On fuzzy difunctional relations

Intelligent Systems NORTH- HOLLAND

On Fuzzy Difunctional

Relations

H. OUNALLI and A. JAOUA Facult~ Des Sciences De Tunis, Dgpartement D'Informatique, Campus universitaire, Le Belvddtre, 1060-Tunis, Tunisia

ABSTRACT Difunctional relations have proved to play an important role in software design and in database theory. On the other hand, the concept of fuzzy set has received increasing attention from researchers in a wide range of scientific areas, especially in computer science. This paper extends difunctional relations in the framework of fuzzy relations with max-min composition for the purpose of gaining a better understanding of their properties, their structure, and their behavior. One motivation for this is to study fuzzy difunctional dependencies in the framework of the fuzzy relational data model. (~ Elsevier Science Inc. 1996

1.

INTRODUCTION

Difunctional relations are a versatile m a t h e m a t i c a l tool, which can be used in software design and specification a n d in d a t a b a s e theory. Work of [2] have revealed t h e usefulness of,difunctional relations in p r o g r a m specification and in defining p r o g r a m correctness. In [5], J a o u a et al. have discussed their application in the relational d a t a b a s e model, O n the other hand, fuzzy set t h e o r y [14] and its applications have been extensively investigated since t h e 1970s [12]. This p a p e r extends the work of J a o u a et al. [3] on difunctional relations in the framework of fuzzy relations with max-rain composition [15] for t h e p u r p o s e of gaining a better u n d e r s t a n d i n g of their properties, their structure, and their behavior. T h e aim of t h e paper is (1) t o characterize in a simple m a n n e r the fuzzy difunctional relations, and (2) to s h o w t h a t m o s t of t h e properties t h a t characterize crisp difunctional relations also hold for fuzzy difunctional relations. INFORMATION SCIENCES 95, 219-232 (1996)

(~) Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

0020-0255/96/$15.00 PII S0020-0255(96)00142-9

220

H. O U N A L L I AND A. J A O U A

This paper is organized as follows. In Section 2, we discuss motivations for extending the difunctional relations in the framework of fuzzy relations. In Section 3, we first recall fundamental operations and properties of fuzzy relations; then we define the notions of difunctional relation and fuzzy difunctional relation. In Section 4, we give some new properties and extend the most important properties of crisp difunctional relations to fuzzy difunctional relations. In Section 5, we conclude with perspectives on the possible impact of the use of fuzzy difunctional relations in the field of database decomposition and information structuring. 2.

WHY FUZZY DIFUNCTIONAL RELATIONS ?

T h e main reasons which have led us to investigate difunctional properties in the framework of fuzzy relations are discussed below: 1. T h e notion of difunctional relation has theoretical, as well as practical, interest. Its theoretical interest stems from its numerous m a t h e m a t i c a l properties. From the theoretical aspect, an interesting question is: Which relational properties remain true with fuzzy difunctional relations? This p a p e r addresses this question by focusing on the properties which are applicable to our work. 2. T h e practical interest of difunctional relation stems from the large number of computer applications to which it proves to be related. In our research, we address the problem of information structuring and database decomposition. We have found the concept of difunctionality useful as a formal tool. In [4], J a o u a et al. use the difunctionality concept to decompose a finite binary relation into a union of r e c t a n g l e s (or a union of small disjoint difunctional relations). A rectangle is the cartesian product of two sets. This kind of decomposition, called rectangular decomposition, is an NP-complete problem for which the authors have proposed heuristics for optimizing the usage of storage space in database application. T h e y apply this decomposition strategy to the structuring of an indexed d o c u m e n t a r y database where R is a binary relation between a set D of documents and a set T of t e r m s used for indexing these documents (i.e., R c_ D × T). In [6], we use the rectangular decomposition as a tool to automatically discover the entity types from an instance (or sample) of an n - a r y relation. We first translate an n - a r y relation Rn to an equivalent binary relation R2. Then, by rectangular decomposition heuristics, we identify a set of all entity types of a given relation instance. An interesting question is: How to modify our heuristic approach to automatically identify entity types from a target instance that involves fuzzy data or fuzzy associations between columns?

