Separation and Information in the Setting of Fuzzy Sets

Copyright © IFAC Theory and Application of Digital Control New Delhi . India 1982

SEPARATION AND INFORMATION IN THE SETTING OF FUZZY SETS C. Dujet Centre de Mathematiques, [NSA , Villeurbar.ne 69100 France, Unitlersite Lyon [, Lyon, France

In the setting of fuzzy sets, new notions of complementation and partition are

Abstract

given, different from the usual ones, and based on the idea of separation in a set by the mean of a fuzzy set ; a measure of this separation is defined by a functional called a "separating power". New orderings induced by a separating power are studied, and the final section is devoted to the connection between separation and information. Fuzzy set - lattice - valuation - separating power - sharpened relet ion - general i-

Keywords

zed information.

INTRODUCTION The starting idea was the following : if we consider a given population

E, and a property

P

attached to this population, it is possible, from a classical two-valued logic, to assign to each member

x

of

E, the number zero or one, depending on the fact, x has the property

or has not. The population by the mean of

E

P

is therefore naturally separated into two complementary subsets,

P

But very often, a given property

P will define a fuzzy set [1]

f

of

E, every member of

the population having the above-said property to some degree (= f(x»

ranging between zero or

one. My idea was then to try to evaluate how much a fuzzy set

E

elements of

E, compared to the ideal case, when

To achieve this purpose,

I

f

defined a functional [2] [3]

upon the lattice of the fuzzy sets of

f

of

was "separating" the

would takes only the values zero or one. derived from an arbitrary valuation

E.

SEPARATING POWER IN E. Let

E be a finite set, and iP(E)

the lattice of the fuzzy sets of

grades being most of the times the chain

(f E ,5f'(E)

[0, I].

~

E, the set of membership

f : E

-+

[0, I]). We recall the

fuzzy inclusion, union and intersection f C g

iff

Vx E E , f(x)

~

g(x)

f v g

is defined by

f v g(x)

f(x) v g(x)

~

is defined by

f A g(x)

f(x)

f

g

~

g(x)

=

417

max (f(x), g(x»

in

[0,1] )

min (f(x), g(x»

in

[0,1])

C. Dujet

418

Let

~(E)

Given

be the class of ordinary subsets of

f E .sf'(E)

AE

~(E)

, we denote by

fA

the fuzzy set of

E

defined by

= f(x), Vx E A

fA(x)

We denote by Let

and

E.

v

A the complement of

A in

E, IAI the cardinal of

be an arbitrary but fixed valuation upon

.sf'(E) , i.e

v

A. is a mapping from

jf(E)

into the real line, satisfying : Vf E .sf'(E)

Vg E .sf'(E) , v(fvg) + v(f" g)

We are ready now to define what

=

v(£) + v(g)

called a "separating power in E".

I

Def. 1.1 a

v-separating power ln E

is the mapping (noted • ) from

.sf'(E) x

~(E)

into the

real line, defined as follows : if

f.A

0

if

A (j. {(6, E}

A E {(6, E}

Example: we can choose for the valuation couple

(f, A)

Properties

where f

f(x).

t~i~ Ev-separating power is maximal for a

is the boolean characteristic function of

A.

For more details, see DUJET [2]

I)

(fvg). A + (f Ag).A

2)

given on

Def. 1.2

~

v: v(f)

According to the intuitive idea of separation in E,

Vf,g E.sf'(E) , VA E~(E).

f.A + g.A

A E~(E), A (j. {(6, E}, the mapping

(f, g)

--+

If.A - g.AI

is a quasi-metric

.sf' (E) .

v-separating index of a fuzzy set

given a v-separating power in noted

f

of

E,

sv(f) , is defined by : s (f) v

(If.AI Remarks

E, the v-separating index of a fuzzy set

max {I f. A I, A E ~(E) }

is the absolute value of the real number

The existence of - f

Sv(f)

is constant on

E

~

is ensured Sv(f)

1j

the finiteness

one of the two numbers

of

E.

sothat for all

f E

0

From now, we choose a valuation v A E ~(E)

f.A).

f.A

or

f.A

~(E)

and for all

is positive.

