DSpace at VNU: Functional dependencies with context dependent Null values in relational databases

DSpace at VNU: Functional dependencies with context dependent Null values in relational databases tài liệu, giáo án, bài...

V N U J O U R N A L OF SCIENCE, Nat Sci , t XV, 1999

F U N C T I O N A L D E P E N D E N C I E S W I T H C O N T E X T

D E P E N D E N T N U L L V A L U E S I N R E L A T I O N A L D A T A B A S E S

B u i T h i T h u y H i e n

F a c u i i v o f Mri thcinat ics, MccliHuics riud InforjjiHtics

C ol l eg e o f N^tìiỉ ỉìl S ci e nc e s - V N Ư

N g u y e n C a t H o

Ij i st i t j i t e o f IiiỉoriỉỊíưioii T e c h n o l o g y Nỉitíoỉiỉìl C e n t e r for S ci cnc e a/id Technol ogy.

A b s t r a c t T h e a n n o f flits p i i p t r /.s to J)7'e,s€7it a n cxtcìLsi.oìi o f t h e c o n c e p t oj

f m i c i t o i i a i d e p c i i d e r i c y in a (iafaha.se Ĩ71 wfitch t h e p r e s e n c e o f c o i i i t s i d e p e n d e n t , n u l l values IS allowed If IS sliowv that the set of A n n s h v i i g ' s nifere.nce rules fo rm s

a s o u n d a n d c o m p l e t e axtoin s y s t e m f o r f u n c t i o n a l (lepeĩìdencies u n d e r a suitabl e

s e i n a i i t i c o f conftJ'-f 7ỈUỈỈS S o m e r u l e s a n d a l g o r t i h i n s f o r n i a m p u l a f i n g c o n t e x t n u l l

values are also nitrodacejl a n d t x a v i i i i t d

1 IN T R O D U C T IO N

In the theory of lelatioiiHl d atab ase (lesion, the iiitPgỉiỉ>’ (‘Onstiaiiits play a crucial role and have been de(*pl>' investigatt'd in tlu' fiaiiK’work of (latat)ase relations without

null values In such a fiaiiiewoik funcitovai (lepcjide/ncies (FDb) are the m ost a a tiiial

and ust'ful The* not ion o f a kr\' ((l('ri\'ctl tioin a Si't ot F D s ) is f u iu l a m e n t a l t o t h e

le lH t.lO lia i ia u d i * L A ^i>Uluì i iu d ( u í i i Ị í l r l t ' (ix'io iii MV.slrui I'oi F D h W(US f i i h t i y i l l [1] aiiv\ iii

known as Armstioiif^’s axiom systi^n Many aiithois, M Levone [11], Lirn[12], Atzoiii and Morfuiii [2], [3j, [4] Maun' [i;3] have coiksidtHiHl FDs in d atah así' rf'lations containing unm arked null values, which semantically int(‘r p n 'to d as "unknow n'’ [11], [\''\^ (-)I

“no infoim atiou” [12], [2], [3], [4] Lien Atzeni an d Morfuni have iutiocluced a sound iUid

complete axiom system for F D s by dropping th e tra n sitiv ity rule an d a d din g th e union and decomposition rules to A rm stro n g ’s axiom system To m aintain th e satisfaction of

FD s in relational d atab a se s w ith incomplete inform ation, Maier [13] has in tro d uce d and investigated m arked null values

T he aim of this pap^'i is to present an extension of rho concept of functional d ep e n ­

dency in a d a ta b a s e in which the presence of co ntex t dependent null values is allowed It

is shown th a t th e set of A rnistron g ’s inference rules forms a so u n d and com plete axiom system for functional (lepeiulencies under a su itable semantic of co ntext null Some rules and algorithms to m an ip ulate context null values are also introduced and exam ined

14

F u n c tio na l d e p e n d e n c i e s w i t h c o n t e x t d e p e n d e n t nu ll va lu es 15

2 BASIC D E F IN IT IO N S

Let R ( A \ , , , ) be a relational scheme defined over a set of a ttrib u te s Ax, , Ar^.

l lio duniaiii of each a tt r i b u t e A, is denoted by Doĩn{A^) T h e dom ain of R consists of the

C aitP isian p ro d u c t Do7n.(/1i) X DoiniA-z) ^ .X D o m ị An ) Sihd denoted by Dom,{R).

We extend each d o m ain D(mi{A,) to an extended dom ain Dorri*{A^) by adding a finite set of null symbols, nam ely D o 77 ỉ*{Aj) — Doĩĩĩ ịA, ) u A,J u A i2 u {due}, where

- is the set of optni context nulls denoted by

- dne riioaus it "does not exit'’ context null.