ON FUZZY DIFUNCTIONAL RELATIONS

221

3. In [5], Jaoua et al. have studied difunctional dependencies as an extension of functional dependencies in the relational data model. This work serves as a starting point to study fuzzy difunctional dependencies in the framework of the fuzzy relational data model [9]. 4. Recently, many database applications such as scientific databases and decision support systems involve a large amount of unknown data and vagueness in data values. It seems natural to extend our rectangular decomposition approach in order to cope with vagueness and uncertainty in database applications. 3. MATHEMATICAL

BACKGROUND

Here we review some definitions and results that will be needed in the sequel. For details we refer to [1, 7, 8, 15].

3.1. FUZZY RELATIONS Let U be a set, called the universe of discourse. An element of U is denoted by lowercase letters from the end of the alphabet. Let [0, 1] be the set of all real numbers (or scalars) a, with 0 < ~ < 1. A scalar is denoted by Greek letters such as a, j3, % . . . and so on, possibly with subscripts. The supremum and the infimum of a set {al, a 2 , . . . , a n , . . . } of scalars are denoted by V { a l , o / 2 , . . . , o / n , . . . } and A { a l , a2,. •., a n , . . . } , respectively. In particular, V { o l i , a 2 } = max{a1, a2} a n d / \ { a l , a2} --- min{al, a2}. A fuzzy binary relation R (or fuzzy relation for short) on the universe U is a function R: U × U ) [0, 1]. For x, y E U, the value R(x, y) --- a is called the grade of membership of (x, y) in R and means how far x and y are related under R. Without loss of generality, we define all fuzzy relations on a fixed universe U. A crisp binary relation R (or crisp relation for short) is a particular fuzzy relation where the interval [0, 1] of scalars is replaced by the set {0, 1 } of integers, that is, R(x, y) = 0 or R(x, y) = 1. A fuzzy relation R is contained in a fuzzy relation S, written R _ S, if R(x, y) < S(z, y) for all x, y e U. For a family of fuzzy relations {R1, R 2 , . . . , R r,}, we define the union [_J~Ri and the intersection [-]~Ri as follows:

222

H. O U N A L L I AND A. J A O U A

Some particular fuzzy relations are defined by the following: 1. T h e identity relation I is a fuzzy relation such t h a t I(x, y) = 1 if x = y and I(x, y) = 0 otherwise, for all x, y E U. 2. T h e universal relation L is a fuzzy relation such t h a t L(x, y) = 1 for all x, y E U. 3. T h e zero relation O is a fuzzy relation such t h a t O(x, y) -- 0 for all

x, y E U . DEFINITION 1. T h e inverse (or transpose) R - I of a fuzzy relation R is defined by

R-l(x,y) = R(y,x)

for all x , y c U.

DEFINITION 2 [8]. T h e semiscalar multiplication ~ R of a fuzzy relation R by a scalar ~ is a fuzzy relation such t h a t (c~R)(x, y) = o~R(x,y). The well-known max-min composition of the fuzzy relation was introduced in [15]. DEFINITION 3 [15]. Let R and S be any fuzzy relations on U. T h e n the product R S is defined by

(Rs)(x,y) = V[R(

,z)A S(z,y)]

for all (x,y) e U × U

zEU

T h e classical composition of two crisp relations R and S is compatible with the max-rain composition: (x, y) e R S e=~R(x, z) = 1 and S(z, y) = 1 for some z E U. W h e n the universe U is finite, say U = {Xl, x2,. • •, xn}, we can represent a fuzzy relation R by a matrix such t h a t each entry R[i,j] = R(x~, yj) = a for some x , y c U. T h e max-min composition can thus be viewed as a m a x - m i n matrix product. The following properties have been proved to hold for fuzzy relations (see [7, 8]). We will refer to these properties by their number in the subsequent proofs: (P. (P. (P. (P. (P. (P. (P. (P.

