In this paper, wepresent a new definitio of fuzzy functo al depencency and fuzzy multiva l-ued dependency based o similarity in fuzzy relational database, for which thresh lds are define
Trang 1T,! p chi Tin hoc va Dieu khien hoc, T.17, S.2 (2001),1 -1
HO THUAN, TRAN THIEN THANH
Abstract In this paper, wepresent a new definitio of fuzzy functo al depencency and fuzzy multiva
l-ued dependency based o similarity in fuzzy relational database, for which thresh lds are defined for each attributes of relation scheme The so ndness and completeness of inference rules, similar to Armstro g's
axioms are pro ed
Tom tit Trong bai bao nay chiin tai trln bay dinh nghia cho phu shuocham va phu thuoc da tr] me)" tren me hln CO " so' dir li~u mo'd 'a tren quan h~tu'ong tv' voi ngu'o'ngtu'o'ng tv' xac dinh rieng cho m6i thucc tinh Tinh xac ding va daydii cila h~ tien detuo'ng tv'h~tiende Armstro g ciingdtro'cchu'ngminh
1 INTRODUCTION
Relational databases have been studied since Codd's Such databases c n only deal with wel
-defined and unambigu us data But in the real world there exist data which can ot be well-defined
in a certain clear s nse and under a certain crisp form (often called fuzzy data) The databases for
the above mentioned data have been investigated by different authors (see [7) The fuzzy database models are an extension of the classical relational model It isbased on the fuzzy set theory invented
by Zadeh to capture the imprecise parts of the real world
In genegal, two ap roaches have been proposed for the introduction of fuzziness in the relatio al model The first one uses the principle of replacing the ordinary equivalence among domain values
by measures of nearness such as simi larity r e lation s hip s , proximity relationship , and dis tingui s h bil ity Junction (see [8) The second major effort has involved a variety ofapproaches that directly use po
s-s ibility di s trib u tions for attribute value (see [5) There have also been some mixed mo els combining
these approaches [121
This paper takes the similarity-based fuzzy relational databases a the reference model in our
study presented here
The data dependencies are the most important topics in theory of relatio al databases Several authors have proposed extended dependencies in fuzzy relational databas models In [1,2,4,6,10,121 have been given definitions offuzzy functional dependencies and fuzzy multivalued dependencies in fuzzy relational data models These dependencies are extension of dependencies of classical relato al model In this article, we give the definitions of fuzzy functional dependency (abbr ( a, t1)-ffd) an fuzzy multivalued depen ency (abbr (a ,,B)-fmvd) These dependencies are extention of dependencies
in classical model and more general than definitions of Rauj, Mazumdar, etc We also sh w that the inference rules of( a, ,B)-ffd,( a, ,B)-fmvd,which aresimilar to Armstro g' axioms for classical relational databases, are sound and complete
This paper is organized as follows Sectio 2 present some basic definitons of the si milarity-based relational databases In Section 3 and Section 4, we introduce an extensio of functional an multivalued dependencies; Armstrong's axioms for fuzzy functional and multivalued dependencies are presented; the soundness and completeness are proved Section 5 concludes this paper with some perspectives of the present work
The similarity-based fuzzy relational database model isa generalizato ofthe original relatio al
mo el It is allowed an attribute value to be a subset of the associated domain Domains for this model are either discrete scalars or discrete numbers drawn from eiher a finite or infinite set The
Trang 2equivalence relation over the domain is replaced by a fuzzy similarity relatio to identify similar tuples exceeding a given thresh ld of simiarity
Definition 2.1 A similarity relation is a mapping s : D x D + 10,1] such that for x,y,zED,
s (x , x) = 1 (reflexivity),
s ( x, y) =s (y , x) (symmetry),
s (x , z ) 2 maxyED{minls(x, y), s(y, z ) ]} (max-min transitivity)
Deftnit ion 2.2 A fuzzy relation scheme is a triple S = (R , s,5), where R = {AI, A2"'" An} is a set of attributes, s = ( s1,s2, , s n) is a set of associated similarity relations, 5 = (a1,a2, , an) is
a set of associated thresholds (a i E 1 ,1], 1:::; i :::;n)
Definition 2.3 A fuzzy relation instance r on scheme S= ( R , s ,5 ) is a subset of the cross product
P(Dd x P(D2 ) X X P ( Dn) , where D ; = dom(Ai), and P(D ; = 2D , - 0.
