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THE DESIGN OF RELATIONAL INFORMATION SYSTEM
ACCORDING TO DIFFERENT KINDS OF DEPENDENCIES
C. DELOBEL (1) & E. PICHAT (2)
ABSTRACT
The purpose of th is paper is to present
(a) a survey of d i f f e ren t approaches proposed for the conceptual design of
log ica l schemas for re la t iona l data base systems,
and
(b) a new approach based upon the study of the decomposition structure of
a re la t ion which ensures the complete j o i n a b i l i t y of data.
(1) - Laboratoire I.M.A.G., Universit~ Scientifique et M~dicale de Grenoble,
B.P. 53, 38041 GRENOBLE Cedex (France).
(2) - Institut drInformatique d'Entreprise, C.N.A.M., 292 Rue Saint Martin,
75141 PARIS Cedex 03 (France).
267
1. INTRODUCTION
The purpose of th is paper is to present
(a) a survey of d i f f e ren t approaches proposed for the conceptual design of log i -
cal schemas for re la t iona l data base systems, and
(b) a new approach based upon the study of the decomposition structure of a
re la t ion which ensures the complete j o i n a b i l i t y of data.
In the design of data base schema there are bas ica l ly two approaches : the decomposi-
t ion and the synthet ic approach.
The decomposition approach has been proposed by Codd [COD 70] [COD 71] in the presen-
ta t ion of the re la t iona l data model. In th is model, the concept of funct ional depen-
dency is an important element when one is considering how to group a t t r ibu tes to
form re la t ions. The properties of funct ional dependencies, studied in [BOI 69],
[COD 71], [DEL 71], [DC 73], [ARM 74], are the basic elements used to define the three
CODD's normal forms and the normalization process. This normalization process can be
viewed as the decomposition of a re la t ion in to two sub-relat ions (or more), so that
the or ig ina l one can be regenerated by the composition operation of the two sub-
re la t ions.
The synthet ic approach attacked the same problem. The object ive is that a designer
should synthet ize the data base schema a lgor i thmica l l y from a given set of semantic
propert ies of data.
The f i r s t attempts based only on funct ional dependencies [DC 73], [WW 75], [SS 75]
led to some imperfections and do not guarantee that the re lat ions are in CODD's
th i rd normal form. The work of Bernstein [BER 75], [BER 77] develops an e f fec t ive
proceaure for synthesizing re la t ions sa t i s fy ing CODD's th i rd normal form. A s im i la r
approach is proposed in [LR 76], [LRP 77] wi th e f f i c i e n t algorithms to derive both
the closure and a minimum covering set of funct ional dependencies by using the
equivalence between the operations on funct ional re lat ionships and the operations in boolean algebra.
I f the funct ional re la t ionsh ip concept plays probably ~he most important role in the
process of def in ing re la t iona l schema, i t is not the only one. From an idea contained
in [BOI 69] we have defined [DL 74] another concept : the f i r s t order h ierarchica l
decomposition which allows to decompose re lat ions independently of funct ional re la-
t ionships. More recent ly , FAGIN [FAG 76] and ZANIOLO [ZAN 76], [ZAM 77] have i n t ro -
duced the notion of mult ivalued dependency which includes, as a special case, the
funct ional re la t ionsh ip .
The fo l lowing sect ion, section 2 of the paper, reviews the de f i n i t i on of a re la t ion
and the condit ion for a re la t ion to be decomposable.
Section 3 presents the d i f f e ren t types of constraints describing a re la t ion :
268
funct ional dependency, mult ivalued dependency, f i r s t order h ierarchical decomposition.
The properties of th is type of constraints are reviewed and the i r role in the process
design is i l l u s t r a t ed through examples.
In section 4, the basic properties of the decompositions of a re la t ion are studied
and i t is shown how i t is possible, from these propert ies, to develop a new approach
for the design of logical schemas. In th is approach, the concept of maximal decompo-
s i t ions for a family of decompositions is an important element of the process.
269
2. THE RELATIONAL MODEL
The i n ten t of t h i s sect ion is to review some d e f i n i t i o n s . F a m i l i a r i t y wi th the fun-
damental concepts of the r e l a t i o n a l model as presented in [COD 70] and [BER 77] ,
and espec ia l l y wi th the concept of func t iona l dependency in [ARM 74] is assumed. We
use throughout the sect ion the terminology found in [BFH 77].
2.1. Relat ions and const ra in ts
In the database terminology, the word r e l a t i o n is sometimes confusing because i t i s
used to denote a set of tuples and a s t ruc tu ra l descr ip t ion of sets of tup les. I t is
important to make the d i s t i n c t i o n c lear : when we ta l k about set o f tuples we shal l
use the word r e l a t i o n and in the other case we sha l l use the word r e l a t i o n a l schema.
With th i s approach a r e l a t i o n is an instance of a r e l a t i o n a l schema.
Attributes are i d e n t i f i e r s taken from a f i n i t e set A1,A 2 . . . . ,A n . Each a t t r i b u t e
A i has associated wi th i t a domain, denoted by DOM(Ai), which is the set of possible
values fo r tha t a t t r i b u t e . We shal l use the l e t t e r s A, B, . . . . f o r s ing le a t t r i bu tes
and the l e t t e r s X, Y . . . . . fo r sets of a t t r i bu tes .
An X-value is an assignment o f values to the a t t r i bu tes o f X from t h e i r domains.
Also, i f X and Y are sets o f a t t r i bu tes (not necessar i ly d i s j o i n t ) , then we wr i te
X u Y, X n Y fo r the union and the i n te rsec t i on o f X and Y, but i f X and Y are
d i s j o i n t , we shal l use (X,Y) fo r the union.
A re la t ion on the set o f a t t r i bu tes {A1,A 2 . . . . . A n } is a subset of the car tes ian
product DOM(A1) × DOM(A2) × . . . x DOM(An). The elements of the r e l a t i o n are ca l led
tuples. A r e l a t i o n R on { A I , A 2 , . . . , A n} w i l l be denoted R(A I ,A2 , . . . ,An) . I f one
wants to d i s t i ngu ish among the a t t r i bu tes the sets of d i s j o i n t a t t r ibu tes . X and Y,
we shal l use the nota t ion R(X,Y . . . . ). In th is case i f X = {AI,A 2} and Y = {A3,A4,A 5}
the tup le (x ,y . . . . ) stands f o r (a l , a2 ,a3 ,a4 ,a 5 . . . . ).
