FUZZY F U N C T I O N D E P E N D E N C I E S IN FUZZY OBJECT-ORIENTED DATABASES
Ho Cam Ha^*)
Hanoi National University of Education
Vu Due Quang
Quang Nam University
^*^E-mail: [email protected]
Abstract. This paper concerns to fuzzy dependencies of attributes on fuzzy object class. Base on the possibility distribution approach, we proposes the formal definition of fuzzy functional and multivalued dependencies on object class. The dependencies in this study allows a sound and complete set of inference rules.
Keywords: object-oriented, database, fuzzy dependencies, fuzzy ob- ject, possibility distribution
1. Introduction
Not all the elements (data) that occur in the real world are fully known or defined in a perfect way. Classical relational database models and its extension of imprecision and uncertainty do not satisfy the need of modelling complex objects with imprecision and uncertainty. So many researchers have concentrated on the development of object-oriented database models to deal with uncertain data [9]. The fuzzy object-oriented approach have recently received increasing attention. Integrity constraints play a crucial role as in traditional databases, and also in fuzzy object- orientation. These constraints are used to define the semantics and guidelines for database design.
Based on the possibiUty distribution approach and resemblance relationship we introduce a measurement of semantic equivalence of fuzzy data. In this paper, we propose formal definitions of dependencies in the fuzzy class. The formal definition fuzzy functional and multivalued dependencies in this study allows a sound and complete set of inference rules. It is also shown that, such extended.dependencies, the classical dependency is a special case.
Ho Cam Ha and Vu Due Quang
The remainder of the ])ap('r is organized as follows: In the next section we recall basic notions from lii/,/.\- set and reseml)lance relationships. Then we deliiie l'u//y fuiKtional dei)eiideiicy. fuz/,\' multi-valued dependency on object class. These formal delinitions of (le])eii(l('ncies allows a scnmd and complete set of inlerence rules, h'inally, wt> present some coin hiding nnnarks.
2. Content
2.1. Representations Fuzzy set and Possibility D i s t r i b u t i o n
.\s pointed out [10], fu//y data is originally described a.s liix/y set. Let /' be a universe of disc-omse, a fuz/y set F is defined in U by a membership funct ion. The membershii) function PF '• U > [0,1] assigns to each element u of U a number, in the (lose interval [0,1]. that, characterizes the degree of membership of ii in F A fn//\- set F can be written as:
F = {//F(ui)/(/i.///r(//2)/v/2....,/fF("7,)/"n} {U is a finite set) or F = I (U is a infinite set).
1/6 f
\\ hen J-IFUI) is viewed to be a inea,sure of the jjossiiiility that a variai^le A'' has the \'alue ii in this approach, where A' takes values in U a fuzzy value is described In- a possibilit>- distribution TTV
TT.v = {7r.v("i)/wi.7rA-('a2)/"2 7r.v("„)/u„}.
where TT.V(?/,), "/ G U denotes the possibilit>- that (/, is true. Let TTX F be the possibility distribution repre.sentation and the fuzzy set representation for a fuzzy value, respectively. It is apparent that Tr.v = F is true.
2.2. Semantic relationship between fuzzy d a t a
Let U - Ul. U2. Uu be the univer.se of discourse, .4 is a variable, a fuzzy data of A on U based on possibility distribution is denoted as follows:
71,1 = {7r^("l)/"l,7r,i(vi2)A/2 ^ . - l ( " n ) / « n } - ", G U.7^A{IU) G [0. l ] .
Definition 2.1 (11). For fuzzy data, its semantics correspond to nn area in space [U X [0,1]). where the universe of discourse i-s its X-mis and possibility is its Y-axis.
The semantics of a fuzzy data TT.^ expressed by possibility distribution corre- sponds to an area in si)ace. the so called semantic space, denoted by 5'5(7r^4) (see Figure 1).
The semantic relationship between two fuzzy data can be described by the relationship between their semantic spaces. Let 55(7r^) and SSins) be semantic
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1.0^^
• ,AUU'
,i.' ' • I : '
.-."-';'
•
/
, . — ' • . . « ^,'
-*;-:''
f
.- SS(~.