Let X, Y be sets of attributes in R , X = ( Ah ) h E I , I ~ {1, ,n }
ax denotes the vector of thresholds for a set of attrib tes X, i.e ax = ( a' , ) h E I
aXY den tes the vector of thresholds for a set of attributes Xu Y (XY for sh r
In order to approximate equality bet een tuples of fuzzy relation, a fuz y measure, a similarity relation r is defined as follows
Definition 2.4 Let r be a fuzzy relatio instance on scheme S = (R , s, 5 ) ' tl and t2 are two tuples
in r The similarity measure of two tuples tl and tz on attribute Ak in R denoted by r(tdAk ] , t2 l Ak ] )
I S grven as
r( tdAk] , t 2[Ak] ) = min { d x, y)} ,
xEd , YEd%
where tl = ( dt , d~ , , d ), t2 = ( df, d ~, ,d~ )
If r ( tdAk ] , t 2l A k ] ) 2 ak then tuples tl and t2 are said to be similar o A k with thresh ld ak.
Defnition 2.5 Let r be a fuzzy relation o scheme S =(R , s,5 ) ' X be a subset of R , tl and t2 are two tuples in r The similarity measure of two tuples tl and t2 on a set of attrb tes X denoted by
r(tdX ] , t21X] is given as
rJt IIX], t2 [X] = (r(t d Ai l] , t21Ai ] )' r(tll Ai 2] ' t2[ Ai 2] , , r(t 1Ai k] , t 2I A ik] )) ,
where X = Ail A i2 Aik ·
If r ( tdX ] , t21X] 2 ax then two tuples tl and t2 are said to be similar o X with thresh lds
ax ·
3 FUZZY FUNCTIONAL DEPENDENCY ( a, ,B)-FFD
Definition 3.1 Let r be any fuzzy relation instance on scheme S = (R, s , 5 ) ' X and Yare subsets
of R with associated thresh lds ax , a y , respectively Fuzzy relation instance r is said to satisfy the
fuzzy functional dependency- ( a, ,B)-ffd, denoted by X , , Y if, for every pair of tuples tl and t 2 in
(o x , O ' y )
r, r(tI I X ], t2 [ X ] ) 2 ax then r (t I !Y ] , t2[ Y ] ) 2 ay
D,efinition 3.2 A scheme S =(R , s,5) is said to satisfy the
instance r on S satisfies X , , Y
( cr :x,a y )
(a,,B)-ffd X , , Y, if every relatio
( ao x,ay )
Remark 1 The definition ffd of Raju et ai. is a special case of Defifition 3.1 (i.e if any instance
r that satisfies ffd X, , 0 Y then r also satisfies (a, ,B)-ffd X , , Y), where ax = (ao, a o, a )
and ay = (ao , ao, ,a ) with a is a constant in [0,1] IXItimes
'-v '
[ y It i m e s
The inference rules for (a,,B ffds
FFD1 (R ef l ex ivity) : If Y ~ X then X , , Y
(o x,a y )
FFD2 ( Augme n t at ion ) If X , , Y then XW , , YW
Trang 3FUNCT DEPENDENCIES AND MULTIV DEPENDENCIES IN FUZZY RELATIONAL DATABASES 1
FFD3 (T ra n si tivit y ) If X ~ Y, and Y ~ Z then X ~ Z
(O ' x ,O y ) ( O ry ,Or z ( a , a z
'I'heor em 3.1. Rul es F D1-FFD3 are so und.
Proof Let r be a relation instance on scheme S = (R, s , a)
Reflexivity: Suppose that Y S; ; X S;; R.
( ox , o : y )
h lds in T
Augmentation: Suppose that X »< r+ Y holds in T, Z S R.