A constraint i nvo lv ing the set of a t t r i bu tes AI,A 2 . . . . ,A n is a predicate on the
c o l l e c t i o n o f a l l r e l a t i ons on th is set. A r e l a t i o n R(A1,A 2 . . . . ,An) obeys the
cons t ra in t i f the value of the predicate fo r R is " t r ue " . A cons t ra in t is def ined
by g iv ing a nota t ion or a language fo r expressing i t and the condi t ion under which
a r e l a t i o n obeys i t .
A cons t ra in t can be seen as an i n t r i n s i c property of the data~ f o r example, suppose
tha t parts in an inventory are described by a r e l a t i o n R(PART-NUMBER, COLOR, PRICE,
.~ . ) . A p r i o r i , any r e l a t i o n o f th i s form can e x i s t in the database. However i f one
spec i f ies a cons t ra in t : the PRICE must range between $0 and $100 a piece, then only
re l a t i ons in which th is cons t ra in t is va l i d can ex i s t in the database. S im i l a r l y
the s p e c i f i c a t i o n that the knowledge of the PART-NUMBER-value impl ies the COLOR-value
is a lso another type of cons t ra in t . In th is case, th i s type of cons t ra in t is ca l led
270
a func t iona l dependency and denoted PART-NUMBER ÷ COLOR.
In t h i s paper, we are i n v e s t i g a t i n g only some cons t ra in t s : f unc t iona l dependency,
mu l t i va lued dependency, f i r s t order h i e ra r ch i ca l decomposit ion which are presented
in sec t ion 3.
2.2. Operat ions on r e l a t i o n s
In his o r i g i n a l p resenta t ion of the r e l a t i o n model [COD 70] , Codd in t roduced the
r e l a t i o n a l a lgebra as a data manipu la t ion language. There are two basic operat ions
t ha t w i l l be of some i n t e r e s t of us : p ro j ec t i on and natura l j o i n .
The projection of a r e l a t i o n R(X,Y,Z) over a set of a t t r i b u t e s X is the r e s t r i c t i o n
of R over the a t t r i b u t e s in X; t h i s opera t ion w i l l be denoted R[X], and def ined by
R[X] = {x l~y ~z : ( x , y , z ) ~ R}.
As a spec ia l case of p r o j e c t i o n we denote by R[x,Y] the p ro j ec t i on of R over Y from
an X-value x, R[x,Y] = {y I~z : ( x , y , z ) ~ R}.
The natural join opera t ion is used to make a connection between a t t r i b u t e s tha t
appear in d i f f e r e n t r e l a t i o n s . Let R(X,Y) and S(X,Z) be two r e l a t i o n s : then the
natura l j o i n R*S is the set of { ( x , y , z ) l ( x , y ) ~ R and ( x , z ) ~ S}; R*S is a r e l a t i o n
def ined over the a t t r i b u t e s {X,Y,Z} .
2.3. Decomposit ion o f a r e l a t i o n
Let R(X,Y,Z) be a r e l a t i o n ; we sha l l say tha t R is decomposable i f there ex i s t s
two r e l a t i o n s S and T such tha t :
(a) S and T are p ro j ec t i ons of R : S = R[X,Y] , T = R[X,Z]
(b) the natura l j o i n of S and T is R : R = S*T.
In o ther way, we sha l l say tha t the p a i r ( ( X , ¥ ) , (X,Z)) cons t i t u t es a decomposition
of R.
This concept o f decomposit ion is very impor tan t in the process f o r des igning a
database schema, because in place of s to r i ng the r e l a t i o n R in the database, we can
s tore only the p ro j ec t i ons of R. The purpose o f t h i s paper is to study under what
cond i t ions we can decompose a r e l a t i o n .
F i r s t , we sha l l r e c a l l bas ic cha rac te r i za t i ons f o r a r e l a t i o n to be decomposable ;
the proofs have been given elsewhere.
~ [~£§~ !2Q_~ : R(X,Y,Z) is decomposable i f f f o r a l l X-values x which are in R
R[x,Y,Z] = R[x,Y] x R[x,Z]
where the opera t ion × denotes the car tes ian product .
271
~ [gPQ~!~_~ : R(X,Y,Z) is decomposable i f f for a l l X-value and Y-value which are
in R :
R[x,y,Z] = R[x,Z].
~ _ ! : Let R(BOOK,CLASS,STUDENT,PROFESSOR) be a re la t ion where the in terpreta-
t ion can be given i n t u i t i v e l y as fol lows. Each CLASS has various STUDENTs, but a
STUDENT is in one CLASS, a CLASS has one PROFESSOR. Each CLASS has a given set of
BOOKS as reference's books. A sample of the re la t ion is given by the table below.
R : BOOK CLASS STUDENT PROFESSOR
b I math peter mike
b I math john mike
b 3 programming jane mike
b 2 math peter mike
b 3 programming, james mike
b 2 math john mike
One can notice that by appl icat ion of proposit ion 1, the re la t ion R is decomposable
into two re lat ions S = R[BOOK,CLASS] and T = R[CLASS,STUDENT,PROFESSOR].
S : BOOK CLASS T : CLASS STUDENT PROFESSOR
b I math math peter mike
b 2 math math john mike
b 3 programming programming jane mike
programming james mike
At the present time, we only make a constatation that re la t ion R is decomposable ;
we shal l see in the next chapter that the decomposition property is related to the
type of constraints.
272
3. THE TYPE OF CONSTRAINTS
3.1. Funct iona l dependencies
3 .1 .1 . D e f i n i t i o n
Func t iona l dependencies form a f a m i l y o f c o n s t r a i n t s . The p r o p e r t i e s o f f u n c t i o n a l
dependencies have been s tud ied e x t e n s i v e l y in [DC 73 ] , [ARM 74 ] , [BER 77] ; t he re -
fo re we r e c a l l here on ly the d e f i n i t i o n .
A functional dependency (abbr. FD) i s a sentence denoted f : X ÷ Y where f i s the
name o f the f u n c t i o n a l dependency and X and Y are sets o f a t t r i b u t e s . A f u n c t i o n a l
dependency f : X ÷ Y holds in R(U) where X and Y are subsets o f U, i f f o r every
tup le u and v o f R u[X] = v [ X ] i m p l i e s u[Y] = v [Y] (u [X] denotes the p r o j e c t i o n o f
the t up le u on X). According to the d e f i n i t i o n f can be seen as the unique app l i ca -
t i o n from R[X] to R[Y] , t h e r e f o r e we can omi t the name of the FD and w r i t e on ly
× ÷ Y .
A fu l l functional dependency X ÷ Y is an FD such t h a t the re e x i s t s no proper subset
X' a X w i th X' ~ Y.