1) 2) 3/ 4) 5) 61 7) 8)

R ( S T ) = (RS)T. R ( S U T) = (RS) tJ (RT). Sr-T~RSE_RT. S~T~SRETR. R_ESc=~aRE_aS. (c~R) -1 _- ~ ( R - 1 ) . a ( R 1 R 2 . . . Rn) -- ( ~ R 1 ) ( ~ R 2 ) ' - " (~Rn). R = S <=~ ~ R = a S .

ON FUZZY DIFUNCTIONAL RELATIONS (P. 9) (P.10) (P.11) (P.12) (P.13) (P.14)

223

R I = I R = R for all hrzzy relation R. ( R S ) -1 = S - I R -1.

( R - l ) -1 = R. (R U S) -1 = R -1 [_JS -1. ( R I T S ) -1 = R - 1 1 7 S -1.

R~S~R

-1 E S -1.

DEFINITION 4. The scalars set of a fuzzy relation R, written O(R), is defined as follows:

e ( R ) = {~ ¢ o I ?(~, y) e u x u, R(~, y) = ~}. In particular, ~(O) = 0, ¢(L) = ~(I) = {1}.

DEFINITION 5. Let R be a fuzzy relation and a E ¢(R). The a - c u t relative to R, written Ra, is a crisp relation such that for all x, y E U: 1, R ~ ( x , y) =

O,

if R(x,y) >_ a, otherwise.

REMARK. Following the decomposition theorem of Kaufmann [7], a fuzzy relation R may be written as follows: R=

U(~in~,)

for all ~ e ~ ( R ) .

E X A M P L E 1. Let R be the following fuzzy relation:

(01 02 U

R=

R0.2 =

y z

(i

0.2 0.0

V

0.3 , 0.1/

1 , 1) 0

Ro.~=

R0.3 =

(!1) 1 1

(i

i)

,

,

~(R) = {0.1,0.2,0.3}, R = 0.1Ro.1 U 0.2Ro.2 [J 0.3R0.a. 3.2.

D I F U N C T I O N A L VERSUS F U Z Z Y DIFUNCTIONAL R E L A T I O N S

The notion and the name of difunctional relation were introduced by Riguet [13]. Difunctional relations have been extensively investigated in mathematics and computer science by [2, 10, 11].

224

H. OUNALLI AND A. JAOUA

Let R be a crisp relation on U. An element of R is denoted (x, y), where x is an argument (or input) and y is an image (or output) of x by R. The image set of x E U, written x- R, is defined by x- R = {z { (x, z) e R}. Formally, R is difunctional if and only if, for all x, y E U, if x . R n y. R ¢ 0 x. R = y. R. This definition is equivalent to R R - 1 R C R ([3]). The difunctionality is an interesting property for information structuring. It means t h a t a finite difunctional relation can be broken down (or decomposed) into disjoint, smaller difunctional relations (or rectangles). This is particularly interesting in information structuring and database decomposition. Before studying the fuzzy difunctionality, let us give an example. E X A M P L E 2. Let U ---- {1,3, 4,5, 6, 7} and let R be the following crisp relation:

.R--{(1,6),(1,5),(2,6),(2,5),(3,7),(4,7)}. We verify that R is difunctional, i.e, 1 • R = 2 . R = {6, 5} and 3 • R = 4.R = {7}. We can decompose R into the two following disjoint difunctional relations or rectangles: R1 = { ( 1 , 6 ) , ( 1 , 5 ) , ( 2 , 6 ) , ( 2 , 5 ) }

and

R2---{(3,7),(4,7)}.