- D o 7 n ( A , ) A , , , A,.^, {(hie) are the disjoint sets.

T he ext en de d d o m ain Dorii*{R) of R consists of the Cartersiaii p ro d uct

D o ì ì ì ^ ( A ị ) X X D oì ĩ l* ( A n )

A relation of a scheiiio /? is a subset of Doĩn^^iR) Such instances are denoted by

lower ca.se letters such as r i ,

A K'latioii c oj it ai ns no or s o m e null values is called a p ar t ia l relafAon T h e set o f all

partial relations over schouio ỈĨ is bv R cl]{R).

A K'latioii w ith o uf null values calltHÌ a total relation, the set of all to ta l relations

ov'(M sclu‘iiH' IÌ is clenottHl bv Rcl{R) A fuplo of an iiKstance r is called ail elem^nit of

\ \ ( ' (h'uotr tuph's by letf(‘is such ais If f is a tuple of a relation ĩ\ then f[A, (Iciioti's th(‘ i onipuiiont of t wliicli coMespoiids to the a ttr ib u te A, If ^[.4,] is not null we

wnio /[-4,]!

VV(' US(‘ rlic n o ta tio n 0 ]>eri to refor an open context imll, and no tation unk to ivĩei

a l l t i u k n c n v i i c o u t i ' x t n u l l

A n u n h i i u w i i U 1 0 Ị M ‘ 11 t o u t c x l n u l l i; I t i l l c i l t i l l i i u U - f i u i l r V d l i u ' i i i n l c\ I I O I I n u l l V r t l i i f

<)1 a fhic null is (‘rilli'd a (l(’íiiiit(' \'alii(’.

3 KU.\X"TIONAL D E P E N D E N C IE S W IT H C O N T E X T NULL

In the classical theory, a functional dependency (FD) is a statem en t f X Y

satisfies J (wi‘ SHV also t hat / holds in ;■ ) if for (‘ach pair o f t uples f \ J 2 Ễ ^ such that

/i[.V] = we havt^ 1 ~ ^‘l ị y ■

Let f] and Ỷ 2 two tuples (m ay contains context nulls) over scheme R and A be

an a ttiib u le \Vf‘ sliall w iirc ^|[.4j —— t- 2 [A]) if

1 h [-4]!, f2[-4]! aiul /ị[.4ì =: or

2 t i [ A] — — ồỊ h i k I / ^ y, o r

3 t \[A] — J p ^2[A] — d, aiul i — 7, 01'

4 ^i[^4] = (hie and

By t [ [ X ==^ we u n d e ista n d fx[A] = = Ỷ 2 A , for all A e X.

ì(i B u i Thi T h a y Hieiiy N g u y e n C ai Ho

'1'ln'ít ' l u i ĩIh' uH'aiiiiijj, ot (■(>iii]>ai isoii í)])('i atoi —“ is just t o ( tli(’ syinì)olic

(IIi(

Cuiitiarv to coiuiiai 1MJ1I ui)í'iai()i - is the c oi n p a i i s o n o p n a t o i Foi ('XiiiUỊilc :ị / = 4 ; ^ = / = = / = , i

-In (latahas('s w i t h iiicoiiipl('l(' i u ĩ oi ui al i on .1 natural (lucstiou arises at tliis jniK tUI('

IS llial; what is tli(' t n i t h valiK' of /■ = ỊJ if' 01 fi 01 both ;u(‘ null'.^ [11 Ị9Ị Co<l(l lias

lu l r o d i u r d a t Incr-valiK'd lo^lc {0.^' 1} foi nsiii^; t o exploi t data from (lat ahascs tliat inav

( O i i l a i i i I i n l i V i i l u c s , l u t l i ( ' c i i s c o l C o i i t c x r n u l l s WI ' a < l o p t a l i v c - v a l i i c d ! ( ) - i c I o c o i u p a i <' 1 l i c

v a lu e s in a contc'Xt n u l l (lat a l ) a s ( ‘ ' ĩ h c f o l l o w i n g trtl)lc a llo w s US t h o t n i t l i v a l u e as,siíi,uiu('iii U) S U( li ( oiuỊ)ai isoii:

Fuị., ỉ

I(’ W(‘ (1('1Ì1U' / ’ a I r u t h \'alu(’ assiji,iiiiunit t o a < uui])Hrisou ì)('t w('('ii iiuy t w o valiK's iu a

c o n t c x l n u U (la t,aha.s(' W(' h a v e r{<i = b) = 0 ; r((i - - (liic) ~ 0 : r ( ồ , = / > , ) = ^ ; ì' (ò, =

t , ) = I '(11 — n) = 1; t ’{di ì c = d i i f ) = 1; .

