a complicated formula, e.g., $A\to B$, there
exists several methods of representing $A\to B$ in terms of $\&$,
$\vee$, and $\neg$: $B\vee\neg A$, $(A\&B)\vee \neg A$ etc. These
formulas are equivalent in classical logic, but they lead to different
numerical values for $d(F)$. As a result, even if initially we had
intervals, and if operations are chosen, if we take the smallest and
the largest of them, we get the {\it interval} of possible values
of $d(F)$. This idea has been proposed and used by I. B. Turksen.
\end{itemize}
In this talk, we will survey the related mathematical results and
applications. Some of them have been described during a special
session of NAFIPS/IFIS/NASA-94 (see \cite{Nafips} and references therein).
\end{abstract} 
  
\auffil{The authors are with the Department of Mathematical Sciences,
New Mexico State University, Las Crices, NM 88003.}

\begin{thebibliography}{99}
\bibitem{Nafips}Vladik Kreinovich and Hung T. Nguyen, 
``Applications of fuzzy intervals: a skeletal outline
of papers presented at this section'', 
In: L. Hall, H. Ying, R. Langari, and J. Yen
(eds.), {\it NAFIPS/IFIS/NASA'94, Proceedings of the First
International Joint Conference of The 
North American Fuzzy Information Processing Society Biannual Conference, The
Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint
Technology Workshop on Neural Networks and Fuzzy Logic, San Antonio,
December 18--21, 1994}, IEEE, Piscataway, NJ, pp. 461--463.
\end{thebibliography}
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