Complex Fuzzy Sets Towards New Foundations

Hung T Nguyen

Mathematics New Mexico State University Las Cruces NM USA hunguyennmsuedu

Abraham Kandel

Computer Science Engineering University of South Florida Tampa FL USA

Vladik Kreinovich

Computer Science UniversityofTexas El Paso TX USA vladikcsutepedu

Abstr actUncertainty of complexvalued physical Then for each the cut for the real part x is an

quantities z x iy can be describ ed by complex interval x x the cut for the imaginary part

fuzzy sets Such sets can be describ ed by member is also an interval y y and hence the cut for

ship functions x y which map the universe of dis the resulting D memb ership function is a rectangular

course complex plane into the interval The b ox x x y y The b oundary of

problem with this description is that it is dicult to this b ox consists of two straight line segments which are

directly translate into words from natural language parallel to the x axis and of two straight line segments

To makethis translation easier several authors have which are parallel to the y axis

prop osed to use instead of a single memb ership func

problems eg when the analyzed In some practical

tion for describing the complex several mem

complex fuzzy number z is the result of applying an exp o

b ership functions which describ e dierent realvalued

nential function to some other complex numb er its cuts

characteristics of this such as its real part

mayhave a more complicated shap e than a rectangle

its imaginary part its absolute value etc The quality

Some such situations can b e describ ed by using the fact

of this new description strongly dep ends on the choice

that in many practical problems it is more convenientto

of these realvalued functions so it is imp ortant to

represent a complex numb er not in the form z x iy

cho ose them optimally In this pap er we formulate

which corresp onds to Cartesian co ordinates in the plane

the problem of optimal choice of these functions and

x y but in the form z expi which corresp onds

show that for all reasonable optimality criteria the

to p olar co ordinates in this plane For such practical

level sets of optimal functions are straight lines and

problems A Kaufmann and M Gupta prop osed in to

circles This theoretical result is in good accordance

use a goniometric representation in which a complex fuzzy

with our numerical exp eriments according to which

numb er is represented by a pair of real fuzzy numbers

such functions indeed lead to a good description of

and with membership functions and In

complex fuzzy sets

this approach for every complex value x iy the degree

x y with which this complex value is p ossible can b e

Many practical problems lead to complex fuzzy

dened as

sets Manyphysical quantities are complexvalued wave

x y min

function in quantum mechanics complex amplitude and

p

imp edance in electrical engineering etc

where x y and arctanyx are the values

In all these problems exp ert uncertainty means that we

of p olar co ordinates of the p oint whose Cartesian co ordi

do not know the exact value of the corresp onding complex

nates are x and y For this representation the cut is a

number instead we have a fuzzy knowledge ab out this

set of all vectors for which is b etween and

number

and the angle is b etween and On a x y

From complex fuzzy numb ers of Kaufmann and

plane this cut is no longer a rectangle it is a rather

b ership function descrip Gupta to Buckleys mem

complicated geometric gure which is b ounded bytwora

tion In order to describ e a z x iy

dial straight line segments corresp onding to

we must describ e two real numb ers its real part x and

and and bytw o circular segments corresp ond

its imaginary part y Thus a natural idea is to repre

ing to and

sent a complex fuzzy number by describing two real fuzzy

Some complex fuzzy numbers haveeven more compli

numbers x and y see eg characterized by the cor

cated cuts or in other words memb ership functions

resp onding memb ership functions x and y In

which cannot be describ ed by the expressions and

this approach for every complex value x iy ie for ev

To describ e such complex fuzzy numb ers it is nat

ery pair x y the degree x y with which this complex

ural to use a general memb ership function x y which

value is p ossible can b e dened as

maps a complex plane C into the interval This ap

proachwas sketched in the ab ovementioned b o ok and x y min x y

thoroughly develop ed in for the latest overview see It turns out that in many practical problems it is use

eg ful to use three or four dierent characteristics For ex

ample we can combine Cartesian and p olar ones into

Why memb ership function description is some

a single characteristic set with t f x y x

p

times not sucient From the purely mathematical

t f x y y t f x y x y and

viewp oint this very general approach in which use a gen

t f x y arctanyx In this case the cut is

eral D memb ership function is very natural However

an intersection of a rectangle corresp onding to and a

there is one problem with this approach Amembership

segment corresp onding to ie a set whose b oundary

function is not something which is natural for a human

consists partly of straight line circular arcs partly of ra

to understand and to use It was invented as a way of

dial straight line segments and partly of segments which

representing human fuzzy knowledge in a language which

are parallel to x or y axes

is understandable for a computer From this viewp oint

after we get the desired memb ership function we must Which realvalued characteristics of complex num

p erform one more step wemust translate it into the nat b ers should we use in this description of complex