W(‘ now ('Xt(‘ii(l the C'0ii('('pt of f unctional clcpíMuloiKẠ' ill a con text null (latal)H.S('.

D i - f l l i i t i o i i 1 L - t n { U ) a I v - i n ■: V c ' ' ■ > > ' - '1 ' ' " t l ' ^

saui t o be (L f i i v c t i o i K i l (IfpciKlc'iKi/ with ((Iiitc.rl iittJIs OIIKV R ( U ) ( c i i F D ) 1Í f o i e a c h

(■(Iiit('xt I iiill K 'la tio u /■ o ve r n ( U) an d fo r I'ach I>air oi' tu p le s /1 /'i € r such th a t I '( i | [ v ] ^ 'ji-V]) > e, W(- havo r ( i | [ y ] = t 2 [y'\) > £■ 11’ f = ^ instead OÍ V )

P r o p o s i t i o n 1 = t '2 Ầ ) > Í it Hijci only it /^i[A] = = t -2 X ■

Proof T he proof is clin'ctly tUHlucod from the (lofiiiition of function r and coiupaiisoii

(j])(n<itor = = .'O’

F i o i i i P r o p o s i t i o n 1 it tollovv.s tiu it D i ' f i n i t i o i i 1 IS C (|iiiv a len t t o t h e follt)wiiifi, o n e:

D e f i n i t i o n 2 Let R{U) bo a K'latiun scheiiiP and X , Y c U \ f : X Y IS soul to

hr a f n nc t lo n a l uniìì coiitc.ii 'ĩi.ìdls over f ì { ư ) ( c i i FD ) if for oacli coiitpxt null

relation r ov('i R{U) and for oacli pail of tuplf's t \ f 2 £ I' sikIi th a t ^i[vY] = = we

have ^1 [ y] = = ^i[ F

In t liis p a p e r , w e 1 ('Strict o m s o l v p s t o i n v e s t i g a t e f u n c t i o n a l d i ' p c r u l e n c i e s midcn

CDiitoxt nulls X ^ Y w ith f = c i.e., cnF D s of the foiiii X —> Y.

F u n c t i o n a l d e p e n d e n c i e s w ith c o n t e x t d e p e n d e n t nu ll v a lu es . 17

Fig 2.

E x a m p l e 1 T[i(> Iplation r given in Fig 2 satisfies functional dependency B.

Let F be a set of (lata dpponclencies, a d a t a dependence / is callod a logical conse­ quence of F if any relation r satisfies F then it also satisfies /

w v cloiiotp F* = { / I / is a logical consoquence of F}

Gi v en a S('t o f infeieiico lilies, set f + = { 7 I / is d e d u c e d from p by m e a n s o f the

iiifpiencp rules }.

Th(> set of iiiforeiKT rules is said to be sound if ( F + c F*) and to be com plete if

(F + = F ‘ )

It is well known th a t for functional deppiicleiicies in th e reJatioiial moiiel without mills, the followiiio is a sound and coiiipletp set of irifpieiice rules;

-4i) K'Hcxivify: If Y c X tlu'u A' ^ Y

From rules ^ 1, A' 2 , A 3 we can deduce the following two rules:

Ẩ.j) union: If X — Y and X -* z then Y — Y Z

.4s) d(H'oniposition: If Y ^ y z then A' —► Y.

A ccoidin” tu Dc'fiiiitioii 1 (or Dctiiiitioii 2) it is easy to S(‘(' t hat icHi'xivitv

auftincuta-tioii, trausiti\'itV uiiiou and iloroniposit iou nih's ai(> s ound also for fuiK tional (lppf'iid('iici('s witli co nt ex t nulls.

For couvcnioiici', W(' lecal l here the following notion: T h e closun* A'+ o f a set of

Httrihutcs A with lespoct to u H('t F of tuuctional (If'pciulencies with coiiti'xt Iiulls is

di'tiiicd as tullows:

l ỉ y the rulf's of union and il('C()iii])ositiuii, it is cloai tluit A' —> is (l('(luc('(i

/■ I n ’ m e a n s ( i t t l i ( ' I i i l i ' s i f a n d o i i l \ ' i f V c A '*”

T heorem 1 The rules A i , A 2 ,A ị foim a sound and complete set o f inference ivies for

ĩnnctioiiRÌ d e p e n d e n c i e s w i t h c o n t e x t 1ÌUỈỈS.