We can r e l a t e the concept o f f u n c t i o n a l dependency to the concept o f decomposi t ion
accord ing to p r o p o s i t i o n 3 which is easy to prove.
~ r 2 ~ ! g ~ . ~ : I f R(X,Y,Z) is a r e l a t i o n such t h a t f u n c t i o n a l dependency X ÷ Y
holds then R is decomposable and we have R = R[X,Y] , R [X,Z ] .
The p r o p o s i t i o n 3 is the f i r s t p r o p o s i t i o n which es tab l i shes an assoc i a t i on between
a c o n s t r a i n t and the decomposi t ion p rope r t y .
3 .1 .2 . P rope r t i es of f u n c t i o n a l dependencies
The p r o p e r t i e s of FD's are impo r tan t in the design o f r e l a t i o n a l schemas and we s h a l
use them l a t e r . These p r o p e r t i e s can be seen as i n fe rence ru les f o r FDs, t h a t is
ru les t h a t deal w i th the i m p l i c a t i o n of new FDs from a g iven se t o f FDs.
Let R(U) be a r e l a t i o n de f ined over a set U of a t t r i b u t e s , and X,Y,Z,W be subsets
o f U.
F1 - R e f l e x i v i t y : i f X ~ Y then X ÷ Y
F2 - Augmentat ion : i f X ÷ Y, then f o r a l l Z X u Z ÷ Y u Z
F3 - T r a n s i t i v i t y : i f X ÷ Y and Y ÷ Z then X ÷ Z
F4 - P s e u d o - t r a n s i t i v i t y : i f X ÷ Y and Y u Z ÷ W then X u Z ÷ W
F5 - A d d i t i v i t y : i f X ÷ Y and X ÷ Z then X + Y u Z.
273
~ ! ~ _ ~ : We can define for the re la t ion R given in example 1, the FDs according
to the re lat ionships propert ies between a t t r ibu tes
STUDENT ÷ CLASS
CLASS ÷ PROFESSOR.
According to F3 we can derive STUDENT + PROFESSOR by t r a n s i t i v i t y .
3.1.3. Decomposition based upon funct ional dependencies
I t is well known that i f someone adopts for the re la t iona l schema the re la t ion R only,
th is re la t ion suf fers from a l o t of anomalies. These anomalies occur when one wants
make an update, inser t ion or delete operation. The problem of designing re la t iona l
schemas is equivalent to replace the re la t ion R by a set of re la t ions which give the
same information wi thout destroying the re lat ionships between a t t r ibu tes . This
problem is equivalent to f ind for the re la t ion R a decomposition such each re la t ion
in the decomposition contains less anomalies than the or ig ina l one. This problem
has been, f i r s t , i den t i f i ed by CODD and leads him to the de f i n i t i ons of normal's
form of a re la t ion . Two main approaches have been recognized as valuable to produce
su i tab le normal forms : the decomposition approach and the synthet ic approach re la-
ted with the irredundant covering technique.
The decomposition approach consists to apply successively the proposit ion 3 to d i f f e -
rent funct ional dependencies. For example, the appl icat ion of STUDENT + CLASS in
example i gives the decomposition of the re la t ion R defined over the a t t r ibutes
STUDENT, CLASS, PROFESSOR, BOOK in to two re la t ions RI defined over the a t t r ibu tes
STUDENT, CLASS and R2 defined over STUDENT, PROFESSOR, BOOK. I t is s t i l l again pos-
s ib le to decompose R2, because the FD STUDENT--~ PROFESSOR holds also in R2, in to
two re la t ions R21 defined over STUDENT, PROFESSOR and R22 defined over STUDENT,
BOOK.
The overal l decomposition process can be represented by a tree as the one shown in
f igure 1. The terminal nodes describe the a t t r ibu te sets of each re la t ion obtained
at the end of the decomposition process. To each non-terminal node we can associate
an a t t r i bu te set equal to the union of the a t t r i bu te sets of t he i r successors. Each
non-terminal node corresponds a decomposition step which is label led according to
a FD.
STUDENT ÷ CLASS
STUDENT,CLASS STUDENT ÷ PROFESSOR
STUDENT,PROFESSOR STUDENT,BOOK
Fig. _I_I- A decomposition of a re la t ion
274
The d i f f i c u l t y of t h i s approach is the p o s s i b i l i t y of m u l t i p l e decomposi t ions. I t
may be seen tha t the r e l a t i o n R has in add i t i on the f o l l o w i n g decomposi t ion as
shown by f i gu re 2.
CLASS ÷ PROFESSOR
CLASS,PROFESSOR STUDENT ÷ CLASS
STUDENT,CLASS STUDENT,BOOK
Fig. 2 - A decomposit ion of a r e l a t i o n
These two decomposit ions possess an anomaly because both they generate the r e l a -
t i on R[STUDENT,BOOK]. The meaning of t h i s r e l a t i o n s h i p between the a t t r i b u t e s is :
s c STUDENT and b c BOOK are re l a ted i f there ex i s t s a CLASS c such the STUDENT
s is in the CLASS c and the BOOK b is used by the CLASS c. In o ther way one can
say t h a t STUDENT and BOOK are not d i r e c t l y r e l a t e d .
The i r redundant cover technique is more systemat ic . I t proceeds in d i f f e r e n t steps.
The f i r s t step is to const ruc t the c losure and an i r redundant cover.
Let F be a given set of FDs, the c l o s u r e of F denoted F + is the set of FDs which
can be der ived from F through the in ference ru les of FDs.
A cover C is a set of FDs from which a l l others can be de r i ved , t ha t means tha t :
C + = F +. An i r r e d u m d a n t cove r F m is a set of FD s such the c losure of F m is equal to
c losure of F, i . e (Fm)+ = F +, and there ex i s t s no proper subset F ~ of F such
(F') + = F +.
The problem of obtaining an efficient irredundant cover algorithm has received wide
attention [De 73], [LR 76], [LRP 77].
As we can see from our previous example 2 , the i r redundant cover of F = {STUOENT+CLASS,
CLASS ÷ PROFESSOR} is equal to F. So, the two r e l a t i o n s b u i l t from the two FDs in
the i r redundant cover , say R1 def ined over STUDENT, CLASS and R2 def ined over CLASS,
PROFESSOR~are not enough to regenerate R.
The second step is to syn the t i ze valuable r e l a t i o n a l schemas from an i r redundant cover
set o f FDs regarded as semantic p r i m i t i v e s . D i f f e r e n t a lgor i thms have been propo-
sed to ob ta in opt imal set of r e l a t i o n s in t h i r d normal form from an i r redundant cover
[BER 75] , [BER 77] , [FLO 77] , [OSB 77] .