Def. 1.3 : v-max separating partition of a fuzzy set Given

f

a fuzzy set of

is an ordered couple f.X

E, not constant on

(X, X)

E, a

v-max separating partition of f

E ~(E) x 9(E) , satisfying the condition

is positive and equal to the v-separating index of

E.

Theoreme I. I The class of the v-max separating partitions of a fuzzy set

f

of

E

is upper (re spec-

419

Separation and Information in the Setting of Fuzzy Sets

tively lower) bounded for the usual operation of union and intersection. Proof not given for brevity. See (4) This theoreme allows to privilegiate a particular v-max separating partition of

~ max(f).

example the upper bound, denoted by

E, we put : ~ max (f)

For a fuzzy set f constant on I

I --v(f ) ~ I - - - v(f )

E

IEI

(respectively

E

IEI

f , for

~

(0, E) (respectively

(E, 0))

if

).

From definitions 1.1 to 1.3 , a fuzzy set of

E

may be characteriged by tree items

- the v-separting index : S (f) I

- the real number

-

Ixl

v (f )

v

x m(f)

is not constant ; or

mf ,called the v-upper average of

noted if

f

f

, if

f

is constant.

Proposition 1.1 The relation noted ~ , defined in f ~ g

iff

~

= ~

max f mf

~(E)

by

max g

= mg

is a relation of equivalence in ~(E). When

f

and

g

are equivalent, we will say they are "co-significant";the set of the equiva-

lence classes will be denoted by co-~(E), the structure of it being studied in section 11. Let us give the definition of complementation and partition in

~(E),

deduced from the

notion of seperation. Dei. I. 4

Given a fuzzy set

f

of

E, a fuzzy set

f'

of

E

is said a v-complement of

f

iff the following conditions are satisfied Cl) C2)

C3)

~ max f = (X,

if

S (f)

v v(f ) + X

X) ,

then .0/ max f'

S (f')

v

= v(l x)

v(f~)

Remark : By a convenient choice of the valuation C' 3)

mf + m , f

=

(x, X)

I ,were h

m , f

. 1S

v, the condition C3) can be rewritten as

th e v- 1ower average

0

f

f'.

Proposition 1.2 The complementation is an involution in the set co-~(E), that is to say if

f'

is a v-complement of

f

if

g'

is a v-complement of

g

if

f ~ g ,then

f' ~ g'

for all

f, g E~(E).

C. Dujet

420

Definition 1.5 fi, i E I

Given a family tition of Sf(E)

iff

of fuzzy sets of

E, this family is said to form a v-par-

there exists a proper subset

A of

E

such as the following

conditions are satisfied :

fi • A = 0

L

P2)

i E I P3)

i v(f ) = v(J A) A i E I L

this definition does not require to have

Advantage

L fi(x)

I , 'Ix E E •

i E I II - IMPLICATION

The idea is to use the concept of separation in Sf(E)

to try to translate the natural impli-

cation, such as "very tall" implies "tall". Def 2.1

Weak implication Let i)

g E

f e,'

(E);let (X,

X)

=~maxf , (Y,

Y)

=~max g

we say that

f ~ g

iff

X C Y

H) mf .,;; mg Hi) m .,;; mg f It is easy to see, ~ is an order in Det. 2.2

co-Sf(E).

Strong implication

With the same notations, we define i) f

~

f

~

g

iff

g

ii) S (f) v

~

S (g) v

if

S (f) # 0 v

Proposition 2.1 The weak and strong implication in

Sf(E) define the same equivalence in Sf (E) , which

is the equivalence of Proposition 1.1 Theoreme 2.1 The set co- Sf(E) , partially ordered by

~

, is a distributive lattice, with a maximum

andaminimum, satisfying relatively to the v-complementation, the involution Law, the contraposition law and the De Morgan Laws. See proof in [41 . Proposition 2.2 The same result can be obtained from the quasi order

~

in Sf(E).