Obviously, the formula R R - 1 R gives the same result. DEFINITION 6. A fuzzy relation R is fuzzy difunctional if and only if it satisfies condition R R - 1 R U R, which is equivalent to R R - 1 R = R (cf. Proposition 1). In the paper, we use one formula or the other, interchangeably. PROPOSITION 1. R E R R - 1 R holds for any fuzzy relation R.

Proof. We must prove t h a t ( R R - 1 R ) ( x , y ) >_ R(x,y) for all x , y e U: ( R R - I R ) ( x , y) = =

V

y) z) A ( n - l n ) ( z ,

(by property P. 1)

y)l

(by Definition 3)

zEU (by Definition 3)

(by DefinitionI) zEU

[wEU

ON FUZZY DIFUNCTIONAL RELATIONS

225

In particular, for z = y, ~ verifies the following inequality:

t3>- A

{R(x,y), we~V[R(w,y) AR(w,y)]}

> R(x, y) A R(x, y) since V R(w, y) >_R(x, y) wEU

>_ R(x, y). Hence, for any fuzzy relation R, we have R E RR-1R.

[]

4. PROPERTIES OF FUZZY DIFUNCTIONAL RELATIONS In this section, we characterize fuzzy difunctional relations and give some of their properties in the framework of fuzzy relations. PROPOSITION 2. Let ~ be a scalar. If R is fuzzy difunctional, then c~R is fuzzy difunctional. Proof.

RR-1R= R (RR- 1R) = aR (~R) (~R- 1) (an) = (~R) (~R) (~R)-I(~R) = (~R)

(since R is fuzzy difunctionM) (by property P.8) (by property P.7) (by property P.6).

Hence a R is fuzzy difunctional.

1:3

PROPOSITION 3. Let R be a fuzzy relation and ~, ~ E O(R). If ~ <_ ~,

then R~ ERa. Proof. ifRa(x,y) = 0 ~ R ~ ( x , y ) = 0 or 1 ~ R a ( ~ , y ) ~ by definition of fl-cut, R(x,y) >_~ s i n c e ~ _ < ~ R a ( x , y ) = l . Hence R~ r- R~ for all x, y E U.

[]

PROPOSITION 4. Let R be a fuzzy relation and ~, ~ E ~(R). (~Ra)(~RB)

= A { ~ , ~}(R~Rfl).

226

H. O U N A L L I A N D A. J A O U A

Proof.

A s s u m i n g t h a t a <_/3, we m u s t prove t h a t for all x, y E U:

( (aR,~)O3R~) )(x , y) = a( (R,~R~)(x, y) ). To show this equality, we will prove t h a t (i) (ii) (o~R,~)OgR~) U c~(R~R~): (i) a(R,~R~)(x,y) nition 3):

= a[Vzeu(R~(x,z ) A Rz(z,y))]

if °L [ V (R'~(x'z) A thenVz

o~(R~R~) E (aRa)(flR~),

and

= 0 or a (by Defi-

--0

R~(x,z)=0

or

R~(z,y)=0

V ( ~ R ~ ( ~ , z) A ~R~(z, y)) = 0; zEU

t h e n ~zo

such t h a t

Rc,(x, zo)

= 1

R~(zo, y) = 1

and

M (~R~(x, z) A ~R~ (z, y)) = A (~R~(x, zo), ~R~ (Zo, y) } zEU

=aR~(x, zo)=a Hence a(R,~R~) E (aR~)(flR~). (ii) ((aR,~)(13R~))(x, y ) = Vzev[(aR,~(x, (by Definition 3):

z))

A

sincea~fl.

(t3R~(z,y))]

= 0 or

it V[(~R~(~, ~)) A (pR~(z,y))] = o zEU

Rc,(x,z)=-O a(R,~Rz)(x,y ) = 0;

thenVz

or

Rz(z,y)=O

it V [(~R~(~, z)) A (zR~(z, v))] = zEU

t h e n 3zo

such t h a t

R,~(x, zo) =

(VIRo(x, zl

1

and

Rz(zo,y) = 1

--5

\z~U

Hence

(o~R~)(flRz) r-- o~(R~Rz).