3.2. Mu l t i va lued dependencies
3 .2 .1 . D e f i n i t i o n
In a func t iona l dependency X ÷ Y, the knowledge of the X-value determines a unique
275
Y-value. In a mult ivalued dependency the X-value determines a set of Y-values. Ac-
cording to the de f i n i t i on given by Fagin [FAG 76] and Zaniolo [ZAN 76], a~I t iva lued
dependency (abbr. MVD) is a sentence denoted g : X ++ Y where g is the name of the
mult ivalued dependency and X and Y are sets of a t t r ibu tes . A mult ivalued dependency
g : X ->+ Y holds in R(X,Y,Z), where X,Y,Z are d i s j o i n t sets of a t t r ibu tes , i f for
every X,Z-value, (x,z) that appears in R, we have
R[x,z,Y] = R[x,Y].
As we do for FDs, here also we usually omit the name g and wr i te only X ~-~ Y.
One can notice that the de f i n i t i on of an MVD uses the same condit ion as proposit ion
2.
Example_3 : For the re la t ion R defined in example 1, one can notice that the MVD
CLASS -~+ BOOK holds in R, because for every CLASS, STUDENT, PROFESSOR-value
(c, t , p) that appears in R we have :
REc,t,p,BOOK] = R[c,BOOK].
~ [ £ ~ 2 ~ Q _ ~ : The re la t ion R(X,Y,Z) obeys the MVD X ->~ Y i f f R is decomposable
into two parts R[X,Y] and R[X,Z].
This proposit ion is a d i rec t consequence of the de f i n i t i on and proposit ion 2.
D
The decomposition given in example i is based upon proposit ion 4app l ied to the
MVD CLASS ~ BOOK.
In the de f i n i t i on of MVD given here, as in Fagin's paper [FAG 76], we require the
l e f t and r igh t sides of an MVD be d i s jo in t . There are two reasons for that :
f i r s t , a t r a n s i t i v i t y property does not always hold i f the res t r i c t i on is l i f t e d ,
and second, in a l l pract ical s i tuat ions a database designer w i l l define this type
of constraint with d i s j o i n t sets of a t t r ibu tes . In [BFH 77] the reader can f ind a
complete study of MVD properties~ nevertheless we shall recal l the basic properties
without proof.
3.2.2. Properties of MVDs
The properties of MVDs are very s imi lar to FDs. In this subsection we discuss
inference rules for MVDs, that is rules that deal with the impl icat ion of new
MVDs from a given set of MVDs.
Let R(U) be a re la t ion defined over a set U of a t t r ibu tes , where X,Y,Z are d i s jo in t
sets of a t t r ibutes contained in U.
276
MO r e f l e x i v i t y : i f X ~ Y then X ÷~ Y
M1 complementation : i f X ~-~ Y then X -~ U-(X u Y)
We have to note tha t t h i s complementation ru le has no equ iva len t f o r the FDs.
M2 augmentation : i f X ++ Y then f o r a l l Z then X u Z ÷+ Y u Z
M3 t r a n s i t i v i t y : i f X ~-~ Y and Y -~ Z then X - ~ Z
M4 p s e u d o - t r a n s i t i v i t y : i f X -~ Y and Y,Z ~ W then X,Z -~> W
M5 a d d i t i v i t y : i f X -~ Y and X ~+ Z then X ÷~ Y u Z
M6 decomposit ion : i f X ÷+ Y and X ~-~ Z then X ~-~ Y n Z.
3 .2 .3 . Decomposition of r e l a t i o n w i th MVDs
We have seen in sec t ion 3 .1 .3 tha t the decomposit ion o f a r e l a t i o n R s t ruc tu red only
by FDs can lead to some anomalies. Now we can repeat the decomposit ion process of the
r e l a t i o n .
R(BOOK, STUDENT, CLASS, PROFESSOR) s t ruc tu red by the FDs
STUDENT ÷ CLASS, CLASS + PROFESSOR and the MVD CLASS ++ BOOK.
I t i s easy to obta in the decomposit ion i l l u s t r a t e d by f i gu re 3.
CLASS ~-~ BOOK
CLASS, BOOK CLASS ÷ PROFESSOR
CLASS ,PROFESSOR CLASS, STUDENT
Figure 3
We can see tha t in th i s decomposit ion ( f i g u r e 3 ) , we obta in a r e l a t i o n where the
a t t r i b u t e s CLASS and BOOK are d i r e c t l y r e l a t e d .
3 .2 .4 . MVDs def ined on a subset o f a t t r i b u t e s
Sometimes i t is poss ib le to i n t e r p r e t the concept o f MVDs as an extension of the
concept of FDs by saying tha t the MVD X ÷+ Y holds in R means tha t the X-value
determines a unique set of Y-value. This i n t e r p r e t a t i o n can be mis lead ing . This f ac t
is more eas i l y c l a r i f i e d by an example.
Exam~e_4 : Consider the f o l l o w i n g r e l a t i o n R over the a t t r i b u t e s BOOK, CLASS,
STUDENT, PROFESSOR w i th a d i f f e r e n t semantic from example I . Each CLASS has d i f f e r e n t
STUDENTs and PROFESSORs but a STUDENT is only in one CLASS. The BOOK is determined
by a PROFESSOR and a CLASS. Then we have the FDs : STUDENT ÷ CLASS and
CLASS, PROFESSOR ÷ BOOK.
An instance of the r e l a t i o n a l schema is given by the tab le below.
277
R : BOOK CLASS STUDENT PROFESSOR
b I math peter mike
b I math john mike
b 2 programming jane mike
b 2 math peter ronald
b 2 math john ronald
b 2 programming james mike
In the re la t ion R given in example 4, the knowledge of CLASS-value determines the
set of PROFESSOR-value, but is not true that CLASS -~ PROFESSOR because :
R[math, PROFESSOR] = {mike, ronald}
Rib1, math, peter, PROFESSOR] = {mike}.
This is due to the fact that the v a l i d i t y of the MVD, X ~ Y in R(X,Y,Z) depends not
only on the values of X and Y, but also on the values of Z. I t is possible that
X -~ Y is not va l id in R(X,Y,Z), but that X -~ Y is va l id in the project ion
R[X,Y,Z'] where Z ~ is a subset of Z.
For example, the constraint CLASS ++ PROFESSOR is only va l id on R[CLASS, STUDENT,
PROFESSOR]. Furthermore we can in te rpre t this constraint as "a l l the students
attending a class are taught by a l l inst ructors" .
