Inexact Information Systems and its Application to
Plamena Tsanova Andreeva
(Institute of Control and System Research
Bulgarian Academy of Sciences
Acad. G. Bonchev st. Bl. 2 Sofia 1113, Bulgaria
Abstract: The inexact information system is based on
linguistic terms which have values lying in the interval
[0,1]. Imprecision has advantages, because fuzzy sets avoid the
rigidity of conventional mathematical reasoning and computer
programming. Fuzzy quantifiers are made explicit by means of fuzzy
logic. Many systems, for example, complex biological processes, cannot
be programmed in a precise way. With fuzzy sets the implicit
quantifiers can be easily translated into machine usable form. This
paper discusses a method for the description of fuzzy quantifiers in
formal languages. A comparison between approximate reasoning and the
method of linear interpolation is made. Inexact information in
biological and medical expert systems, and the reliability inferences
based on it, are also discussed.
Key Words: Linguistic approach, fuzzy implication, fuzzy quantifier,
fuzzy number, approximate reasoning, information system.
Category: I.2.1, H.4, J
In convectional information systems there is strong operation
consequence following a given algorithm. The direction is from
definite digital for any kind of quantitative information to a fixed
computer program. For the process of decision making to be successful
there must be an exact and full description of the problem to be
solved. In an inexact information system linguistic terms are used the
values of these linguistic terms are inexact, and there is sometimes
only a vague idea of how to interpret them.
Humans tend to use words rather than numbers to describe how systems
behave. Words are a form of inexact information appropriate to
communication, and used in complex biological, economical and expert
systems. The measurement of this information is both quantitative
and qualitative. Many different senses may be fitted to a single
word. The question which arises is how exactly to interpret this
Fuzzy sets simplify the task of translation between human reasoning
and operation of digital computers. Such translations are made by
providing the membership function that defines linguistic values -
such as "very", "highly", "young", "like", "healthy". This is
particularly important in expert systems, where the instructions to be
programmed are essentially rules of thumb. In fuzzy logic fuzzy
quantifiers are made explicit. A fuzzy quantifier such as "most" may
be represented as a fuzzy number, that is, a fuzzy set that defines the
degree to which any given proposition matches the definition of
"most". The capability to translate concepts such as "usually" in a
consistent way would be an advantage
for expert systems. The translation between human usable linguistic
terms and formal languages operators might be expressed by fuzzy
numbers, including the context factor and prehistory.
The inference process from imprecise or vague premises is becoming
more and more important for knowledge-based systems, especially for
fuzzy expert systems (see [Mamdani 77], [Yager 84], [Zadeh 75]).
In approximate reasoning there are several kinds of inference
rules, which deal with the problem of deduction of conclusions in an
2 Concept Foundation
Fuzzy logic is the tool which gives programs the capability of making
an approximate logical deduction from incomplete or imprecise
knowledge. The process of inference which is used by linguistic
approach is called fuzzy implication. The exact calculation formulas
for the compositional rule of inferences, which has the following
global scheme (see [Zadeh 75]):
Global scheme of Generalized Modus Ponens:
Relation: If X is A, Y is B
Observation: X is A'
Conclusion: Y is B',
where the membership function of the conclusion is defined by some
sup-t-norm composition of and relation matrix R.
Let there be two linguistic variables X and Y defined over the
universe U = and , respectively, and let us consider the two
propositions p1 and p2, expressed in natural language. Also
where A and are fuzzy
subsets over U, B is a fuzzy subset over V and the relation R is a
fuzzy subset of the Cartesian product U x V. The fuzzy implication (1)
is chosen by some user defined approach. Using the compositional rule
of inference we obtain
where denotes the well known max-min composition operator.
We assume that we have the input fuzzy information X is . Then the fuzzy expert system may
infer that Y is by
(2). Let there be n fuzzy implications of type (1) which specify the
rule in some system. The final inference may be Y is as result of
separate rule based computing and their consequent union. If we
assume that we are given some inexact (i.e., not crisp) input
information, and if we do not take this fact into account, the final
inference may be quite different from the real one. One solution to
this problem is the defuzzification of fuzzy sets using any of several
existing methods. We propose to perform calculations using fuzzy
implicit quantifiers which have the possibility of being dynamically
3 Method of Fuzzy Quantifiers Description
One of the first attempts at the quantification of word meaning was made by
Mosier (1941) in [Hersh and Caramazza 76]. Mosier hypothesized that the
meaning of a word may be considered as containing two components: a constant,
reflecting the overall meaning value and a variable component representing the
variation in the meaning of the word due to context.
The method however requires the availability of a simple
interpretation of every one case. This requirement is difficult to
fulfill in real situations where many of the medical cases for each
concrete person will be appearing for the first time. In this case
more often than not, there is a vague idea about activity, which must
then be estimated subjectively. Facing this situation one solution is
The proposed method for the description of fuzzy quantifiers in formal
language involves two steps. The first step is to estimate the context
(dependance from related Data-base), this is C value, which is in
connection with previously quantifier's state . The second step gives
the calculated value of this quantifier used at the current moment. We
then obtain the meaning value of the current fuzzy number
(quantitative measurement) from the sum of the results of the first
and second steps.
Since the grade of membership is both subjective and dependent on
context, there is not much point in treating it as a precise
number. How then do we calculate the fuzzy quantifier "most", or
"usually" in the preposition: "If John is ill, John is in the
Relation: "If John is ill, John is in the hospital."
Observation: "John is ill."
Conclusion: "John is in the hospital."
This means "usually" with 0.6, membership grade in which the "usually"
may be translated.