[]

ON FUZZY D I F U N C T I O N A L RELATIONS

227

This proposition also holds for an arbitrary number of arguments:

for all x , y e U

The following proposition gives a necessary and sufficient condition for a fuzzy relation to be fuzzy difunctional. PROPOSITION 5. R is fuzzy difunctional if and only if Ra is difunctional

for all a E ~(R). Proof. 1. R is a fuzzy difunctional; then R~ is difunctional for all a E O(R):

a(RaR~IRa) = (CeRa)(aR~ 1) (c~Ra) (by property P.7) (by monotonicity E RR-1R of composition) (since R is fuzzy difunctional) (since R~ = [.J{P[P is crisp and aP U_R}).

ER

Hence, Ra is difunctional. 2. R~ is difunctional for M1 c~ E ~(R) ~ R is fuzzy difunctional:

RR-1R = ] [ (aRa)(/3R~)-I(~/P~) ( b y Definition 6) ~,B,-r

K U(aRa)(aRa)-'(aRa) -

~

E U a(RaR~-lna) ~- U aR~ = R

(by Proposition 3 with a =- A { a , fl,-r} and by monotonicity of composition) (by property P. 7) (by the decomposition theorem [7]).

[]

Let us give an example to illustrate this property. E X A M P L E 3. The fuzzy relation R of Example 1 is not fuzzy difunctional since Ro.1 and R0.2 are not difunctional, i.e., RR-1R ~ R: U

x

RR-1R= y z

0.2 0.2 0.2

V

0.3 0.1

,

R=

x y z

tt

V

0.2 0.0

0.2 0.3 0.1

.

228

H. OUNALLI

AND

A. JAOUA

Consider now the following fuzzy relation S:

S ---- y z

u

v

0.1 0.1

0.3 0.5

S0.3 =

~

,

S0.1 --

,

~0.5 -----

•

We can see that S is fuzzy difunctional, i.e., S0.t, So.3, and So.s are difunctional. We can verify in this case that S S - 1 S = S. DEFINITION 7. A f u z z y relation R is max-rain transitive (or transitive for short) if and only if R R E_ R, or more explicitly,

for ~ll ~, y, z e u,

R(~, z) > (R(~, y) A R(y, z)).

PROPOSITION 6. If R is fuzzy difunctional, then R R -1 and R - 1 R are transitive. Proof.

R R - 1 R ~_R R - t ( R R - 1 R ) E_R-I(R) ( R - 1 R ) ( R - 1 R ) E (R-1R)

(since R is fuzzy difunctional) (by property P.3) (by property P.1).

Hence R - 1 R is transitive. The proof for R R - t is similar.

[]

DEFINITION 8. A f u z z y relation R is symmetric if and only if for M1 x, y • U, R(x, y) = R(y, x ) . P R O P O S I T I O N 7. If a f u z z y relation R is symmetric and transitive, then R is fuzzy difunctional

Proof. R R - 1 R E_ ( R R - 1 ) R R R - 1 R E_ ( R R ) R

E (R)R ER Hence R is fuzzy difunctional.

(by property P.1) (since R is symmetric) (since R is transitive) (since R is transitive).

[]

PROPOSITION 8. I f R and S are fuzzy difunctional, then so is their intersection R N S.

ON FUZZY DIFUNCTIONAL RELATIONS

229

Proof. Let T = R N S: T T - 1 T --- ( R ~ S ) ( R N S ) - I ( R ~ S) = (R n S)(R -1 n S-1)(R n S) E_ (RR-IR) n (sss)

(by property P.13) (since R N S ___R and

RnsEs) ERNS

(since R and S are fuzzy difunctional). []

Hence R U S is fuzzy difunctional.

PROPOSITION 9. A fuzzy relation R is fuzzy difunctional if and only if R - i is fuzzy difunctional.