ural language fuzzy numb ers As shown in the eciency of the

new description in solving practical problems with com

This translation is dicult even for real variables how

plex numb ers strongly dep ends on the appropriate choice

ever forrealnumb ers wehave accumulated a lot of in

of the realvalued characteristics which are used to de

tuition and we are often able to describ e dierent D

scrib e the corresp onding a good choice can

memb ership functions by naturallanguage words suchas

drastically improve the quality of the result It is there

small close to etc Unfortunately complex num

imp ortant to nd out which functions are the best fore

bers are much less intuitive and there are few terms of

here This is the problem that we will b e solving in the

natural language which can b e naturally used to describ ed

present pap er

the knowledge ab out complex numb ers

Preliminary step reformulation in terms of sets

A new approach to describing complex fuzzy sets

The memb ership function t corresp onding to a char

f

which combines the generality of Buckleys ap

acteristic f C R can be describ ed by the extension

proach with the intuitiveness of KaufmannGupta

principle

descriptions Since the original memb ership function

t max t

C is dicult to interpret directly the au

f

z f z t

thors of prop osed a new approach In this approach

to describ e a fuzzy knowledge ab out a complex number

Thus to b e able to compute all the values t we do not

f

weuse instead of single memb ership function which de

need to compute know the exact characteristics f t it is

scrib es the number z x iy several two or more

sucient to b e able to describ e their level sets fz j f z

memb ership functions which describ e realvalued quanti

tg So instead of cho osing the best characteristics we

ties whicharefunctions of this complex numb er suchas

can cho ose a family of sets

its real part Rez its imaginary part Imz its absolute

For the ab ove characteristics f x y these level sets

i

value jz j its phase etc

are straight lines and circles

In other words instead of describing a single D mem

Of course the more parameters we allow in the descrip

b ership function x y we describ e several twoormore

tion of a family the more elements this family contains

memb ership functions t t corresp onding to

and therefore the b etter the representation So the ques k k

dierent realvalued characteristics t f x y t

tion can b e reformulated as follows for a given number of k

f x y of this complex number z x iy In this ap

k parameters ie for a given dimension of approximating

proach for every complex value x iy the degree x y

family of sets which is the b est family In this pap er

with which this complex value is p ossible can b e dened

we formalize and solve this problem

as

Formalizing the problem All prop osed families of sets

x y min t t

k k

have analytical or piecewise analytical b oundaries so

it is natural to restrict ourselves to such families By

where t f x y

i i

denition when wesay that a piece of a b oundary is an

If we use two characteristics t f x y x and

alytical we mean that it can b e describ ed by an equation

t f x y y then we get complex numb ers of typ e

F x y for some analytical function

in which cuts are rectangles

If we use two characteristics t f x y

x y a b x c y d x e x y f y F

p

x y and t f x y arctanyx then

we get complex numb ers of typ e in which cuts are So in order to describ e a family we must describ e the

ab ovedescrib ed segments corresp onding class of analytical functions F x y

For example the level set f x y x t can b e de Denition By an optimality criterion we mean a

scrib ed an equation F x y for an analytical function transitive relation on the set of all ddimensional fam

F x y x t the level set f x y y t can be ilies We say that a criterion is nal if there exists one

describ ed by an analytical function F x y y t the and only one optimal family ie a family A for which