Proof, a) Souiuliu’ss of tlu'se rult's has hccii shown above.

b) Coiijplctf'iK'ss: Let F he a set-of iuuctional (lf'p('iul(’ncÌPS with context nulls over a srliomo R(U ), F* = { / I / is logical consoqiH'iicf' of F }, = { / I / is ileducpd from F by

iiK'aiis o f A i , A- 2 /l.'i}- SiiK't' 4 ; , ,4;j ai(' s o i i i u l , W(> h a v ( ‘ F~^ c . It r e n i a i i i s t o p r o v o

th a t F* c F + , ^ h a t means wo need to show: if / € F* thon / G F + This is equivalpiit to show th a t if / ^ then Ị ị F *

Assunu' tliar (Ị : X —+ 1 is a ciiF'D and (j Ệ Let ;■ be « two tuple iT'lation

• ^ 2 } , w I k ' K '

V/1, G u, tht'ij ^i[.4,j = 1 or

ii[A ,l = Ố

OI-18 B u i Thi T h u y H i e n , N g u y e n C a t H o

/i[.4,] = d, IJI

^ Ị/1,Ị = line.

\/A, e A'+ tlicn /■.>[.4,] = = /1 4,

V.4, € u \ -V+ tli('ii / j[.4,] = / = h[.4,

(1) r satisiics all til*' tlejK'iulcncies in /•’: Let ( \ ' -* u ) G f , as.sinní' V' u is not

satisfied by r i.e., ^[V"] = = tọịv] and /i[v r] = / = ^2ỊU■j From ^i[V'] = = t-,[V] we h a w

V c From / i [ i v ] = / = it iinpli('s tliat II' nmst cont ai n at least ou(' att,iihiiti'

in Ư \ say 4 Thvis 3.4 G \ v such th at A ị A' ■ (*) Since r c A '' W(' concliulc that

so w c A'^ Blit, it is iiupossiblc l)(‘caus(* of (*) Hence, r satisfies all the (lcp('iul('iinc>

in F

(2) r does not satisfy fi : A s s ui u c the c o i it ra iv th at /• SHtisfi<'S (/ : X y From

implies by defimtioii of t i t ’i th a t Y c x ^ So, (A' ^ y') G F a (‘on ti ad ict ion.

Therefore F* c Coinbinin^ with c F* \ VP hav(‘ F* — F ^ \ ộ

4 SO M E RULES T O M AINTAIN T H E SATISFA CTIO X

O F F U N C T IO N A L D E P E N D E N C IE S IN C O N T E X T NULL DATABASES

A c co rd in g to the Si'iiiantir a pproach to co n te xt nulls, c-ontext null valvK'S is (Iciinod

b v well -known information T l u ‘ set OÍ f unct ional (l(*p(‘iulenci(\s a i ( \ o f course, very i m p o r ­

t a n t well -known i nf o in i at io u to defino conti^xt nulls T h a t m ea n s, c‘ont(*xt nulls have to be

defined and haiulled to onsiin* tliat the (latal)ase w ith cOiiti'xt nulls uiulri cousideiatioii still satisfies a given set of fiuK'tioiial (li'pi'ndiMicies Hoỉicí\ wliik' im phnnenting tlie (lata

u p d a te procedures, tlu' system has to m aintain thí' satisfaction of functional (l(^pend(Mi(‘U‘s

in the database To obtain íliih' obj('Ctiv(' S011U' Mili's for iiandliug contoxt nulls luvil to

ho oV>oyr*il

D e f i n i t i o n 3 l e t r he in R d ^ { R ) X -* A hv a n i F D OYVI R J i and Ỷ 2 be two tupN's of

r such th a t /ỉ[X]

- If ^i[>4]! t ‘i[A]\ and f\\A] ~ / = ^)[-4Ị then r has a hard violation of X —^ 4 airi

o pen context null then r has a hard violation O Ĩ X A.

least one of two side of (*) is null fhen r has a soft violation OĨ X A and, /i and f .2 are said to cause a soft violation 0 Ĩ X A.

E x a m p l e 2 Let >1 is a cnF D over R an d consider two tuples an d t 2 of r and

- if ti[A] = = 3 then r has a h a rd violation OĨ X A

- if f\ A] = dne, to[A] — 1 then r has a hard violation of X —^ A

- if ^i[^] ~ (Inc, to[A] = l ị then /' has a soft violation of A' —* A

F u n c t i o n a l d e p e n d e n c i e s w i t h c o n t e x t d e p e n d e n t null v al u es 19

T h e function VIOLATION in A lgorithm 1 will ch(Tk whothoi two tuples /1 a n d Ỷ 2

causo a violation of the ciiFD X A.

A l g o r i t h m 1 V I O L A T I O N { r , f x , Ì 2 s X > 4)

I n p u t : V E Re l i ^ ( R ) , X A is ‘A c n F D ov('r R J \ and Ỷ 2 are an v two t uples o f r

such th a t ^i[A'] = = t-2[x

O u t p u t ; 2 if /] and cause a liaid violation of X —» /i; 1 if /ỉ aiui Ì 2 cause a soft

violation of F ; 0 otherwise.

