A model that makes use of fuzzy linguistic scales (FLS) is considered in this report. The definition of FLS fuzziness and its major properties are given in the report. Definitions that are concerned with information loss and noise are also presented

Ryjov, Alexander P.

English

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THE AXIOMATIC DEFINITION OF A LINGUISTIC SCALE
FUZZINESS DEGREE, ITS MAJOR PROPERTIES
AND APPLICATIONS
Alexander P.Ryjov
/<*
Abstract.Model of human estimate of real objects as measuring
procedure in fuzzy linguistic scales (FLS) is being considered
in the report. The definition of FLS fuzziness degree and its
major properties is given in the report. Definitions of infor-
mation loses and noise while user works with data base (or
knowledge base), containing linguistic description of objects
are being introduced and described, and proven, that this va-
lue gives linear connection with degree of fuzziness. T .,, /(\
Key words: estimate of real object, fuzzy linguistic scales,
degree of fuzsiness, quality of information search.
INTRODUCTIONS
Model of human estimate of real objects as measuring pro-
cedure in fuzzy linguistic scales (FLS) /!/ is being conside-
red in the report. While describing objects some human being
can't use any measuring devices, he makes it in terms of some
sensible properties, and he has some doubts while giving some
value to a property.
If there are a lot of property's values the trouble of
choice is that there are some of them, which are "just equal-
ly" suitable for the object description. And if there are lit-
tle of values the trouble is that all of them are "just equ-
ally" unsuitable to describe some object.
General study object of this works is a set of scale's
value of a linguistic scale /!/• Example of scale's value for
linguistic scale "Height" is given an Fig.l.
small
(u)
medium
(u) M- (u)
high
-> U
Fig.l
21
https://ntrs.nasa.gov/search.jsp?R=19930020331 2020-03-17T04:47:03+00:00Z
Such structures can be also interpreted as a set of dif-
ferent alternatives in problem solving and decision-making
/2,3,4/ or a descriptions of classes in fuzzy classification
and clustering /5,6/ or a representation of term-sets of lin-
guistic values /?/ and etc. However, first interpretation (in
the same way /8/) is the most preferable for application in
information systems.
1. FLS FUZZINESS DEGREE: DEFINITION, EXAMPLE AND PROPERTY
The definition of FLS fusziness degree is given in the
paper under some matter-of-fact restrictions on membership
function form, and the set of such functions, which create
the FLS.
Let's assume, that membership functions for FLS 1 (where
1t - number of scale values) are defined on some segment U e R
and meets following requirements:
1) normal /9/: Vj (l<j<t) 3 ul*0,
1 1
where U- = {u<sU: M-.-(u) = 1>, U. are segment;
«J J J ^
2) increasing from the left U. and decreasing from the
right V\.J
The requirements are quite natural for membership functi-
ons of notions gathered in some FLS's scale values set. Actu-
ally, the first means that there's at least one object for
each scale value, which is typical or ideal for the notion;
and the second may be interpreted as requirement of gradual
changing of the notion limits.
Characteristic functions we'll be mentioned in the artic-
le. Let's assume, that :
3) those functions can have not more than two break
points of second sort.
Let's assume that L is the set of functions satisfying
requirements 1)- 3). The set L is a subset of a set of functi-
ons integral able on some measurable set of functions L^, and
therefore , a measure can be introduced on L. For example :
d(f,g) = lf(u) - g(u)ldu, f e L, g « L.
U
Let's introduce some restrictions on a set of functions
from L, which are creating a set value of FLS 1 . . And let's
\i
assume that a set of such functions suit following require-
ments:
4) completeness: Vue U 3j (l^ j^ t): M-.(u) * 0
22
t
5) orthogonally: V
 u e u E M-.(u) = 1
d = l J
These restrictions are quite natural too. Assuming that
4) isn't true then a set U '= {u « U: Vj (i<j<t) jJ-.(u) = 0} mayj
be harmlessly deleted from , therefore a set U\U.* may be con-
sidered instead of universum. That means that there's no scale
values associated with any point from U' set, and scale has
improper definition.