One can check that the MVD CLASS -~+ STUDENT is va l id in R ; then the re la t ion R
is structured by the FDs : STUDENT ÷ CLASS, CLASS,PROFESSOR ~ BOOK and the MVDs
CLASS ÷~ STUDENT and CLASS-~ PROFESSOR which are only va l id on the set o f a t t r i -
butes CLASS, PROFESSOR, STUDENT.
To capture this new type of constraints, Fagin cal ls this case embedded multivalued
dependencies. To denote an embedded MVD we need to specify precisely the sets of at-
t r ibutes which occur.
This concept of embedded multivalued dependencies is s imi lar to the concept of f i r s t
order hierarchical decomposition which has been previously defined in [DL 74] and
which is more general than the concept of MVDs.
3.3. F i r s t order hierarchical decomposition
3.3.1. Def in i t ion
A first order hierarchical decomposition (abbr. FOHD) [DEL 77] is a sentence denoted
X : YIlY21... IYk where X,YI,Y 2 . . . . . Yk are d i s j o i n t sets of a t t r ibutes.
A re la t ion R(X,YI,Y 2 . . . . . Yk,W) where X,YI,Y 2 . . . . . Yk,W are d i s j o i n t sets of a t t r ibu -
tes, obeys the FOHD i f for every X-value we have
R[x,Y1,Y 2 . . . . . Yk ] = R[X,Yl] x R[x,Y2] x . . . x R[X,Yk].
278
According to p ropos i t ion 1, i t is easy to prove tha t th is cond i t ion is equ iva lent to
the condi t ion :
R[X,Y1,Y 2 . . . . . Yk ] = R[X,Yl l * R[X,Y 2] * . . . ~ R[X,Y k]
which expresses the decomposabi l i ty of the p ro jec t ion R over {X,Y 1 . . . . . Yk } i n to k
p ro jec t ions .
The d e f i n i t i o n of FOHD given here is more general than the d e f i n i t i o n of decomposition
tha t we have introduced in sect ion 2.3.
In the FOHD, X : Y I lY21 . . . IYk , X is ca l led the root segment, Y1,Y2 . . . . . Yk are ca l led
segments and the pa i r (X,Yi) is ca l led a branch. We shal l say that a FOHD is a f u l l
first order hierarchical decomposition or a generalized multivalued decomposition
(abbr. GMVD) i f W is the empty set. We shal l denote a GMVD as X -~ Y I IY21. . . IYk . In
the case where the root X is empty, th is means that the p ro jec t ion o f R over the set
o f a t t r i bu tes {YI,Y2 . . . . . Yk } is the car tes ian product o f the pro jec t ions o f R respec-
t i v e l y on YI,Y2 . . . . . and Yk"
For the r e l a t i o n R given in example 4, the FOHD
CLASS : STUDENTIPROFESSOR
holds. We have also seen that the MVD CLASS -~ STUDENT holds in R, thus propos i t ion
4 can be appl ied to give the FOHD :
CLASS : PROFESSOR,BOOKISTUDENT.
As one can see, the d e f i n i t i o n s o f FOHDs and MVDs are two facets of the same problem :
under what condi t ions is a r e l a t i o n decomposable ? E f f e c t i v e l y , each time that a
cons t ra in t on a r e l a t i o n can be paraphrazed by a sentence l i k e " a l l the students
at tending a seminar are taught by a l l i n s t r u c t o r s " , we have a cons t ra in t of the
fami ly of FOHDs, and th is sentence refers to the condi t ion of p ropos i t ion 1. On
the other hand, a sentence l i ke "the type of seminar determines uniquely the set of
books" can generate a cons t ra in t of the fami ly of MVDs, and th is sentence refers to
the cond i t ion o f p ropos i t ion 2.
From the d e f i n i t i o n o f FOHD and MVD, i t is c lear that we have the fo l l ow ing r e l a t i o n -
ships : a GMVD represents a set of MVDs, fo r example i f X -~+ YIZIU holds in
R(X,Y,Z,U) then we have X ~-~ Y, X ~-~ Z and X -~+ U and the converse. Nevertheless
i f X ÷÷ ¥ holds in R(X,Y,Z,U) we have X - ~ YIZ,U. And i f X : YIZ holds in R(X,Y,Z,U)
we cannot deduce any MVD.
3.3.2. Propert ies of FOHDs
In th is sec t ion , we discuss inference proper t ies fo r FOHDs from a given set o f FOHDs.
The proper t ies are only reviewed here since they have been presented in [DEL 77].
Throughout th is sect ion, we assume that the r e l a t i o n R is def ined over a given
d i s j o i n t set of a t t r i bu tes X,YI,Y 2 . . . . . Yk,W.
279
HO permutation : i f X : Y I lY21 , . . iYk holds in R and ~ a permutation of 1 , 2 , . . . , k ,
then X : Y ( I ) IY ( 2 ) I . . . I Y (k) holds in R.
HI add i t ion of branches : i f X : Y I lY21. . . IYk holds in R then X : Y1 u Y21.. . IYk
holds also in R.
H2 de le t i on of a branch : i f X : Y I IY21. . . IYk holds in R then X : Y21. . . IY k holds
in R.
H3 p ro jec t ion of a branch : i f X : Y I lY2 [ . . . IYk holds in R then X : Y~IY21.. . IY k holds
in R where Y~ is any subset o f YI '
H4 root augmentation : i f X : Y I I Y 2 i . . . I Y k holds in R then X u Y~ : Y1-Y~IY21...IYk
holds in R, where Y~ is any subset o f YI"
H5 p ro jec t ion of the roo t and decomposition of a branch :
i f X : YIiY2 u Y31.. . IYk and X u Y1 : Y2 IY3 holds in R then
X : YI lY21Y31.. . IY k holds in R.
H6 decomposition : i f X : Y I lY21 . . . IY k and X : Z I l Z 2 [ . . . I Z r hold in R wi th
YI n Z I # ~ and YI n Z 2 # ~ then X : Y1 n Z I I y I n Z21Y21.. . Iy k holds
in R.
3.4. Ax iomat iza t ion fo r FDs and MVDs
Armstrong [ARM 74] proved that the proper t ies o f r e f l e x i v i t y , augmentation and t ran-
s i t i v i t y cons t i t u te a complete set of in ference ru les f o r FDs. The completeness
theorem can be stated as fo l lows : f o r every set F of FDs there ex is ts a r e l a t i o n R
such that the FDs which are va l i d in R is exac t l y F +.
This property is only va l i d fo r the FDs ; we now turn our a t ten t i on to a more general
problem. What could be a set o f in ference ru les fo r a set F o f FDs and a set M of
MVDs ? In the previous sec t ion , we have seen proper t ies fo r MVDs but these proper t ies
are not enough to def ine a complete ax iomat iza t ion since cer ta in combinations of FDs
and MVDs imply add i t i ona l FDs that cannot be der ived by the use o f the above ru les.