Begin

if (/i[.4Ị! and ^>[.4]! and ^i[.4] - / = /2[.4j) then V I O L A T I O N ;= 2

else if (^i[/l] — Ị ~ /'2Ị-4]) and ((/i[*4] diic

then V I O L A T I O N 2:

else if (^[.4] - / = t 2 [A]) then V I O L A T I O N 1;

End,

D e f i n i t i o n 4 A conicxt null (latabase is said to bf' consistent with a given set F of

iuiK’tiuiiiil ílt'Ị)íUKlí'ii('ị('s if iIk 'k ' is not any hard 01 soft violation uf F in thí' database*.

D e f i n i t i o n 5 Li't D B 1)(’ a cuiitoxt null databa8(' and r h(‘ H n^lation in D B aiul F he

a set o f c i i F D s o\'('i R A ss ui i u' tliHt t \ a n d /2 ai(' a n y t wo t u p l i ' s ill r t h a t <‘aiis(^ a soft

\ iolati<m of a n .V — 4 in /■’ A soft v i o l a t i o n i‘(‘iuo\' al t h a t is cHusi'd 1)V t i a n d f ) in r (or

111 i ) i U is (l('fiiK'd a.'> follows;

1 II' tíiit' OÍ tlio twt) valiK's t \ { A ] a n d is n ot null, (sav / i [ / l ] ) aiul t h (‘ o t h e r (/íỊ.4]) is ('iilit'i ;iii I i n k n o u ’ii coiiicxT mill in a n OJX’II con tt 'xt null, t h e n v\' (n \ occurroĩicí^ DÍ’ thi' mill t y[A] iu /■ (or iu D B ) is I'lianj^ed l)V

‘J 1 Í < >1 ii (>1 t 1 1 < t w o \ a l u c : / J A a l it I t Ỉ / 1 ] i (■ it 1 1(1 a <i m i m l l t n <111 111 k n o w 11 t o u t c : v t

null (sav /i[.-l]) a]i<l tlir uIIh'i vahu* (/2[.4]) is an oị:ìími coiiU'xt ĩiull tlu'ii (’V('IV occiiin'iK’i'

OỈ ili(' mill (/ )[/i]) in /■ (or in 1)1^) is ỉ>y /][/l

3 11 1)01 li ;ui(í r ai(' (‘it lu’i Iiiiknuvvii contf'xt nulls OI couti'xt nulls,

tlif'ii ('\‘('iy tx'curn'iict' ui thf’ 01H' with iiuli'X (sa\' i } /1Ì) ill r (or in 1)B) is cluiii^cd

bv the o th e i w i t h s maller index (^i[-4j).

E x a m p l e 3 Ill íhí' cas('s ỉ)('!t)\v tlìí' soft \io laiio n s ill r (or D R ) will be iPiiioved as

tolluws:

If and /'iỊ/l] = llicii ocí iỉi'iriic(' of h, ill /■ (or ill D B ) is rhaugíHỈ h\'

h[ A]

If a n d f ị [ A] = ii, tlii'ii V t x ' c m n ’Tic(' OÍ J , HI /■ (or in D B ) is cliaii^^rd 1)V

If /] [-4] ” d i i v a n d ^2[.4] = li, t h e n ('vt'i'v o c c u n e i u o of :^ị in /’ (^or 111D B ) is cliau^íHÌ

h\' (ỉììe.

If and ^J[/1Ị = ‘i/ ilu'ii ('Very ov c n n v n c c OỈ ijj ill r (or ill D B ) is chauj.Ợ'(l

1) V Í) ,

20 Bill Thi T h u y Hieii, N g u y e n C a t H o

If f \ [Ạ] — iuiil Ỉ < j IIhmi v v o i v uccul I ('net' of dị ill r (or in Ỉ ) B ) is changed by Ồ,.

I f / i [ - 4 ] = /:/, , ^ 2 [ * 4 ] = iii a i n l / < J t h r u ( ' v n v o c c ' i i i r c i u ' ( ' OỈ .i J i l l r ( o i i n D Ị Ì ) IS

c hanged 1)V i,.

prf'sriitiii^ an algoiithin for rPinoviiio a soft violation th a t a p p ra rs in a

relation or in a DB l(‘t us iiist show an algoiitliiii wliicii cliang<\^ ọxviy ocv un vu vv of a

mill value ill a li'latioii (or in a DB) to a ck*tinit(' valiu' t)i to a luorc inioMuation null valiu'

T h e algoiiihiu C H A N G E ill Algorithm 2 changes all the value HÍ th(‘ colum n of attrihiiK'

A ill M { M — I' Ơ1 M = D B ) to valiu' Ị.Ị.