•
0 U : U ; U; Uj; ;_-
Figure 1. Semantic space of fuzzy data.
spaces of two fuzzy d a t a TT,.! and TTB- respectively-. If SS{Tr_j^) .^ SSITTB), T^A seman- tically inchides TTg or Ttg is semantically included by 7r.4. If SS{TXA) 3 SS{7TB) and
SSITTA) Q SS{TrB)- 7r.4, TTB are semantically equivalent to each other. We employ semantic inclusion degree to measure the semantic relationship of fuzzy data.
Definition 2.2 (11). Let TTA end TTB be two fuzzy data, and their semantic spaces be SS(7rA) und SS(TTB)- respectively. Let SID (T^A^ "I^B) denotes the degree that TTA
semantically includes TTB- Then
SID(TXA, TTB) = (SS(TVA) D SS(TrB))/SS(TrB).
For two fuzzy data TTA and TTB- the meaning of SID(T:A^ T^B) is tfie percentage of the sema/ntic .space of TTB which is -wholly included in the semantic space of TT/.
Definition 2.3 (11). Let TTA and TTB be two fuzzy data. SE(TCA- T^B) de-note the degree that TTA and TTB are equivalent to each other, then
SE(TTA- T^B) - rmn(SID(TXA- T^B): SID(TTB- T^A))-
2.3. Evaluation of Semantic Measures
Definition 2.4 (11). Let U = {ui_,U2. Vn] be the universe of discourse. Let TTA v TTB be two fuzzy datas on U based on possibility distribution and 7r.4(i/.,;), (/,.; G U denote the possibility that Ui is true. SID{-KA,'^A) is then defi.ned
SID(TXA..T^B) = V'miii(7rs(i/.,:),7r.4('a,;))/V'7r£?('u,;).
* * 1 / ; ^ / / ' *
• i = l
' Ujeu
i-l
Definition 2.5 (11). Let U = Ui, U2, .u„ be the universe of discourse. Let 7r^ v TTB
Ho Cam Ha and Vu Due Quang
///( pos.sibdity thai ll, ts true Let lies be a resemblance relation on domain U- a for ( X n < 1 be a Ihre.sliold eorre.spirndrng to lies. SIIMTTA^T^B) is then defined by
II "
,S7a,(7r I. TT/y) - V mill (7r/3("J,7r,4(".;))/5Z^B("')-
^ — ' II,.III<. l!:Hr,s{ii,.iii] •I\ . ,
i \
lu t here, resemblance relat ionsliip Pes on domain U is ma.i)|)iiig I' / U > [0.1]
whicli salislies Iwo leiius as follows:
( 1 ) V/( C ( ' .He.s(//,//) - 1;
( 2 ) Mu,. Ill G (• H e . s ( / / / , " / ) ^ \U'S{UJ. II,).
2.4. Fuzzy functional dependencies a n d fuzzy multi-valued dependencies
./Vssumption:
OlD is a set of object-identities.
C is a set of classes.
A is a set of names of attributes.
nil \s used to denote a special coiustaiit. that presents uncertainty vahie.
dom includes all crisp atom values and fuzzy vahus.
Fuzzy Object: A fuzzy ol))(>ct is a pair of {id. c). in there, id is the object identifier taken from the OlD. r is a fuzzy (or clearly) \'alue on the OID. The GID
^'alues is signed as val(OlD) and defined as follows [1]:
(a) an element oi' doin. au element of IOD and the nil values itu OID
(b) if ('i- ro, r„ are values and .4i. .Ij 1„ are names of attributes, the tuple value is {Ai : /'i.ylj : ''j 4„ : /'„) and the .set value is {vi. Co. c,,} on OID.
Fuzzy class [11]: The objects having the same properties are gathered into classes. A class is fuzz>' because of the following several reasons. First, some objects are fuzz\' ones, which have similar properties. A c la.ss defined by these objects may be fuzzy. These objects b(4ong lo the class with membership degree of [0,1], Second, when a (lass is defined, the domain of an attribute may be fuzzy aud a fuzzy class is formed. Third, the sui x lass produced by a fuzzy (lass b\' means of specialization and the sujx'rclass i)ro(luced by some classes (in which there is at least one class which is fu/z>) b>' means of generalization are also fuzzy.