( a x, a )
Vt1 , t2 E T , r(t d XZ ] , t 2[ XZ ] ) ~ axz
We have r ( tdX ] , t 2[ X ] ) ~ ax and r (t l [ Z ] , t2 [ Z ] ) ~ az
Since X , Y holds in T then r(tl [ Y ] , t 2[ Y ] ) ~ ay
( O' x ,O y )
Combining (1) with (2), we obtain r(tdYZ ] , t2 [ YZ ] ) ~ ay z
Hence XZ ~ YZ holds in r
(a xz, o : y )
(1) (2)
Tr ansi ti vi t y: Suppose that X ~ Y, and Y ~ Z hold in T.
( a ,C f y ) ( a y ,a z )
V t1 , t 2 Er , r (t d X ] , t 2 [ X ] ) ~ ax.
Since X ~ Y holds in r then r(tl [ Y ] , t 2[ Y ] ) ~ ay
( a ,a )
Since Y • • Z holds in T then r(tl [ Z ] , t 2[ Z ] ) ~ az,
C o , az }
Therefore, Y ~ Z holds in T.
(c r y , o z
/
The following rules are easily obtained from FFD1-FFD3
(a x ,o y ) (a x , O ' z ) ( ax ,Cl: yz)
FFD5 (P seu d o- tran s iti v ity): If X ~ Y and YW Z then XW Z
(o x ,o ) ( OYW, O " ) ( c rxw O " )
FFD6 (D ec om p s iti on ) If X ~ Y and Z C Y then X ~ Z
'I'heor em 3 2 Rul es FFD1-FFD6 ar e co mpl e t e on sc h e me S =(R , s , a) when the f o owing condition
h old s :
F o r eac h Ai E R , th e re exis t s at lea s t on e pair of e l e m e nt s ai, b , E d m(Ai) such t h t S i ai, bi :S ai
Pr oo f Let F be a set of (a,,B)ffds on scheme S = (R ,s,a ) , and suppose that f = X ~ Y does
(O ' x,O' y )
not follow from F by the rules FFD1-FFD6
Consider the relatio instance T o scheme S with two tuples as follows
Attributes ofX + Other atributes
a1 a ak ak+1 . an
It is easily shown that all (a, ,B)-ffds in F are satisfied by r, and f is not satisfied by T.
We conclude that whenever X ~ Y does not follow from F by the rules FFD1-FFD6 then F
(a x c r y )
does not logically imply X , . Y That is, the rules FFD1-FFD6 are complete
( a a y )
4 FUZZY MULTIVALUED DEPENDENCY (a, ,B )-FMVD
R , wih associated thresh lds ax, ay, respe tively Relation T is said to satisfy the fuzzy multivalued
dependency (a,,B)-fmvd, denoted by X ~ Y if, for every two tu les t1,t2 in T, r(tdX] , t2[X]) ~
(a x ,C lr : y )
a x then there exists a tuple t3 in T such that r( td X ] , td X ] ) ~ a x, r(t d ] , t3[ Y ] ) ~ ay, and
r( t 2 [ Z ] , t3 [ Z ] ) ~ az , where Z = R - XY
Trang 4Definition 4.2 A scheme S =(R , 5,& is said to satisfy the ( a, fJ)-fmvd X, , ,-. Y ifevery relatio
( n ·O ' y)
instance r on S satsfies X , , ,- Y.