A l g o r i t h m 2 c H A N G E ( r , A, M y)

I n p u t ; r 6 R c ỉ ị ( R X M is a cont ext Iiuli databasí' or M is a rel ation r , A is an

attril)iito (’ohiiiin undi'r consideration T and y ai(‘ two \'ahu\s (may b(' null)

a t tIk' a t n i h u t o c ol uiii n A

O u t p u t : Chaiij>(' vvvvv IM'CUM(‘lire of \-alu(' /■ at a tĩriìm li' coliiiiiii 4 ill M to vahu* ij

Begin

For each relation r in M do

For each tuple f in Ì' do

if ^[.4] = = :r then f[A] //;

End,

T h e algoiithin for a soft violation in a If'lafion oi ill a DB is presenti'd as Algorithm, 3

A l g o r i t h m 3 R E M U V E V I O ( M J X A)

I n p u t ; r G R e l \ { R ) , M is a (‘ontt'xt null claíabas(' or M is a relatio n f \ X ^ is a

c i i F D over R and, /i and t -2 ai(' tuples vvliicii causf' a soft violation of X ^ A

O u t p u t : Reniov'i' tlio soft violation caus('(l by and Ỉ 2 -

Begin

begin /■ f 2 [ A ] ; C H A N G E ( r , A , M j ị [ A ] ) , end;

if t ‘ 2 [A]\ and ^i[.4] = = iink then

begin r :='ti [ A ] \ C H A N G E ( i \ 4, M , end;

if ^i[-4]! and /2(^4] oj)e7i then

begin X := t 2 [ A \ ' , C H A N G E { x , A , M M [ A ] ) \ end;

if /j[^4]! and / ị [ ^ ] ‘= — open then

begin :r : = t i [ A ] : C H A N G E{: ĩ \ A M J - M ] ) ’ end:

if ^i[^] “ = due and /2[^4] —= opev then

begin X t 2 [ A ] ; C H A N G E { : r , A, M , d n c ) ; end,

if /ịỊ/1] = — u v k and t 2 [Ả] = — opcm then

begin :r t 2 [A];y := C H A N G E { x , A, M^y): end;

if ^2[>l] (Ỉ 1 ĨC and tị[A] = = open then

begin T t i [ A ] ; C H A N G E { T , A M^dn e); end,

if /2(^1 = — and f\{A] —= open then

begin X \= t\[A]; y := f -2 A]: C H A N G E { x A A/, y)\ end;

if (^i[-4] = — and ^2[.4Ị —= u.ilk) or (^[[/l] = = opt'ii and (fpcif) then

begin

if tn(lc:r{.r) > ịn.(ỉ(í.r(y) then C Ỉ I A N G E { r , A, M , Ịj)

else C H A N G E { i j A / M , r )

end;

End

L e m m a 1 Let r he in ĩỉ('/J (R), F i>e a set UÍ ciiFDs over R I f two tiiples f] and Ỷ 2 o f Ì' Cr ỉ i / s e a s o f t vìoÌHtioii o f F Ỉ^ÌKÌ r ' is t h e r c h t i o i i ( ỉ c d ì ì c c d f io iii r h y m o v i n g t h e s o f t VÌOÌHÌÌOÌI CHìiseil hy fị riiid tj then r' > r ộ

Proof Bv Definition 5 if a soft violation is removed then:

(i) Tlio definite values in /■ an' nuchanged

(ii) Each null value in r is either unchanged or clianged to a definite value or d ian g ed

to a more iiifoiIllation null value

Conibiniug (i) a n d (ii) we have r' > T.Ộ

T h e o r e m 2 Let D D i>e H context liỉỉìì (ÌHtHÌmse, r he ri relỉĩtỉon ill D B a/id F ỉ)e a set

o f ciiFDs ovei R I f the toilowiug conditỉoỉis hold:

1 In D B , f/jei'c is ÌÌOĨ a n y ÌIÌÌKỈ v i ohi tion o f F.

2 AU tiic soft vioiritioiis o f F t h a t r ippear s in D B CHii he veiiioved Siich t ỉ ìãt ill D B

t he r e is n ot ỉiỉiy hĩirt \'io/atiơJi o f F.

rhcji Dĩỉoìti c DBncu ■

Proof D i K ' d l v (1(h1uc('c1 iruui Linnnia 1.Ộ

L e n i i i i a 2 L a r i)€ ]ii R r l ị ( R ) F he ỈÌ set o f ciiFDs o w i n j J ) e a tìipìe over R I f the

following coiKÌiiious hold:

- hctwceii f a n d r t l ic r c is not ỈÌÌIV ỉ ì ã i d vioỈHtion o f F\

' / > e t \ v e c i i t ỉiỉKỈ r t ìì Cĩ v is a s o f t vioÌHtỉoìi o f F ,

ỉ ì í t cỉ i v i i i u v i n g t i i i s s o f t Vì oì i ì t ì on. VV'C ijHve:

(i) ill /■ tììCỉv is n o t HỉiỴ ìiHid vioìrìtioii o f F,

(ii) b e t w e e n f a n d r t h e r e is n o t a n y h a r d v i oi a t i o n o f F.