opt

p

B A B Wesay that a criterion is shift corr

level set f x y x y t can b e describ ed byan

opt

rotation and scaleinvariant if for every two families A

analytical function F x y x y t and the level

and B A B implies TA TB where TA is a shift

set f x y arctanyx t can be describ ed by an

rotation scaling of the family A

analytical function F x y y x tant

Since weareinterested in nitedimensional families of

Theorem d For every nal optimality criterion

sets it is natural to consider nitedimensional families of

which is shift rotation and scaleinvariant the border

functions ie families of the typ e

of every set dened by the optimal family A consists of

opt

d

straight line intervals and circular arcs

fC F x y C F x y g

d

General comment This result is in go o d accordance with

i

where F z are given analytical functions and

the fact that for the ab ovedescrib ed characteristics

C are arbitrary real constants So the ques C

d

whichhave actually b een used to describ e complex fuzzy

tion is whichofsuch families is the b est

numb ers the b oundary is indeed of this t yp e In partic

When we say the b est we mean that on the set of

ular we get a new theoretical justication of the gonio

all such families there must be a relation describing

metric approachdevelop ed in

which family is b etter or equal in quality This relation

This result is also in go o d accordance with the exp eri

must b e transitive if A is b etter than B and B is b etter

ments describ ed in according to which such sets indeed

than C then A is b etter than C This relation is not

provide a go o d description of complex fuzzy sets

necessarily asymmetric b ecause we can havetwoapprox

Practical comment In practical terms our conclusion is

imating families of the same quality However wewould

that we should use c haracteristics whose level sets are

like to require that this relation b e nal in the sense that

general straight line intervals and circular arcs in other

it should dene a unique best family A ie the unique

opt

words in addition to the ab ovecharacteristics x Rez

family for which B A B Indeed

opt

y Imz jz j and z we should also use f z

If none of the families is the b est then this criterion

Rez z z and f z jz z j for arbitrary complex

is of no use so there should b e at least one optimal

valued constants z and z with jz j these character

family

istics describ e arbitrary straight lines and circles The re

sulting general description of a complex fuzzy number can

If several dierent families are equally b est then we

include several D memb ership functions corresp onding

can use this ambiguity to optimize something else

to suchcharacteristics

eg if wehavetwo families with the same approxi

Pro of of the Theorem This pro of is similar to the

mating quality then wecho ose the one which is easier

ones from

to compute As a result the original criterion was not

Let us rst show that the optimal family A is itself

nal we get a new criterion A B if either A

opt

new

shift rotation and scaleinvariant

gives a b etter approximation or if A B and A

old

Indeed let T be an arbitrary shift rotation or scal

is easier to compute for which the class of optimal

ing Since A is optimal for every other family B we

families is narrower We can rep eat this pro cedure

opt

have A T B where T means the inverse trans

there is only until we get a nal criterion for which

opt

formation Since the optimality criterion is invariant

one optimal family

we conclude that TA T T B B Since this is

opt

It is reasonable to require that the relation A B should

true for every family B the family TA is also optimal

opt

not change if we add or multiply all elements of A and

But since our criterion is nal there is only one optimal

B by a complex numb er in geometric terms the relation

family and therefore TA A In other words the

opt opt

A B should b e shift rotation and scaleinvariant

optimal family is indeed invariant

Now we are ready for the formal denitions

w that all functions from A are p oly Let us nowsho

opt

Denition Let d be an By a d

nomials

dimensional family we mean a family A of all functions

Indeed every function F A is analytical ie can

opt

d

of the typ e fC F x y C F x y gwhere

d

b e represented as a Taylor series sum of monomials Let

i

z are given analytical functions and C C are F

a b

d

us combine together monomials c x y of the same degree

arbitrary real constants Wesay that a set is dened by

a bthenwe get

this family A if its b order consists of pieces describ ed by

equations F x y with F A F z F z F z F z

k

where F z is the sum of all monomials of degree k Let can now use the fact that the family A is rotation

k opt

us show by induction over k thatforevery k the function h transforms b cinto invariant let T b e a rotation whic



F z also b elongs to A the xaxis then we conclude that F Tzb x A

k opt opt

Let us rst provethat F z A Since the family and hence x A Similarly y A So the family

opt opt opt

A is scaleinvariant we conclude that for every A contains at least linearly indep endent functions a

opt opt

the function F z also b elongs to A For eachterm nonzero constant xandy

opt

k

F z wehave F z F z so If d then the D family A cannot contain any

k k k opt

thing else and all the pieces of b orders F x y ofall

F z F z F z A

opt

the sets dened by this family are straight lines

If d thenwe cannot haveany cubic or higher or

When wegetF z F z The family A

opt

der terms in A b ecause then due to Part wewould

opt

is nitedimensional hence closed so the limit F z also

have both this cubic part and a linearly indep endent

b elongs to A The induction base is proven

opt

quadratic part and the total dimension of A would

opt

Let us now supp ose that wehave already proven that for

be at least So all functions from A are

opt

all ks F z A Letusprove that F z A

k opt s opt

quadratic Since dim A and the dimension of

opt

For that let us take

and D parts is the dimension of p ossible parts

Gz F z F z F z

of second degree is Since A is rotationinvariant

s

opt

the quadratic part d x e x y f y must b e also

We already knowthat F F A so since A

s opt opt

rotationinvariant else wewould havetwo linearly inde

is a linear space we conclude that

p endent quadratic terms in A the original expression

opt

and its rotated version Thus this quadratic part must

Gz F z F z A

s s opt

b e prop ortional to x y

The family A is scaleinvariant so for every

Hence every function F A has the form F x y

opt

opt

s s

the function G z F z F z also

a b x c y d x y and therefore all the pieces of

s s

b elongs to A Since A is a linear space the function

b orders F x y of all the sets dened by this family

opt opt

s

H z Gz F z F z F z

are either straight lines or circular arcs The prop osition

s s s

also b elongs to A

is proven

opt

When wegetH z F z The family A

s opt

Acknowledgments This work was supp orted in part

is nitedimensional hence closed so the limit F z also

s

by NASA under co op erative agreement NCC by

b elongs to A The induction is proven

opt

NSF grants No DUE and CDA by

Now monomials of dierent degree are linearly inde

United Space Alliance grant No NAS PWO

p endent therefore if we have innitely many nonzero

CCA bytheFuture Aerospace Science and Tech

terms F z wewould have innitely many linearly inde

k

nology Program FAST Center for Structural Integrityof

p endent functions in a nitedimensional family A a

opt

Aerospace Systems eort sp onsored by the Air Force Of

contradiction Thus only nitely many monomials F z

k

ce of Scientic Research Air Force Materiel Command

are dierent from and so F z is a sum of nitely many

USAF under grantnumb er Fby the Na

monomials ie a p olynomial

tional Security Agency under Grant No MDA

Letusprove that if a function F x y b elongs to A

opt

then its partial derivatives F x y and F x y also b e

The authors are thankful to anonymous referees for

x y

long to A

their valuable comments

opt

Indeed since the family A is shiftinvariant for ev

opt

ery h we get F x h y A Since this fam

opt

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opt k

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opt

a nonzero linear function F x y b x c y We