We need to introduce three add i t i ona l proper t ies :
FM1 : i f X ÷ Y then X ~-~ Y
FM2 : i f X~+Y and W ÷ Z where Y ~ W with Y n Z = ~ then X ÷ Z.
Berr i and al [BFH 77] proved that the rules {F1,F2,F3,M1,M2,M3,FM1,FM2} are a
complete set o f in ference ru les fo r the fami ly of func t iona l and mul t iva lued depen-
dencies. The completeness theorem can be stated as fo l lows : f o r every set F of
FDs and M of MVDs there ex is ts a r e l a t i o n R such tha t the set of FDs and MVDs va l i d
in R is exac t l y (F u M) +.
The problem of determining a complete set of inference rules fo r FOHDs iS an open
research problem. I t has been solved only in the special case o f f u l l FOHD.
280
4. DECOMPOSITION PROCESS USING DECOMPOSITION STRUCTURE
~ . i . General analysis of the decomposition approach
The d i f f e ren t approaches for the design of logical schemas presented in section 3 are
based on the properties of FDs and MVDs. In a l l of these approaches, the process
starts with a l i s t of dependencies given by the designer. From a designer view point ,
i t is not necessary to consider a l l the given dependencies.An irredundant cover is ade-
quate for this purpose, since the remaining dependencies are derivable by the in fe-
rence rules. In the previous section (section 3.4), we have remembered that we have
to consider 8 basic inference rules.
Instead to look at the dependencies, there exists an a l ternat ive approach where the
family of decompositions of a re la t ion is d i rec t l y studied. This approach presents
the fo l lowing advantages :
- to study the decomposition properties of a re la t ion we need only 4 basic inference
rules
- the family of decompositions for the re la t ion R can be ordered by an order rela-
t ion in order to s impl i fy the designer's task for f inding a minimal cover. To
achieve this goal, f i r s t we shall define a *-operation that generates from two
decompositions a new one, and second to avoid the m u l t i p l i c i t y of decompositions
we introduce an order re la t ion denoted c and according to which the *-operation is
isotonous. This property is very important when one is looking for a minimal cover
- to consider only decompositions which ensures the complete j o i n a b i l i t y of data.
A s imi lar idea has been introduced by Rissanen [RIS 77] with the notion of inde-
pendent component and by Zaniolo and Melkanoff [ZAM 77] with the concept of
complete r e l a t a b i l i t y .
4.2. The decomposition of a re la t ion
Let R be a re la t ion over U. A decomposition of R is a pair (X,Y) of subsets of U with
X u Y = U such that
R = R[X] , R[Y]
A decomposition can be considered as a Gi4VD : × n Y -~ ×-YiY-X with two branches.
By s im i l a r i t i es the sets X-Y and Y-× w i l l be called the segments of the decomposition
and × n Y the root segment.
Using the de f i n i t i on , Armstrong and Delobel lAD 77] proved that the fo l lowing in fe-
rence rules (or axioms) for decompositions are complete.
DI empty decomposition : (~,U) holds in R
D2 symetry : i f (X,Y) holds in R then (¥,X)
D3 augmentation : i f (X,Y) holds in R and Z ~ U then (X u Z, Y u Z)
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04 i n te r sec t i on : i f (X,Y) and (Z,W) hold in R with Z n W = Y then (X n Z, W).
The completeness theorem can be stated as fo l lows : f o r every set D of decompositions
there ex is ts a r e l a t i o n R such tha t the decompositions which are va l i d in R are exact-
ly D +. The very important po in t wi th th is ax iomat iza t ion is tha t i t captures d i r e c t l y
the decomposition s t ructure of the r e l a t i o n R.
We introduced now some add i t i ona l ru les tha t we have found to be useful f o r the mani-
pu la t i on of decomposition :
D5 absorpt ion : i f X c U then (X,U) holds in R
D6 : i f (X,Y) and (Z,W) hold in R wi th Z n W ~ Y then
(X n (Y u Z), Y u W) holds also in R.
The proofs are given in [AD 77] and they are d i r e c t consequence of proper t ies D1-D4.
4.3. Analysis of an i n i t i a l set o f decompositions
To study the decomposition s t ruc ture of the r e l a t i o n R we need at the i n i t i a l stage
o f the process to have a set of decomposit ions. This can be done only by the designer
in i n t e r p r e t i n g , from his understanding o f the case at hand, the dependencies in
terms of decompositions. The fo l l ow ing example w i l l exp la in the i n i t i a l stage.
Example_5 : Let R be a r e l a t i o n over the a t t r i bu tes Student, Class, Professor, Book
and Hour. On an i n t u i t i v e background the r e l a t i o n R expresses a time tab le . A
student is only in one class. A class has d i f f e r e n t students and professors.A profes-
sor at one hour is teaching to only one class. A class uses d i f f e r e n t books. Then
we can consider tha t the r e l a t i o n R is s t ructured by the f o l l ow ing set F of depen-
dencies (FDs and MVDs), F = { f l , f 2 , f 3 , f 4 }
f l S ÷ C
f2 C ~ S
f3 C -~ B
f4 H,P ÷ C
By app l i ca t i on of p ropos i t ion 3 successively to f l - f 4 , we der ive the fo l l ow ing
decompositions d l -d4 :
d l S ~* CIB,P,H
d2 C ++ SIB,P,H
d3 C ~M, BIS,H, P
d4 H,P -~ CIS,B
I t is from d l , d2, d3 and d4 that the designer w i l l study the various decompositions
of the r e l a t i o n R.
282
We now present an e f f e c t i v e methodology fo r the design of r e l a t i o n a l schema using
the decomposition approach. The d i f f e r e n t stages of th is methodology are :
- s ta r t i ng from an i n i t i a l set o f decomposit ions, der ive a l l the possib le decomposi-
t ions
- construct a minimal cover fo r the i n i t i a l set o f decompositions
- se lec t from the minimal cover a decomposition that ensures complete j o i n a b i l i t y
of data.
4.4. An a]9or i thm fo r f i nd ing the maximal decompositions
4.4.1. The * -opera t ion
The reference ru le D4 fo r decomposition can be general ized i n to two d i rec t i ons by
de le t ing the condi t ion and by consider ing a decomposition in more than two branches.
This can be done by consider ing the propos i t ion 5 appl ied to GMVDs.
