%From mantaras@iiia.csic.es Wed Dec 28 02:43:51 1994
%From: mantaras@iiia.csic.es (Ramon Lopez de Mantaras)

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\title{From Intervals to Fuzzy Truth-Values:\\
Adding Flexibility to Reasoning Under Uncertainty}
\author{Ramon Lopez de Mantaras}

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

\maketitle

\auffil{The author is with IIIA--Artificial Intelligence Research Institute,
CSIC--Spanish Research Council,
Campus Universitat Autonoma de Barcelona,
08193 Bellaterra, Barcelona, Spain.}

Early uncertainty management systems like certainty factors 
\cite{Shortliffe and Buchanan 1975} 
and probability based approaches 
\cite{Adams 1976,Duda Hart and Nilsson 1976}, were dealing only with
one single real value chosen within a
predefined range to represent uncertainty. These approaches had serious
problems, in particular it has been argued that human numerical estimations
of uncertainty have little interobserver and intraobserver consistency
\cite{Beyth-Marom 1982,Horvitz and Heckerman 1986}. Even worse, most experts
are unable to make a fair estimation of the inaccuracy of their judgments,
making far larger estimation errors than the boundaries accepted by them as
feasible \cite{Freksa and Lopez de Mantaras 1984}.

A first step out of this problem, that added some flexibility to the
uncertainty representation problem, was proposed in several interval valued
approaches, some generalizing probability based approaches, like Quinlan's
consistency and plausibility masures or Dempster-Shafer belief and
plausibility measures \cite{Shafer 1976}; and others outside the probabilistic
setting like the approach of the possibility and necessity measures 
\cite{Dubois and Prade 1988} 
based on the notion of fuzzy set \cite{Zadeh 1965}. Besides
making easier for the expert to preserve consistencies, these two-valued
approaches allow for an explicit representation of ignorance a concept not
easily representable with single-valued approaches; for example in single
valued probability based approaches the lack of certainty about an event
implies the certainty of its negation even in situations where the poverty
of information would suggest that the best we could say is that we are
uncertain about both, the event and its negation i.e. we would like to
explicitely express our ignorance.

Even though the two-valued (interval valued) approaches have been a
valuable step, they still lack the capability of dealing with imprecision,
a very important source of uncertainty mainly in artificial intelligence
and particularly in expert systems.

\section{Uncertainty and Imprecision}
At this point it may be interesting to point out the differences and
relationship between uncertainty and imprecision. For example, given the
imprecise information: {\it the temperature of the reactor is over 200
degrees}, the proposition: {\it the temperature of the reactor is exactly 300
degrees} is uncertain but very precise. On the other hand, if the known
information is very precise, like for example: {\it the temperature of the
reactor is 250 degrees}, then there is no room for uncertainty, that is, we
will always be able to say whether a proposition is true or false. However,
if the known information contains terms with a vague meaning such as:
{\it the
temperature of the reactor is high}, then, in general the only thing we
will be able to say is to what degree a given temperature is a high
temperature. This degree is precisely the degree of membership of the given
temperature to the fuzzy set HIGH TEMPERATURE.

\section{Posibility Theory}
It is worth noting that a given membership value has two different
meanings. On the one hand, it denotes the degree of truth of a proposition
about a variable when the exact value of this variable is known (for
example, let us assume that 0.8 is the degree of membership of the
temperature value 600 degrees in the fuzzy set HIGH TEMPERATURE, then 0.8
is the degree of truth of the proposition {\it high temperature} for a
temperature value of exactly 600 degrees). On the other hand, a membership
value also denotes the degree of certainty about the fact that the variable
takes a concrete value (i.e. 0.8 is the degree of certainty of the
temperature 600 degrees knowing that the temperature is high). This second
interpretation is the one associated with the concept of possibility
distribution \cite{Zadeh 1978}. Possibility theory is at the basis of
{\it Possibilistic Logic}.

\section{Possibilistic Logic, Many-Valued Logic and Fuzzy Logic}
Possibilistic logic and Many-Valued logic are particular cases of fuzzy
logic. In the case of possibilistic logic, we are interested in the
evaluation of the truth of a crisp (non vague) proposition given some vague
information like in the temperature example above. When we are interested
in the truth evaluation of a vague proposition given a crisp information,
we are in the setting of many-valued logics. In the more general case in
which we evaluate the truth of a vague proposition given a vague
information we are in the setting of fuzzy logic. In that case the result
of the evaluation is a fuzzy set called fuzzy truth-value. Fuzzy
truth-values allow to integrate the vagueness as well as the uncertainty
resulting from imprecise information within the same framework. Fuzzy
truth-values are fuzzy subsets of [0,1] taking values on [0,1] that are
referred to by means of linguistic terms such as: {\it absolutely false, very
false, false, fairly false,..., very true, absolutely true}. They provide a
very flexible way of representing uncertainty and reasoning under
uncertainty.