T y p e s ( C ) is defined as follows:
(1) Base t\'pe: integer, string, bool, float and the fuzzy type is generated base on the base types.
20
(2) Class name in C is fuzzy (clearly) ty])e objec:t type (3) If T is a fuzzy l.ype, set(T) is fuzzy (clearly) s(M. type.
(4) If Tl, Tj. ,T„ are t.ypc^s and /I], /I2,.-., At are names of attribute's in A, the tuple type is (/li : Ti, A : T,..., A, : T„).
2.4.1. Fuzzy Pimctional Dependencies
Definition 2.6. Let c be a class. Suppo.se thai U ts sel of attributes of c and X,V C U .A fuzzy functional dependency .V -'> ) is said to hold in cla.ss c tf for every pairs of objeels 0^,02 of class c. loe have SE{0i.X.02.X) < SE(0i.Y.0.2.Y).
Note that: The eciuivalent measurement of semantics is defined as follows:
\ (i) If Y - 4,..4. /U, then
SE{Oi.X, O2.A) - inin{SE{0i.Ai,0.2.Ay), SEiOi.A., 02.^2),..., SE{Ox.Ak, O2.A.)) (ii) If Ai refers to class Ci{Ui), type of .4, is object t3qje, then
SE{0i.X.0-2.X) = 1 when Oi.A - Oo.X,
E{0i.Ai,0.2.Ai) = SE{0i.A.i.U,,02.A.i.U.i): otherwise,
(in) If type of .4, is set type, Oi.Aj — {si, .S2, s->„ .... .s,,..., .s,,} O.2.A = {si,.'<2^% •''•, s„i}, then
SE{Oi.A.i, 02-Ai) = inin{mm{meoCi{SID{sj, .s,)), inin(inaxj(5'/D(6-i, Sj))).
i=l..n,j=l..TU j=l..m.; = l..n
(iv) If type of A is tuple type, Oi.Ai = (^i, W2: ",3, ...,/',•,..., u„),
02.A., = {u'l. 102. tu:i. ••••,Wj...., w.,^}, then SE(Oi.A.i,. O2.A;) = mm{SE{v.i, Wj)).
i,j = l..ii
The equivalent measurem,ent of seman.tic on Y and on X is defined similarly.
Example 1: Given c{U), where c is class c and U is set of attributes of c.A'. Y C U. Res{X) and i?e6(y) are .similarity related on A' and Y\ with a-i = 0.9, (T;2 = 0.95 are threshold of Res{X) and Res{Y) respectively.
Res(X;
a b c d e
a 1.0
b 0.2 1.0
c 0.3 0.92 1.0
d 0.2 0.4 0.1 1.0
e 0.4 0.1 0.3 0.2 1.0
Res(Y:
f g h i j
f 1.0
g 0.3 1.0
h 0.2 0.4 1.0
i 0.96 0.2 0.3 1.0
j 0.2 0.3 0.1 0.4 1.0
Ho CaiJi Ha and Vu Due Quang Instances of objc^cl (lass is shown as follows:
hi ldi
1(12
Tl3
X
{().7/a,0.4/b70.5/(i}
{{)J)/H,{).\/yA).H/i\}
{().;5/(l, ().8/e}
Y
{0.9/f. O.G/g^l.O/l|}
"{0'6/gV0.9/h.fJ.9/ij' {()l3/li,"0.4/i.0.l7j}' For e\'ei\- objects 0 | . Oj of c we have:
SE{Ch.X.()2.X)=-niin{SlD{0\.X.()2.X).SID{0.2.X.Oi.X))
= ;/;//;(0.<S2 1.0.875) - 0.824 v
SE{(),.Y.().2.)') - niin{SID{0,.y O^.Y). SI DiCh-Y Oi.)')) .--•- //////(I.O, 0.90) = 0.90.
It is easy to realiz(> that SF{0i.Y.()2.Y) > SEiOi.X,().2.X) from the definition of fuzzy functional (lei)endenc>' A' •'--•-> )
L e m m a 2.1. Lei c{U) is a class with set of attributes U Suppose X C.U.AE. U. If X -^'^ I holds and :\: tuple{Ai : Tj. /U : T. A,, : Tk) then X —> -4,,/ = 1,2 k.