~ 9 ~ ! ~ Q ~ _ ~ : I f Y -~ B I lB21 . . . IB J and Z ~-~AI !A21. . . IA I hold in R then :
(1) Y u (B k n Z) ~÷ BI IB21. . . IBk_ I IB k n AI lB k n A21. . . IB k n A I I B k + I I . . . I B a,
wi th k ~ J, holds also in R.
Proof :
By se t t ing B k n Z = C, B k n A I = C 1 . . . . . B k n A I = C I then we have to prove that :
Y u C - ~ B1iB21. . . IBk_11C1IC21. . . iC I IBJ+11. . . IB J , , , , , , , , ~ , . (2)
Let
(3)
( 4 )
( y ,C ,b l ,b 2 . . . . . b k _ l , C l , C 2 , . . . , C l , b k + l , - - - , b J)
( y , c , b { , b ~ . . . . b ' c ' ' . . ' b ' . . ' ' k - l ' 1 ' c2 '" ' c I ' k+1 '" , b j )
be two tuples in to R.
Y - ~ B I IB21 . . . IB J impl ies Y -~* BklB I . . . . . BK_I,Bk+ I . . . . . Bj according to HI ; then we
can i n f e r the existence of the tuples :
' c ' ,b j ) (5) (y,c,b~,b~ . . . . . bk_ 1, 1,c2 . . . . . Cl,bk+ 1 . . . .
(6) ( y ,C ,b l ,b 2 . . . . b , c ' , c ' , . . . ' . . . • k-1 1 2 ,Cl,bk+ 1, ,bj)
Z ~-~AI IA21. . . IA I impl ies Z ÷+ Al IA 2 . . . . . A I according to H I ; then we can i n f e r from
(3) and (6) the tup le :
(7) ( y ,C ,b l ,b 2 . . . . . bk_l,C~,C 2 . . . . . Cl,bk+ I . . . . ~bj)
and from (4) and (5) the tup le :
' . . . b ' , c l , c ~ . . , c i , b ~ + I , b j ) " ( 8 ) ( y , c ,b { ,b 2, ' k-1 . . . . . . .
the tuples (3) , (4) , (7) and (8) imply the existence of the GMVD
283
(9) Y u C ÷~ CI lB I ,B 2 . . . . . Bk_I,C 2 . . . . , C I , B k + I , . . . , B J.
I t is easy to prove tha t we have also by rep lac ing successively C 1 by C 2, u n t i l C I
by C I :
Y u C ~ C21B1,B 2 . . . . . Bk_I,CI~C 3 . . . . . CI,Bk+ 1 . . . . . Bj (1o) :
Y u C -~+ CIIB1,B 2 . . . . . Bk_I ,C1,C2, . . . ,C I_ I ,Bk+ 1 . . . . . Bj.
From GMVDs (9) and ( i0 ) we can der ive :
(11) Y u C - ~ BI,B 2 . . . . . Bk_I ICI IC21. . . IC I IBk+ 1 . . . . . B a
according to inference ru le H6. But the two branches B1,B2, . . . . Bk_ I and Bk+ 1 . . . . ,Bj
in (11) can be decomposed since Y -~ B I IB21 . . . IB J ; then we get f i n a l l y
Y u C - ~ B I l B 2 1 . . , I B k _ I l C I I C 2 1 . . , I C I I B k + I I . , . I B J
which is equal to (2) .
We sha l l denote by ~ the operat ion :
(Y ~ B I I B m I . . . I B j ) ~ (Z ~+ A I I A 2 1 . . . I A I ) =
(Y u (B k n Z) ÷~ B k n A I I . . . I B k n A I I B I l B 2 1 . . . I B k _ I I B k + I I . . . I B j ) .
The * -opera t ion can be considered as an app l i ca t i on from GxG to G, such the image by
the , -ope ra t i on of two GMVDs g l and g2 given wi th respec t i ve l y i and j branches is a
GMVD wi th i + j - 1 branches. This operat ion looks l i ke the consensus operat ion in boo-
lean algebra def ined in [KP 77].
Rema[k : Let g l and g2 be GMVDs and g = gl ~ g2 ;then the root o f g contains at leas t
one of the roots o f g l or g2"
4 .4 .2. D e f i n i t i o n of an order r e l a t i o n
We sha l l def ine an order r e l a t i o n , denoted c, on the decompositions of a r e l a t i on
as fo l lows :
(Y ~+ B I I B m I ' ' ' I B j ) 2 (Z -~+A I ]A21 . . . !A I ) < ~
(Y ~ Z) A (¥i = 1,2 . . . . . I , ~j ~ {1,2 . . . . . J} : Bj u Y ~ AI)
~[2~g§~Q~_~ : The r e l a t i o n ~ is a pa r t i a l order.
The proof is given in appendix.
284
~ ! ~ : A,D ~-, BIC,E £ A ++ B,DICIE
A,D ++ BIC,E £ A ++ BIDIC,E.
Remarks :
( I ) i f an GMVD g' is derivable from g by addit ion of branches (inference rule H1)
then g' ~ g.
(2) i f an GMVD g' is derivable from g by augmentation ( inference rule D3 or H4)
then g' £ g.
These two remarks show that to obtain the closure of a given set of decompositions
i t is not necessary to consider the addit ion of branches and the augmentation in fe -
rence rules. We shal l prove in the next proposit ion an important property that
relates the *-operat ion and the order re la t ion £.
~£~P£~I~9~_Z : The , -operat ion is isotonous according to £.
The proof is given in appendix.
This property means that i f (Y ++ BI IB21. . . IB j ) £ (Z -~+A I IA21 . . . iA I ) then
(X ++ CIIC21...ICK) * (Y ++ BI IB21. . . IB J) £ (X -~ CIIC21...ICK) * (Z ~-~ A I lA21 . . . IA I ) .
4.4.3. Maximal decompositions algorithm
The determination of a maximal covering of decompositions is great ly s imp l i f ied
according to proposi t ion 7 because we have only to consider the maximal decomposi-
t ions according to the order re la t ion £. This w i l l be . i l l us t ra ted on an example.
Let G be an i n i t i a l set of decompositions as in our example 5, G = {dl ,d2,d3,d4}
d l S +* CIB,P,H
d2 C ~-~ SIB,H,P
d3 C ~-~ BIS,H,P
d4 H,P ++ CIS,B
From d2 and d3, we derive according to proposit ion 5
d5 C -~ SlBIHP.
As d2 ~ d5 and d3 ~ d5 we can delete d2 and d3 from G and we introduce d5 in G. At
the end of the process, we have only the maximal decompositions MAX(G*). In the
present example, we have MAX(G*)= {d5,d6,d7}
d5 C --~ StBiH,P
d6 s+~ ClBIH,P d7 H,P ~-~ CISIB-
NOW we have only to search an irreaundant covering among the maximal decompositions.