Proof. Let us prove that if R is fuzzy difunctional, then R - i is fuzzy difunctional: RR-iR G R ( R R - i R ) - i E R -1 ((RR-i)R) -i (R-l(RR-i)-i) R-l(R-i)-iR-i R-i RR-1

G R -i E R-i E R-i E R-i

(since R is fuzzy difunctional) (by property P.14) (by property P.1) (by property P.10) (by property P.10) (by property P.11).

Hence R - i is fuzzy difunctional. We can prove in the same manner that if R -1 is fuzzy difunctional, then R is fuzzy difunctional. [] DEFINITION 9. Let R and S be two fuzzy relations. We say that 1. R is more deterministic than S if and only if R - i R G s - i S . 2. R is deterministic if and only if it is more deterministic than the identity I, i.e., R - 1 R E I. PROPOSITION 10. If R is deterministic, then R is fuzzy difunctional. Proof.

R-iREI R ( R - i R ) E R(I) RR-iR E R Hence R is fuzzy difunctional.

(since R is deterministic) (by property P.3) (by property P.9). []

DEFINITION 10. A fuzzy function is a fuzzy relation R such that for all a E @(R), Ra is a crisp function.

230

H. OUNALLI AND A. JAOUA

PROPOSITION 11. If R is a fuzzy function, then R is fuzzy difunctional.

Proof. R is a fuzzy function ¢=~ R~ is an ordinary function (by Definition 10) <=~R~ is a crisp difunctional relation (since any ordinary function is difunctional) ¢=~ R is fuzzy difunctional (by Proposition 5). [] 4.1.

CLOSURE OF FUZZY DIFUNCTIONAL RELATION

By analogy with other closures such as transitive and reflexive closures, we define difunctionM closure of a fuzzy relation R as the smallest fuzzy difunctional relation containing it. Formally, the difunctional closure of fuzzy relation R, written as R*, is defined by:

1. R E R*, 2. R* is a fuzzy difunctional relation, 3. For all fuzzy difunctional relations S that contain R, R* E S. T h e following proposition gives a constructive formula for the difunctional closure. PROPOSITION 12. The difunctional closure of R is given by

R* = R U R ( R - 1 R ) U R ( R - 1 R ) ~ U . . . U R ( R - 1 R ) n U . . . . Proof. Clause (l) of definition is trivial. Clause (2):

=

U

(R(R-1R)i)(R(R-1R)J)-I(R(R-1R)})

i>0, j~0, k>O

=

LJ i>0, j~0, k~0

_-

p~l

r- R*.

(R(R-1R)i+J+k+I)

ON F U Z Z Y D I F U N C T I O N A L R E L A T I O N S

231

Clause (3): The proof t h a t T is the smallest fuzzy difunctional relation containing R can be done by induction on the terms of R*. T h e basis of induction is the hypothesis t h a t T contains R. T h e induction step stems from the difunctionality of T. [] 5.

CONCLUSION

In this paper, we have characterized in a simple manner the fuzzy difunctional relations and have a t t e m p t e d to extend most of their properties in the framework of fuzzy relations. T h e s t u d y of difunctional relations has exhibited the concept of rectangle as an atomic object for crisp relations decomposition. In future work, we will focus on the concept of rectangle for fuzzy difunctional relations decomposition. The concept of fuzzy r e c t a n g l e will allow us 1. to modify our approach of automatic entity types extraction from an n - a r y fuzzy relation, 2. to s t u d y fuzzy difunctional dependencies, which generalize functional and difunctional dependencies, in the framework of the fuzzy relational d a t a model [9]. The authors are grateful for the many excellent comments and suggestions made by the anonymous referees. This work was supported by S E R S T (Secretariat d'etat de la Recherche Scientifique et de la Technologie de Tunisie); PNM/93 (Projet National Mobilisateur: Organisation Inerementale de L 'information).

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(1971). Received 1 March i995; revised 17 February 1996