Restriction 5) was described in /2/. Scales built under
5) are not only useful for theoretical analysis, but they must
be the most spread in use, because the restrictions mean that:
- used notions (scale values) are quite differ from each
other;
- they do not describe the same objects.
Let's call a set of FLS with scale values under 4) and 5)
G(L) - scales.
We can introduce a measure on G(L) too.
Lemma 1. Let's assume that
(M-^(u), M.O(U), ... , M-t(u)} - a set of scale values 1^;
A. I A £. A tl x\ \t
{^ (u), M-2(u), ... , M-^-(u)} - a set of scale values 1^;
d(f,g)- a measure in L .
A W A
Then P(l,.,lt) = E d(M-. ,U. ) - is a measure int t
 i = 1 11
G(L).
To formulate axioms we should define a scale, which is
based on some FLS and is "unfuzzy", meaning that the scale's
value is a set of characteristic functions, produced with mem-
bership functions of FLS.
Thus,assuming that 1 e G(L), is a FLS defined on U and
consisting of membership functions p.1 (u) , ... , p.. (u). Let's
A J. A t-
construct some "unfuzzy" set value lt. lf - is a set of chara-L <i
cteristic functions h^(u), ... , h (u), where
f 1, if max U.(u) = |l.(u)
h i(u) = j 1*j2£t * 1
0, otherwise
A.
Call 1^. - the nearest "unfuzzy" scale, based on FLS
lte G(L).
Let's assume that fuzziness degree of FLS, whose scale
values are defined upon universum U, is the value of
functional ?(!.), defined on the membership function scale
b
values set and satisfying following axioms:
23
Al. 0^ |(l)^ l V ie G(L).
A2.
A3.
 ldt)=0 «-» V ueu 3i
(I. (u) = M-. (u)= max }l.(u).
X1 12 1<j<t J
A4. Let's assume that FLS 1, and 1' are defined on uni-
t
 t .
versumes U and U' correspondingly; t and t can be equal and
not equal and not equal to each other.
* A
€(lt) ^  !(!'.), if P(lt,lt) * P(l*. ,1*. ),1
 t r t t
where P( ' , ' ) - some metric in G(L).
Axiom Al defines domain of values for functional £(!,),
t>
or fuzziness measuring borders.
Axioms A2 and A3 describes the scales where £ ( 1 . ) assumes
L>
minimal and maximal values, or maximal "unfuzzy" and maximal
"fuzzy" scales correspondent.
Axiom A4 defines the fuzziness degree comparison rule
for each pare scales. It may be expressed in such a way: the
nearer given FLS to its nearest unfuzzy scale, the less it's
fuzziness degree.
Let's give an answer for question of existence a functio-
nal satisfying those axioms.
Theorem 1. Assume that 1. e G(L). Then functional
1 r
+)= - f(P-
U I r i l l *|U|J i ^i
u
here p. (u) = max (J-.(u), p. (u) = max P-.(u),- , .
* *i jt
f satisfies following requirement:
Fl: f(0) = 1, f(l) = 0;
F2: f decreases,
is fuzziness degree 1 , i.e. satisfies Al - A4 .
It's easy to prove, that the only linear function satis-
fying Fl, F2 is a function f(x) = 1 - x.
A subset of polynomials of degree 2, satisfying Fl, F2,
can be described. Those are expressions of the following type:
f (x) = ax2 - (1 + a)x + 1.
3.
24
Subset of functions of other types (logarithmic, trigono-
metric etc.) satisfying conditions Fl, F2 may be defined in a
similar way. Let's use those functions in formula for £(!.),
L>
and get some functionals, satisfying Al - A4, i.e. it is a
fuzziness degree.
FLS fuzziness degree properties for linear f are being
described in the report. In this case
1 r
t)= -
fc J
- (M- *(u) - fi (u)))du,
U i? i*
u
here Jl (u) = max p..(u), M- -.(u) = max M-.(u),
i* i<j<t 3 i*
This fuzziness degree measurement functional was introdu-
ced at the first time in /10/ for the task of optimal quality
properties values set choice in human-machine systems.