\section{Reasoning Within Fuzzy Truth-Valued Logics}
A fuzzy truth-valued logic is a logic whose truth-values are fuzzy
truth-values (i.e. linguistic values). In this logic, propositions are
fuzzy truth-qualified and the generalized modus ponens inference rule is as
follows: Given a fuzzy truth-qualification T1 of a rule {\it If X is A then Y
is B} and a fuzzy truth-qualification T2 of the fact {\it X is A}, one can
deduce a fuzzy truth-qualification T3 for the conclusion {\it Y is B},
where T3
is computed from T1 and T2 (see \cite{Lopez de Mantaras 1990} for details).

We have developped an expert systems building tool based on this approach
and several large scale expert systems have been developped mainly in two
medical applications \cite{Godo et al 1989,Belmonte et al 1994} the results
obtained are remarkable since in all the applications the experts felt
extremely confortable with the possibility of expressing their uncertainty
by menas of linguistic truth-values. Furthermore the performance of the
espert systems was compared, by mans of a cluster analysis, with several
physicians (6 in the case of pneumonia diagnosis and 12 in the case of
rheumatology) and in both cases the expert systems clustered with the best
physicians.

\section{Conclusions}
Looking at the chronology of the approaches to deal with uncertainty we
have noticed a sustained tendency to increase its flexibility in order to
avoid problems of inconsistency. The most flexible approach to date is that
of the fuzzy truth-values which is based on fuzzy logic. We believe that
the fuzzy truth-values approach to reasoning is the most powerful idea
within fuzzy logic both from the theoretical and the practical point of
view.

\begin{thebibliography}{99}
\bibitem{Moore 1966}
R. E. Moore, {\bf Interval analysis}, Prentice Hall, Englewood Cliffs,
NJ, 1966. 

\bibitem{Author 1994}
A. B. Author, ``Bad applications of interval computations'', {\it
International Journal of Bad Applications}, 1994, Vol. 8, pp. 666--669.
    

\bibitem{Adams 1976}
J. B. Adams, ``A probability model of medical reasoning and the MYCIN
model'', {\it Mathematical Biosciences}, 1976, Vol. 32, pp. 177--186.

\bibitem{Belmonte et al 1994}
M. Belmonte-Serrano, C. Sierra, and R. Lopez de Mantaras, ``RENOIR :
An expert system using fuzzy logic for rheumatology diagnosis'',
{\it International Journal of Intelligent Systems}, 
1994, Vol. 9, pp. 985--1000.

\bibitem{Beyth-Marom 1982} 
R. Beyth-Marom, ``How probable is probable (a numerical taxonomy
translation of verbal probability expressions),'' {\it 
Journal of Forecasting}, 1982, Vol. 1,
pp. 257--269.

\bibitem{Dubois and Prade 1988}
D. Dubois and H. Prade, {\bf Possibility Theory}, Plenum Press, N.Y., 1988.

\bibitem{Duda Hart and Nilsson 1976}
R. O. Duda, P. E. Hart, and N. J. Nilsson, ``Subjective Bayesian methods
for rule-based inference systems'', in {\it Proceedings of the AFIPS National
Computer Conference}, 47, 1976, p. 1075--1082.

\bibitem{Freksa and Lopez de Mantaras 1984}
C. Freksa and R. Lopez de Mantaras, ``A learning system for
linguistic caregorization of soft observations'', {\it 
Proc. Colloque Association
Recherche Cognitive, Orsay}, 1984, pp. 331--345.

\bibitem{Godo et al 1989}
L. Godo, R. Lopez de Mantaras, C. Sierra, and A. Verdaguer, ``MILORD: 
The architecture and the management of linguistically expressed
uncertainty'', {\it International Journal of Intelligent Systems},
1989, Vol. 4, pp. 471--501.

\bibitem{Horvitz and Heckerman 1986}
E. Horvitz and D. Heckerman, ``The inconsistent use of measures of
uncertainty in artificial intelligence research'', in Kanal and Lemmer (eds.),
{\bf Uncertainty in Artificial Intelligence}, North-Holland, 1986, 
pp. 137--151.

\bibitem{Lopez de Mantaras 1990}
R. Lopez de Mantaras, {\it Approximate Reasoning Models}, 
Ellis Horwood Publ., 1990. 

\bibitem{Shafer 1976}
G. Shafer, {\bf A Mathematical Theory of Evidence}, 
Priceton University Press, 1976.

\bibitem{Shortliffe and Buchanan 1975}
E. H. Shortliffe and B. G. Buchanan, ``A model of inexact reasoning in
medicine'', {\it Mathematical Biosciences}, 1975, Vol. 23, pp. 351--379.

\bibitem{Zadeh 1965}
L. A. Zadeh, ``Fuzzy Sets'', {\it Information and Control}, 
1965, Vol. 8, 338--353.

\bibitem{Zadeh 1978}
L. A. Zadeh, ``Fuzzy sets as a basis for a theory of possibility'',
{\it Fuzzy
Sets and Systems}, 1978, Vol. 1, pp. 3--28.
\end{thebibliography}

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       _/ _/ _/_|   Ramon Lopez de Mantaras
       / _/ _/ _|   IIIA - Artificial Intelligence Research Institute
        _/ _/___|   CSIC - Spanish Scientific Research Council
       _/ _/____|   Campus Universitat Autonoma de Barcelona
       / _/    _|   08193 Bellaterra, Spain

                    voice: +34-3-580 95 70    fax: +34-3-580 96 61
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