In the present case i t is ~AX(G*).
285
The designer has the choice between three d i f f e r e n t decompositions dS, d6 and d7 ;
we have to choose the ones which s a t i s f y a special cond i t ion ca l led "complete j o i na -
b i l i t y of data".
4.5. Complete j o i n a b i l i t y o f data
The not ion of normal forms introduced by CODD provide a gu ide l ine to study the
decomposition of a r e l a t i o n .
Many authors have modif ied the d e f i n i t i o n of the t h i r d normal form by looking at
the type of anomalies in a r e l a t i o n , so we have Boyce, Codd, Kent, I r reduc ib le and
four th normal forms. We propose a more general c r i t e r i o n fo r a decomposit ion,
the complete jo inabi l i ty of data. We shal l say tha t a decomposition has the complete
j o i n a b i l i t y of data, i n t u i t i v e l y , i f t h i s decomposition preserves a l l the j o i ns .
~ ! Q ~ ! 2 ~ : Let F be a given set of dependencies (FDs and MVDs) fo r a r e l a t i o n R
and D the i n i t i a l set o f decompositions der ived from F as expla ined in subsect ion
4.4. Let Y++ A I IA21 . . . IA I be a general ized decomposition of R belonging to D ; then we consider
FyuAk k = 1 , 2 , . . . , I as the set of dependencies which hold over the a t t r i b u t e set
Y u A k. To each FyuAk k = 1,2 . . . . . I , we associate DyuAk which is the set o f
decompositions der ived from FyuAk.
We sha l l say tha t the general ized decomposition Y-~+ A I l A 2 1 . . . I A I has the complete j o i n a b i l i t y property i f f :
I ( U ) + = D + k=l DyUAk
gxam~le : I f we consider decomposition d6 S ~ CIBIH,P, we have
FS, C = {S ÷ C, C -~ S}
FS, B =
FS,H, P = and
DS, C = {S ~ CIB,P,H , C ~ SIB,H,P}
DS, B =
DS,H, P = ~.
One can check (Ds, C u DS, B u DS,H,p) + ~ D +, then d6 has not the complete j o i n a b i l i t y property of data.
286
Now, i f we consider d5
and
then as DS, C u DC, B u Dp,H, C o f data.
C +~ SIBLH,P, we have
FS, C = {S ÷ C, C ~ S}
FC, B = {C ~ B}
Fp,H, C = {P,H ÷ C}
DS, C = { d l , d 2 }
DC ,B = {d3}
Dp,H, C = {d4}
= D, the decomposi t ion d5 has the complete j o i n a b i l i t y
287
5. CONCLUSIONS
The main object ive of th is paper was to establ ish a rigorous foundation for the design
of conceptual schemas for a database system. Because of i t s formal nature the re la-
t ional data model was used as a framework.
In a f i r s t par t , we introduced an analysis of the state of the a r t , by considering
the d i f f e ren t kind of dependencies : funct ional dependencies, mult ivalued dependen-
cies and h ierarchica l dependencies. We also recognized that dependencies and i rredun-
aant covers supply a power tool for the ~nalysis and design of re la t iona l schemas.
In a second par t , we proposed the decomposition approach where the i n i t i a l re la t ions
are broken in to subcomponents. This decomposition approach is based upon the concept
of generalized mult ivalued dependency. The formal properties of these dependencies
y ie lds a number of new concepts, such the *-operat ion which of fers the p o s s i b i l i t y
of der iv ing new decompositions from an i n i t i a l set of decompositions ; a technique
is proposed for f ind ing the maximal decompositions among the whole set of decomposi-
t ions.
288
APPENDIX
~ [ g P Q ~ ! 9 ~ _ § : The r e l a t i o n
i . £ i s r e f l e x i v e .
2. Let us show t h a t E
(x .+ c11c21... Ic K)
(Y-~+ B I I B 2 1 . . . I B j )
(X ++ C I IC21 . . . ICK)
£ i s a p a r t i a l o rde r .
i s t r a n s i t i v e , t h a t i s :
£ (Y - ~ B I I B 2 1 . . . I B j ) and
E (Z ~ - ~ A I l A 2 1 . . . I A I )
£ (Z + - ~ A I l A 2 1 . . . I A I ) .
Indeed X ~ Y and Y ~ Z---=> X ~ Z
¥i = 1,2 . . . . . I , ~ j ~ {1 ,2 . . . . . J} : Bj u Y £ A i ; ~k c { I , 2 , . . . , K } :
C k u X £ B j .
Hence : ¥i = 1 , 2 , . . . , I , ~k c { I , 2 , . . . , K } : C k u X £ B j u Y E A i .
3. c i s a n t i s y m e t r i c because
Y ->* BIIB21...IB J £ Z -~+AIlA21...IA I and Z +->AIIA21...IA I E g i v e , thanks to the a n t i s y m e t r y o f ~ : Y = Z and
¥i = 1,2 . . . . . I , ~ j c {1 ,2 . . . . . J} : Bj u Y E A i
or Bj £ A i because A i and Y are d i s j o i n t ,
t h a t i s by symetry : ¥i = 1,2 . . . . . I , ~ j c { l , 2 , . . . , J } : Bj = A i .
~[Q~9~}!~Q_Z : The , - o p e r a t i o n i s i so tonous accord ing to £ .
Let us show t h a t
(Y +-> B I I B 2 1 . . . I B j ) £ (Z ~ A I l A 2 1 . . . I A I ) - - > (X ~-~ C l l C 2 1 . . . I C K )
(Y ~ B I l B 2 1 . . . I B j )
(x ~ c11c21.. . Ic K)
(z +* A I IA21. . . IAz) .
At f i r s t Yk = 1,2 . . . . . K, i so tonous n and u r e l a t i v e l y to £ g ive :
Y £ Z ~ X u (C k n Y) £ X u (C k n Z ) .
Secondly ¥i = 1,2 . . . . . I , ~ j c { 1 , 2 , . . . , J } : Bj u Y £ A i ,
hence Vk = 1,2 . . . . . K : C k n (Bj u Y) £ C k n A i ,
a f o r t i o r i : (C k n Bj ) u X u (C k n Y) £ C k n A i .
Y ~-~ B I I B 2 1 . . . Bj
[BOI 69]
[COD 70]
[COD 71]
[DEL 71]
[DC 73]]
[DL 74]
[ARM 74]
[WW 75]
[SS 75]
[BER 75]
[FAG 76]
[RIS 76]
289
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