KMiioviug a soft violation i)Otwf‘cn f arid 1' , th ere aro two the following possibilities:

C a s e Ỉ: T h e t u p l e t IS to be c h a n g e d a t t h e v a l u e t A] a n d t h e t u p l e f ' IS k e p t unchanged:

In this case, t h e rol ati on 7' is not chaug(H] B y t he first con di ti on o f tho a s s u m p t i o n

wo have (i)

To prove (ii), we suppose the coiitiaiv, t h a t there is a tuple f\ of 7' such th a t f and

tị cause a hard violation (HV) of F Since t is only be changed at value t\A] and in the

initial relation r th ere is not any HV, so if f and f\ cause a HV of F then such HV must

bo HV of cnF D X -+ A Since after lonioving th e soft violation (SV) between t and f' wo

F u n c ti o n a l d e p e n d e n c i e s w i th c o n t e x t d e p e n d e n t n u ll v a lu es 21

22 B u i Thi T h u y Hieriị N g u y e n C a t H o

also must caus(‘ a HV of c iiH ) .Y ^ A This conitaclicts (i).

Ca.se 2: The tuple f' is changed at value f' A a n d the faj)la t I,.s kapi Uiichaiigcd:

(i): On th(' cout iHiw s u p p o s e th(‘ ass('itioii (i) d o e s not hold.

By thí' first comlitioii of the assuni])tioii, tlu'ie is not anv HV in r tliei(*foi(\ if a

HV that appt'ais ill V ai'ti'i loinoviiig the s v between f and f th en such IIV m ust bí' ĩli('

HV aiul a tuplí' of Sinct* is only c hanged at valu(‘ /^[^4], tho HV betw(‘('ii

and t\ must ho of ciiFD A' —^ A Siiici‘ after liunoviu^ rli(‘ s v h(‘t\v('(‘u f and t' \vv hav(‘

/[ 4] /^[^4] a n d ^[A'] —— if a n d f ị caus( ' a H V o f c i i F D A' 4 t Hiul /|

CMUSÍ' also a HV o f c i i F D X — 4 Cli'ailv, is \ hv t uple o f / Ix'lou' iciii oviu^ tln' s \ ’

'.)x t\xvvu t aiul f ' This fon tim licts assmiipriun th a t f and V do not cause any l i v of F.

(ii): Siucí' tlio tuple i is kept micliaiigíHl so for any f\ G r \ and if Ỷ ^ t] do

MUÍ t aius<* anv HV of F It remains to rlHH'k th a t whoth(‘i f and V cause a HV of F or not'.''

Suppose / Hiul caus(‘ a HV of ciiFD Y B Because aftci reuioviiig ĩlu' s v ỈK‘t\v(*(*ii

also a HV of c i i F D Y ^ B T h is co i it ia di c ts the s(H-oncl f o u d i t i o n o f t he a s s n i n p t i o i i ộ

T h e o r e m 3 Let r />e j]j / ? c / | ( / ? ) , F he tlic set o t c u F D s o v e r R , t b e ft t u p l e o v e r s c h e m e

R I f the following coiiditioiis m e satisfied:

- in V theve is n o t riiiv soft o r ÌÌỈÌHÌ violati on o f F.

- /jcfu'cej] f riiid r t he r e is n ot Hỉi\' lìỉtid \'iuIrìtioỉj oí F tlìciì iiftvi ivỉii uviti g rill tliv

soft vioỈHtioiìs ỉìppcíìi hewecn i i-ìiicì r, \\v liỉìvc:

(i) In r t l i c i v is ììot rỉ/u' ìiHrd vioìiìtioỉi oi F.

(iì) Between t aiỉd r tììeiv is ỉiut Hiiy lìnid viohìtiun ot F,

(Ui) ỉn r there is not i^ìiv soft violiìtioỉi o f F.

Y j \ L i U11U<V 2 t i i t i i u m o v i n g : , V j \ \ a Uv i i i i / , \ i n \ / ' , Vv'<’ \ i a v c

- in /■ íliOK' is not aii\' IIV.