(ax · o )
Rema r k 2 The definiton fmvd of Mazumdar et al 1 1is aspecial case of Definition 4.1 (i.e if relation
r holds fmvd X, - -.or o Y then r also h lds ( a,fJ)-fmvdX , ,,- Y , where ax = ( a , ao, , ao ) and
I YIt i m c s
By Defifition 4.1 it is easy to sh w folowing remarks
(ax ,O ry)
there exists a tuple t 3 in rsuch that rhlX]' t31X] ~ ax , r(t 2 Y ] ' t 31 Y ] ) ~ a y, and r(tdZ ] , t31ZI ~
Rem ar k 4. Relato r satisfies X , - - Y if, for every two tuples tl, t 2 in r ,r(tdX ] , t 2 X ] ) ~ a x then
( ox ,c r y )
there exists a tuple t 3 in rsuch that r(tl I XY ] ' t 3 I XY ] ) ~ aX Y , and r(t 2I X(R - Y) ] , t 3 I X(R - Y) ] ) ~
a X(R - Y) '
The inference rules for ( a , fJ)-fmvds
( ox , O ' y } (O y ,o z ) lc r O" Z y » ·
P roof Let r be arelatio instance o scheme S = (R , 5 , 5
Complementation: Suppose X , , ,- Y holds in r
Vt1,t2 E r ,r(tlX],t2IX] ~ ax
Since X, - - Y then ::l3Er, such that
( o , o )
r ( 2I X ] ' t3 X] ~ ax ,
r ( 2 I Y ] , t 31 Y 1 l ~ ay ,
r lZ]' t 3 Z ] ) ~ az, where Z =R - XY
Combining (1) wih (2), we have r(t d X ] , t 3I X ] ) ~ ax
Since W = R - XZ ~ Y, and by (3) then r(t2IW],t3IW] ~ aw.
From ( 5 ) and ( 6 ) it follows that X , , ,- Z holds in r
( ox , c r z )
( o ,c r y )
Vt1, t 2 E r ,r( td XW ] , t 2I XW ] ) ~ axw
Since X, - Y h lds in r ,and by (7) , we have ::lt3E r ,such that
(o x · o )
rtlIX]' t31X] ~ a x,
r ( td Y ] ' t31 Y ] ) ~ ay,
r ( 2I Z ] , t3I Z ] ) ~ az·
From ( 8 ) , ( 9 ) , (10) and (11) we have
Since r ( t1 I W ] , t J [ W ] ) ~ a w and V ~ W, then r(tl l V ] , t3 1 V ] ) ~ a v o
From (12), (13) an (14), i follows that XW , - YV holds in r ,
(1)
(2) (3) (4) (5) (6 )
(7) (8)
(9 ) (10) (11) (12) ( 13) (14)
Trang 5FUNCT DEPENDENCIES AND MULTIV DEPENDENCIES IN FUZZY RELATIONAL DATABASES 17
Transitivity: Suppose X , , Y, a nd Y , , Z hold in r.
(o x, O 'y) (Qy , oz )
( ax ,a
Since Y , , Z holds in r, then 3t 4 Er , such that
( a , O 'z)
First, we show that r(tl [ X ] , t4 [ X ] ) 2: ax.
Since X - YZ ~ R - Z , and by (5), it follows that r(t 3[ X - YZ ] , t 4[ X - YZ ] ) 2: aX-YZ
By (17) and X - YZ ~ R - Z then r(t 2[ X - YZ ] , t4[X - YZ ] ) 2: aX-YZ.
Combining (20) with (21), we obtain r(tl [ X ] , t4 [ X ] ) 2: ax.
Next, we show that r(t d Z - Y ] , t 4[ Z - Y ] ) 2: aZ-Y.
By (18), it is e sy to see that r(t d Z - Y ] , t 4[ Z - Y ] ) 2: aZ-Y.
Final, we show that r(t2 [ W ] , t4 [ W ] ) 2 : aw , where W =R - X(Z - Y)
Since R - XYZ ~ R - Z, and R - XYZ ~ R - Y , by (18) and (19),
From (16), (18) and (19), we have r (t 2[ Y ] , t4 [ Y ] ) 2: ay (23)
Since W ~ Y(R - XY) , by (22) and (23), it follows that r(t 2[ W ] , t4[ W ] ) 2: aw
Consequently, X , , Z - Y holds in r
(Qx,O:z-y)
The following rules are easily to obtained
( a x,cry)
( ox , a y ) ( ax , 0 ' z ) ( a x ,neZ-Y»
( "x '' ' ( y-Z »
(" XW '''( - YW »
Rules relate ( a, ,B ffds an ( a,,B fmvds
(ax ,c r y) ( a x la y)
then X • • Z'
(O' X 'O'z')
Definition 4.3 Let F and G be sets of ( a, ,B)-ffd and ( a, ,B fmvd on relation scheme S =(R , s, a).