Proof X '"> A implies that SE{0^.A.02.A) > S£'(Oi..V, OoA'). for every O1.O2 hi fuzzy object class. A1.A2 A^ takes the place of A. so that SE{0i..A.02.A) = min{SE{0i.Ai,()-2.Ai)). i = l..k. Obviously 5 £ ( 0 i . / l , . O2..4,) > 5 £ ' ( 0 , . . \ ' . Oo.A'), / — l..k. Thus, according to definition of fuzzy functional dependency, we have
.V -^'^.4,./:- 1.2 k. D
Tlieorem 2.1. Th.e functional dependency in relational database is a special case of the fuzzy functioind dependency in object oriented database (given above).
Proof Suppose A —> Y holds in relation r{U),X.) QU It means: for every pairs of tuple /i,/2 G /• if /i[.Y] = /2[A'] then ti[Y] = t^lY]. If consider U as the set of attributes of (4ass e. for ever\- pairs obj(>ct of r ( 0 i . 0 2 ) . we have SE{Oi.X.02X) =
l.SE{()^.y C)..)') -= 1. Thus. SE{C)i.Y O.A') ^ SE{0i.X.0.2.Y) ^ 1. It means
A •''' y liolds. D
2.4.2. Decomposing tuple type attribute to fuzzy function dependencies The grou]) of some attributes in a class into the other one which has tuple tyi)e. This can lead to binding semantics or fuzzy functional dependencies between attril)utes are not preserved. In this case, we need to decompose these attributes to preserve the semantic constraints.
28
Suppose A^: tuple(.Y, : i;), / =- 1, 2, 3,.... A-(^- > 1). If A; takes the place of A*
f • k
and X —> Y holds, where A' = U A',', A'.' C A',, then A* can be decomposed into
( = 1
attributes as follows:
Bi : t.uple(A;\X; : Tx,\x0^i - 1, 2,3... A: - 1.
B, : tuple(A,\A'j;Ar : rv,vY[\v). / - 1 . 2 , 3 A: - 1.
Ci : tuple(A7:r.v;)./ = l,2,3....,A-.
D : tuple(Y' : Ty).
2.4.3. Fuzzy Multi-valued Dependencies
Definition 2.7. Let c be a class. Suppose that U is set of attributes of c. Suppose
fc
also X. Y QU Z = U\X) A fuzzy multivalued dependency X -^~^Y is said to be held in cUiss c if whenever O^, O2 are two objects of class c, contains object O.j so thatSEiO-^.X.O^.X) ^ SE{0^.X,02.X) and SE{0:^.Y.O^.Y) ^ SE{0,.X,0-2.X).
SE{0:i.Z,0-2.Z) ^ SE{0i.X,0.2.X)
Example 2: Given class c.{U), X.Y C U, Z = U\XY Res{X),Res{Y) and Res{Z) are three similarity relations of X. Y and Z respectively, as in Example 1.
Q'l = 0.9. a'2 = 0.95, a-i = 0.90 are three thresholds of Re.s(X) and Res{Y), Re.s{Z) respectively.
Instances of object class is shown as follows:
Ol
02 Oi
Id Idi Id2 Id3
X
{0.4/a, 0.6/b, 0.7/d}
{0.4/c, 0..5/d, 0.2/e}
{0.4/a, 0.5/b, 0.6/d}
Y
{0.6/g, 0.9/h, 0.8/i}
{0.3/h, 0.6/i, 1.0/j}
{0.6/g. 0.7/h, 0.8/f}
Z
{0.5/a., 0.7/c, 0.4/e}
{0.2/b. 0.5/c, 0.9/e, 0.8/f}
{0.6/d. 1.0/e, 0.7/f}
SE{Oi.X, 0 2 - ^ ) = m.in{SID{Oi.X, O^.X), SID{02.X, d.X))
= m'm(0.818,0..529) = 0.529
SE{Oi.X, Oti.X) = mm{SID{Oi.X, O^-X), SlD{Oi.X, d-X))
= mm(1.0,0.882) = 0.882 > SE{0i.X.02.X) SE{Oi.Y.O.i.Y) = m.in{SID{Oi.Y 0:i.Y),SID{0:i.YOi.Y))
= m'm(1.0,0.913) = 0.913 > SE{0i.X,0-2.X) SE{02.Z, 0:i.Z) = min{SID{0-2.Z, O-s-Z), SID{0:i.Z, O2.Z))
= mzn(0.913,0.875) = 0.875 > SE{Oi.X,02.X)
Ho Cam Ila and Vu Due Quang
Thus, accoi'ding to Ihe definilion of fuzzy nuiltivalued dependcmcy, A '->Y holds in fuzz>- objcMl class c
T h e o r e m 2.2. Tlic midlt-ualued dependency in relal.tonal database is the .special case of Ihe I'lizzji mvUividiied depen.deney in object oriented database (as given above).