421

separation and Information in the Setting of Fuzzy Sets III - SEPARATION AND INFORMATION. Firs~,

we are intended to deal with information from a semantical point of view ~(E)

It is possible, to interpretate

, partially ordered by ~ , as a system of ~(E)

propositions. We can then define [5) [6) on

Kampe de Feriet [7) by the mean of a functional i) J(O)

and

J(I)

ii) for all

f, g E

a generalized information -+

J :~(E) ~ R

I

of

,satisfying:

0

=

(E), f

~

g

implies

J(f)

~

J(g)

Proposition 3.1 The functionel for all

~(E)

on

J

f E

E , J(f)

defined by : S (f) / mf

=

v

if

(I-m(f)) / m(f) on

mation

if

f

is constant,is a generalized infor-

, ~).

(Y(E)

~(E)

Now, if we consider a relation in

which do express intuitivaly some"less fuzziness", we

can deal with a kind of localization information. To achieve this purpose, we first use the sepa rating power to define in of the "sharpened" relation indroduced by Trillas Def 3.1

~ (E)

a generalization

[8).

The generalized sharpened relation Let

f

and

f C g) s

iff

g

two fuzzy sets ; we say that

f

is "less fuzzy" than

g

(noted

i) 9' max f = 9' max g ii) mf ~ mg iii) m .;;;; m f g Advantage : it can be used when the set of membership grades is any distributive lattice, and the number

21

is not privilegiate.

Proposition 3.2 The relation co- ~ (E))

is a quasi order (respactively an order) in

C

~ (E)

(respectively in

s

.

Proposition 3.3 The functional

h

: ~ (E) ~

R defined by

h(f)

1 - Sv(f)

is a s ymmetric entropy

measure [9) . It is easy to s e e i) ii)

h

is isotone for the ordering h(f)

=0

iff

f

C

is the boolea~ characteristic function of

is equivalent to : f

is minimal for

a subset of

E (that

C). s

We point out that another connection between separation and information was developed in Dujet [2), the v-separating power, depending on the choice of difference between t wo entropies.

v , being interpretated as the

C. Dujet

422

CONCLUSION The notion of separation in the setting of fuzzy sets turns out to be rich and useful, specially because of the induced orderings and the partition derived from it, which may help for problems of automatical classification. On the other hand, the conveyed information by this notion is hoped to be enlarged and to open the way for concrete applications.

REFERENCES [I] ZADEH L.A

Fuzzy Sets; Information and control 8 , 338-353 (1965)

[2] DUJET Ch.

Valuation et separation dans les ensembles flous, Journees Mancelles Information et Questionnaires, Publications CNRS (Sept. 1980).

[3] DUJET Ch.

Separation dans les ensembles flous et complementation induite. Actes de la Table Ronde CNRS sur le flou

[4] DUJET Ch.

LYON (Juin 1980)

Une generalisation des ensembles flous : les ensembles co-flous ; Seminaire "Mathematique Floue" , Publications Departement de Mathematiques

Universite

LYON I (I 980-81) [5] LOSFELD J.

Information generalisee et Relation d'ordre

Actes des Rencontres de

Marseille-Luminy (Juin 1973) [6] SALLANTIN J

Systeme de propositions et informations

C.R. Acad. Sc. Paris, t. 274

(20 mars 1972). [7] KAMPE DE FERIET J. [8] DE LUCA A and

Mesure de l'information fournie par un evenement

Colloques Internationaux

CNRS (1969) Ed. CNRS (1970) 191-221. Entropy and Energy measures of a fuzzy set

Advances in fuzzy set theory

and applications, North Holland (1979)

TERMINI S [9] TRILLAS E. and RIERA T.

Entropies in finite fuzzy sets

Inform. Sci. 2 (1978) 159-168.