Let's define the following subset of function set L:
L- a set of functions from L, which are part-linear and
linear on ^
U = {u e U: Vj (1 < j < t) 0 < M-.(u) < 1);
•* «
L - a set of functions from L, which are part-linear on U
(including tf).
d
Theorem 2. Let 1 « G(L). Then £ ( 1 ) = - , where
w T»
2IUI
d = IU I = l(u e U: Vj (i < j <
 t) > Ji.(u) * 1)1
«)
d
Theorem 3. Let 1, « G(L). Then £ ( 1 ) = C - , where
U t
IUI
C < 1, C = Const.
The fuzziness degree of a fuzzy set induced by £(!+-) i-s
defined as fuzziness degree of a trivial FLS, determined with
a fuzzy set |J.(u):
1
 r(1 - I2|i(u) - II )du
J|u|
U
It's easy proved, that l(fJ-) satisfies all the axioms for
the set's fuzziness degree /ll/. It may show that the introdu-
ced in the report more general notion £(!..) had been correctly
U
defined.
It's easy shown, that the functional may be considered as
an average human doubts degree while describing some real ob-
ject (situations) /4,12/.
25
2. APPLICATION FLS FUZZINESS DEGREE TO INFORMATION SEARCH
The results were published at the first time in /ll/.
Definitions of information loses and noise while user
works with data base, containing linguistic description of
objects are being introduced and described in the report. Whi-
le interacting with the system user formulates his query and
gets an answer according to the search request. And if he knew
real (not linguistic) values of object characteristics, he,
possibly, would defeat some of displayed objects (noise) and
he would add some others from data base (loses). Information
noise and losses appear because of fuzziness of scale ele-
ments.
Because of volume restrictions and taking into account
the illustrative character of the chapter we stop at the main
results. In the next work we are going to describe the prob-
lems of formalization of fuzzy database information retrieval
quality rations in complete.
Theorem 4. Assume that 1. e G(L), £ (1. ) - degree of fuz-
ziness of 1. ; II (U), H (U) - average information loses and
u X X
noise, appearing during information search with search attri-
bute value set X, equal to 1. -scale values set; U - universum
U
1. ; N(u) - number of objects, whose definitions are in a data-
U
base and which having a real characteristic value equal to u,
- is a constant. Furthermore, assume that all of property va-
lues are equally preferable for user, meaning that request
probabilities for all the property values are equal. Then
2N
II (U) = H (U) = — ?(!«.), N = Const.
x x
 3t
Theorem 5. Assume that 1 € G(L), Nfu) = N = Const and
L>
request probabilities for all the property values are equal.
Then
H (U) = H (U) = -C(M-),
x x
 t
where c - a constant, which depends on N only.
Thus,OL% - fuzziness degree decrease leads to the same
decrease of average information loses and noise if the number
of property values is constant. Simultaneous fuzziness degree
decrease of properties values number lead to even more
substantial decrease of information loses and noise.
The following method of property values set choosing for
fuzzy databases, can be evaluated from the given results:
1. To generate all possible sets of property values.
2. To represent each of with FLS scale values set.
3. To evaluate the degree of fuzziness for each of the
property values sets according to (*).
26
4. Chose the set of property value set, which has the
minimal ratio of fuzziness degree and number of elements. Your
choice will provide the minimal information loses and noise of
information retrieval using the property.
CONCLUSIONS
Some method to calk the fuzziness degree of the
combination of fuzzy sets (defined upon the same universum)
has been given in the article. The axioms for such measure of
uncertainty have been formulated, its interpretation has been
given. The theorem of existence has been proven and some
properties of fuzziness degree have been described.
The problems of using of the results in information
applications (fuzzy retrieval systems) have been discussed. It
is described that the fuzziness degree has linear dependence
with the indicator of retrieval quality. Taking into account
the result the methodic of choosing the optimal values has
been suggested. Using the method some user may describe
objects to achieve better results of finding information in
fuzzy data bases. Under these circumstances a person - a
source of information - would suffer minimal difficulties
(uncertainties) to describe real objects.
The results may be used also in some tasks to construct
knowledge bases, decision-making tasks under fuzzy conditions
and pattern recognition.
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The axiomatic definition of a linguistic scale fuzziness degree, its major properties, and applications

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