Thus to calculate the implicit fuzzy quantifier in the preposition
given above it is proposed to calculate them using the fuzzy numbers
in the formula
where and are parametric components, over people; is the membership
function value against previously x; is the current value of the fuzzy
quantifier, and C is a context dependant value in the interval [0,1].
In (3) the "+" operation is the fuzzy sum operation, defined by Zadeh
We use the product operation given by
When we use a linguistic term for the first time then we have the following
reduced formula obtained from (3) by ignoring
The value X is assumed to be the current value of the fuzzy
quantifier , and is a weight coefficient which indicates how many times
it was used. The algorithm for calculating the fuzzy quantifier is
dynamic and in each subsequent consideration of , the value of is
altered. Thus, in the end we have a calculated value for a fuzzy number
that implies the context, the prehistory and the recursive accumulated
use of such linguistic terms. How then do we calculate with fuzzy
numbers? This is matter of deffinition and furthermore assumes that the
interpretation of connectives in fuzzy logic is generally
context-dependent rather than universal. Following [Wood, Antonsson
and Beck 90] we use the triangular function for the representation
of linguistic notions.
4 Example Analysis and Comparison with Interpolation Method
Instead of assuming that an ill-known value should be represented by a
probability distribution, a fuzzy number may be more appropriate. In
general, calculating the membership function is a non-trivial
problem. However, in certain cases it is possible to calculate
relatively easily. For example, let us consider the fuzzy numbers
x = "about 3" and y = "roughly 6", for which the membership functions
Linear interpolation is valid between the point x = [2 : 0,3 : 1,4 : 0], and
the point y=[0 : 0,6 : 1,9 : 0] [see Fig. 1]. We can easily calculate the
difference byres=y-x=[- 4: 0,3: 1,5 : 0].
The resulting membership function is obtained by:
The sum may be calculated in an analogous manner.
Multiplication and division are similar, although linear interpolation
is no longer valid. Fuzzy numbers are an approximation because data is
not known accurately, and we should not calculate results to a greater
accuracy than is justified by the original data. The imprecision in an
inferred result is greater than the imprecision contained in each
premise, just like in error calculus where as soon as computations are
performed, imprecision increases.
In [Raha and Ray 92] it is demonstrated that
instead of inferring by performing approximate reasoning using a
relational matrix R formed from a compound proposition p1 and a
simple proposition of the form p2:
p1 Relation: If X is A then Y is B,
p2 Observation: X is A',
Figure 1: The fuzzy numbers x, y, and the difference y - x
inferences may also be made by constructing a simple conventional relation of
the form y = f(x), a value of Y for a particular value of X.
In most cases the output of an approximate reasoning system is
defuzzified either conceptually (in case of medical consultancy, etc.)
or physically (in case of process control, etc.). In [Raha and Ray
92] it is proposed that instead of defuzzifying the output, we can
defuzzify the vagueness of the linguistic statements at the structural
level, construct a simple conventional relation that also captures the
experience and intuition of an expert, and apply the method of
interpolation for inference. But why do not we use the fuzzy number
and fuzzy arithmetic appropriate for a given situation? If we do this,
we allow the user to build a knowledge base representing the contents
of a technical case study.
The defuzzifying of information before making any conclusion may be
useless in a real-world application.
A Fuzzy set, as its name implies, is a class with fuzzy boundaries:
the class of small numbers, e.g., old men. Basically the grade of
membership is subjective in nature; it is a matter of definition
rather than measurement. Humans have a remarkable ability to assign a
grade of membership to a given object without a conscious
understanding of how the grade is arrived at.
A fuzzy quantifier such as "Most" may be represented as a fuzzy number
- a fuzzy set that defines the degree to which any given proposition
matches the defition of "Most".
Thus "Vegetarians are healthy" may really mean "Most vegetarians are
healthy". The proportion of fuzzy set elements is represented by the
fuzzy quantifier "Most". For example, from the statement: "Usually
lean people are vegetarians" and "Most vegetarians are healthy", one
could deduce that "usually most lean people are healthy". In this case
"usually.most" represents the product
of fuzzy quantifier "most" and "usually". But this resulting quantifier is less
specific than "most" or "usually" in the premises [see Fig. 2].Figure 2: The fuzzy quantifiers "most", "usually" and their product.
In this paper a method for the description of fuzzy quantifiers was
discussed. A comparison between fuzzy reasoning and interpolation is
made. We have shown that the difficulty in Fuzzy arithmetic arises
because of the algebraic structure of fuzzy numbers. We examine a
numerical example with a self build convolution for computing with two
In the last several years expert systems have emerged as one of the
most important applications of Artificial Intelligence. Reflecting
human expertise, much of the information in the knowledge base of a
typical expert system is imprecise, incomplete, or not totally
reliable. For this reason the answer to a question or the advice
rendered by an expert system is usually qualified with a "certainty
factor", which gives the user an indication of the degree of confidence
that the system has in its conclusion.
To arrive at the certainty factor, the existing expert systems employ
what are essentially probability-based methods. However, since much of
the uncertainty in the knowledge base of a typical expert system
derives from the fuzziness and incompleteness of data, rather than
from its randomness, the computed values of the certainty factor are
frequently lacking in reliability. This is still one of the most
serious shortcomings of expert systems when the reliability of the
conclusions - as in the case of medical diagnostic systems - is of
Fuzzy logic provides a natural framework for the design of expert
systems. The design of expert systems maywell prove to be one of the
most important applications of fuzzy logic in knowledge engineering
and information technology.
The linguistic approach may well prove to be a step in the direction
of a lesser preoccupation with exact quantitative analyses, and a
of the pervasiveness of imprecision in much of human thinking and perception.
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