- Ix’tWiH'ii r aiul / ílicK' is not any 11V.

Applying*, Li'iniiui 2 to the tuplí' / and thí' ri'lation r until all tli(‘ SVs hotwoon t

r arc Kuuoved, we ob tain (i) and (ii)

(iii): Assume the contrary, th a t after removing all the SVs between t and 7’, thviv

m v a c i i F D A' A in F aiitl two t u p k ’s i' and ^1 of r such that Hiid f\ faus(' a s\^ OÍ

X 4 TIhmi, = = ^ 1 ~ / = (U- Siiicr is not any s v 111 tli('

initial K'latioii /', t h e app(‘HiaiiC(‘ of tlu' s v l)et\veeu ì' and s h o w s tiiat at such a tiiiií'

eitluT ị' and i or and t tlo caiusi* a s v of ciiFD X —+ A Iud(H*(l suppose* the co n tia rv

th a t, b oth f' and t\ and t (Iocs not cause any s v of ciiFD X A u shows t h a t thv

values /[.4] aiul f\[A] art' not cliaii^otl Since tholo is not any s v in tlio initial relation r, if

/ '[ x j Hieu ^'[.4] — /i[*4j which contradicts (1) Therefoie at such a tiluo (Mtlii'r

all tlii^ SVs l)(»tw(‘(ni f and r liavo Kunovetl an d A[yY] we ịyot /[,4] = = f' [A and f A] = = tị A ỉt follows th a t f \ A ] = — a contradiction to ( l ) ộ

Bv tho TheoiiMii 2 and th(‘ Thi'OK'Mi 3, in Older to m a in t ai n tlu' <oii.sist(Mu V of a

c o n t e x t null da t ab as o, it is nocessaiV to introcluce t he followiijg t wo rules foi u p d a t i n g and inserting (lata:

R u l e 1 (For updaUnq data)

Lr*t D B 1k' a (*ont(‘xt luill (JHtal)as(' and r !)(‘ a lolafion in DB; F b(‘ a set of cnFDs

ov(n R , [j'i f b(* a t uplo of V that ne(*(ls to u p d a t e d to b e c o m e /j:

(i) If th eiv is a tiipl(‘ f' in r \ {^} th a t ti an d cause a h ard violation of F the

system will not iinpleuHMit th(' u p d a te procedure for the tuple f.

C o n ve i s (‘ly, if (i) is not satisfied:

(ii) For each X A in F tlio system will iiuplement checking:

For (*ach tuplp f' in r \ {^} if /] and caiiso a soft violation of F then the svsĩern

will renioví' that soft viol ation.

When all soft violations between r \ {/} and the tuple /i have been removed, the

system will im plem ent u p d atin g the tuple t to become the tuple ti.

T h e a im of R ul e 1 is t o m a in t ai n the d a t a b a s e under cons idera ti on to b e consist ent

with a givcni set F of fmictioiial dep(‘ndeiicies, i.e., Rule 1 ensures th a t u p d a tin g procedure

d o e s not caus e aiiv v i o l a t io n o f F Hence, tho R ul e 1 is said to be coiTPCt if it realizes this

aim.

P r o p o s i t i o n 2 TììC R u l e Ỉ is coiiect.

Proof The proof of (i) is straightforwarcl.

For (ii): L('t / — r \ {f} Rule* 1 shows th at:

- TI k ' h ' is not a n y HV and a n y sv in

- / and r d o v s not caus e any H V of F,

13y a pp licati on OÍ Tlii'oiein 3 to t h e rolatÌDii r and thí' tỉiph' all t h e S V s hetwpeu

r and t can \)V all r em o v e d so that:

( i ) ' ĩ l i o r o n o t ‘A n y H V o f F i l l r '

(ii) Thero is not any HV of F between r and f.

(iii) T hcro is not a n y s v of F in r '

Rule 1 is c'oiK'ct, by HuKHoin 3.Ộ

Algoritlim 4 below uses Rule 1 to u p d a te a tuple t of relation r to become a tuple

A l g o r i t h m 4 Ư P D A T E { ĩ \ F j j i )

I n p u t : r G / ĩ c / |( / ? ) , a livA of caP'Ds F ovf'i /?, a tuple tị over /Ỉ, a tuple t of r that

need to b e u p d a t e d to b e c o m e t h e tupl(' ^1

O u t p u t : U p d ate th e tuple f to become fị if there is not any HV of F t h a t appears

between r \ and ^1, do not u p d a te if otheiwise

Begin

r '

For each A” 4 in F do

For each tuple f in r' do

then

F u n c ti o n a l d e p e n d e n c i e s wi th c o n t e x t d e p e n d e n t null va lu es . ‘23