The closure of F uG, denoted by (F , G)+, is the set of all ( a,,B)-ffds and (a, ,B)-fmvds that can
Theorem 4.2 Let S = (R , ii, & ) be a scheme relation, X be a subset of R, then we can partition
R - X into sets of att r ibu t es Yl,Y2, , Yk , such that if Z ~ R - X, then X r r + - + Z holds if and
( Qx ,az )
on l y if Z is the union of some of t h Y; 's
Trang 6P r oof Similar to classical case, see the proof in [11].
D the depen d n y b s is for X (with respect to D)
sc hem e S = (R, s, 5 ) wh e n th e following condition holds:
For eac h A i E R , th e r e ex i s t s at l e s t o ne p a ir of e lem e nt s ai , b, Edom(A i s uch that s; (a i, bi) :S ai
(aX,o A )
m
(F, G ) + By Theorem 4.2, we have R - X* = U Wi , where W i E {Yl,Y 2, , Yd·
i = 1
(ai )iEI o (ai)i f 1 (ai )i E I2 (a;)i E I m
( ai ) iE I o (b;}iE (a i )i I 2 ( a ; } iE Im
( ai ) iEf o ( ai ) iE I1 ( b; }i E f 2 ( ai ) iE I m
(a ; iE I o (b;)iEI1 (b ; )i I2 (ai)i E I m
(a ; }i E I o (b i iE I1 (b ; }i E f 2 (bi)i E I m
where X* = (Ai)iEIo , WJ = (A ; E I ' I) ~ {1,2, ,m}, f = D, ,m
( c r x w ,cr v )
(c rx ,o : w ) (ox , n, )
then X , X * W E (F, G) +. From FFD-FMVD3, X ~ (V -X * W) E (F, G) + , implies
that t r U ] = t 2[ U ] , t I W] = t2 [ W] , tdX * ] = t2[X*] so r(tl[X * W] , t2[X * W]) 2: ax.w. Hence
t(t d V ] , t 2[ V ] ) 2 : a Therefore, r satisfies (a, 8)-ffd U , , V
(au la )
We have X , X * W E(F, G)+ , by FMVD3 then X , - (V - X * W) E(F, G) +
( o x, O x* w ) (O 'X , C l ( V_X * W »
i E I1
t 3[ V - X ' W] = tllV - X* W ] , and t 3[ R - (V - X*W) ] = t 2[ R - (V - X * W) ]
t 3 [R - UV ] =t2 [ R - UV ].
Hence r(tdU ] , t 3[ U]) 2 : a , r(tdV ] ' t 3 [V]) 2 : av, r(t2[R - UV ] , t3[R - UV]) 2 : aR-UV ·
(O u l a )
We now show that d does not hold by r,
( a x a y )
Trang 7FUNCT DEPENDENCIES AND MULTIV DEPENDENCIES IN FUZZY RELATIONAL DATABASES 19
By constructio of r, there exist two tuples tl t 2 E r, such that t d Y ) -= I t 2[ Y Furthermore,
we have r(t d Y ) , 2 [ Y ) Lay But t d X ) =t 2 [ X ) then r(t d X ) , 2 [ X ) ~ ax Hence, r does not hold
Now assume that d=X ~ Y tf. (F, G) +, and d holds on r,
(a x a y )
By construction of r, it is easy to sh w that Y = X ' uW ' , where X' C X * , W'
J <; ; {l , , m}
we have X ~ Y E (F , G ) + , contraditio Thus d does not hold o r The proof is complete
(O x ·O y )
This paper deals with fuzzy data dependencie in fuzzy relational databases We give the defi
ni-tio s' of fuzzy functional and multivalued dependencies Furthermore, we discuss the inference rules
of these dependencies The soundness and completeness are pro ed A futher study involving the
definitions of fuzzy join dependency, normal forms for the fuzzy relational databases has been o
gomg
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H o T hua n - I ns t ute o f I n f o rmation T ec hn o l o y
Tran Thien Thanh - P e dag o i cal In s titut e of Q u y N h on