I'roof. Siipi)()se .V -> >) holds ill relalicm/(^Z). It means V/i./2./i[A'] —/2[A''], -I/3 such that l-,{X] --- /|[A1. •/;i[)'| = /i[V'], l-^Z] = l-^lZ], where Z ^U X) Ccni.sider I be as the sel of al tributes of class c h'or evfn'y l)aii- of objects of r((.)i, O.-j). exits objed (h C r we have Si:{(h.X.Oy.X) = SE{(h.Y.()^.y) = 1. SFACh.Z.Oi.Z) =
SE{0i..\.C)2.X)== I. D Obviously, Si:{(h.X.()i.X) ^ SE{()^.X 0.2.X) ^ 1.
.S7-.'((;•,.)• O l . ) ' ) ^ SE{0x..\,().2.X) ^ 1
and SE{(h.Z.0.2.Z) ^ .S7;(0,.A' O-i.X) ^ 1. Thus. .Y ^' >) holds.
2.4.4. T h e inference rules of fuzzy d e p e n d e n c i e s
The inference lules of fuzzy functional dependencies:
Rule 1: ReJIe.rivity. If A D ) then A ^''^Y
Rule 2: .\ugmenta.tion. If A ^-'>Y then XZ^' YZ Rule :i: Trans ill mty. If A -^' Y and Y - Z then A ~> Z Rule 4: Union, li X ^'">) A -^^ Z then A ^ YZ
Rule /7.- P.seudo transtl.iuily. If V - - ) ' ) ' H ' - Z then . Y i r ^ ^ > Z Rule 6: Deeompo.silion. If A" -"> Y Z C. V theuA' - Z
The iiifereiK (> rul(\s of fuzzy multivalued depeiichmcies:
Rule 7: ReJIexirily. If ) C A then V '•'' ^)
Pide (V.- Complemenlalion. If A -('-'->)' then .Y - > % r^ A ' ) ' R.ule 9: .Augnie Illation. If .Y - / ' ' . ) • and Z C W then A n i ' - f ^ V Z /I'vf/c //7.- Tran.sH.irily. If A ( - > ) ' and ) - / - > Z then A' -4--> Z - T y?//ic n.- 6'///(///. If X -M- )• and A 4 ^ Z then X 4 - > Z y
/?7/,/r' /.'j.- Pseudo Iran.sHi.vity. If X - ( ' - . V and i ' l F - / % Z then A i r ( % Z - >TF
Rule L'y.- If A - >-•>)' and A 4 . 1 ^ Z t h e n A 4 i ^ Z n T
Rule 14: If A - i ' . Y and A' 4 - > Z then X JUY- Z and X 4U Z-Y 30
Theorem 2.3. The inference rule's (rule 1- ndc I4) in fuzzy class arc sound and complete.
We suppose that procedure of j^roof for the soundness and completeness (jf above inference rules is similar to the ca.se of fuzzy relational databa.s(> [7, 11].
3. Conclusion
When \^v design an object-oriented databas(^ schema, we need to eliminate data redundancy. The reason is that as a conseqiuMice of the redundanc:y, the int.egrity constraints are broken. It is the same in classical database, the fuzzy dependencies are c(mceptuall>- meaningful, which are integrity constrains in fuzz>' oljjc-^ct-oriented database design. This j^aper describes an on going work. In order to continue, we have already begun some studies of normalization in OODB design.
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