Proof.
Note that Y = Y n (YU
X)
= ((Y -'-X) u Y) n(
(Y -'-X) u X) = (Y -'-X) u (Y n X).X u W --+ Xx::;xuw X u w ____.. X X ____.. y X u w --+ y n XYnx::;xuw X u w ___.. y_,_x X U W ----* Y n X x u w ___.. y X U W ----* Y U V X u W --+ Vv::;w::;xuw X U W ----* V 0
It follows from the previous lemmata that reflexivity axiom, extension rule, transitivity rule, implication rule, Brouwerian complement rule, pseudo-transitivity rule, mixed pseudo transitivity rule, multi-valued join rule and mixed meet rule form already a sound and complete set of inference rules for the implication of FDs and MVDs. We are now going to show that this is in fact a minimal set, i.e., each of the rules is independent from the others.
Lemma 4.19.
The reflexivity axiom is independent from the set
m ={ extension rule,
transitivity rule, implication rule, Brouwerian complement rule, pseudo-transitivity rule,
mixed pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
The reflexivity axiom is the only inference rule that allows one to infer dependenciesfrom the empty set. o
Lemma 4.20.
The extension rule is independent from the set
m ={reflexivity axiom,
transitivity rule, implication rule, Brouwerian complement rule, pseudo-transitivity rule,
mixed pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
LetN
=L(A, B),
E ={L(A)
--+L(B) }
and a =L(A)
--+L(A, B).
The closure ofE under m is represented by the following tables in the following way. An FD X --+ Y is in the closure E� if and only if there appears a cross x in row X and column
Y
of the left table. Correspondingly, an MVD X ____.. Y is in the closure E� if and only if there appears a cross x in row X and column Y of the right table.--+ ____..
A
XA
X XL(A)
X X XL(A)
X X X XL(B)
X XL(B)
X X X XL(A, B)
X X X XL(A, B)
X X X XIt can be seen that a tf: E�. However, a can be inferred from E using the extension
Lemma 4.2 1 .
The transitivity rule is independent from the set
91 ={ reflexivity axiom, ex
tension rule, implication rule, Brouwerian complement rule, pseudo-transitivity rule, mixed
pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
LetN
=L(A, B),
E ={A
---+L(A), L(A)
---+L(B)}
and aclosure of E under 91 is represented by the following tables.
A
---+
L(B).
The ---+A
A
X XL(A)
X X X XL(B)
X XL(A, B)
X X X X -A
A
X X X XL(A)
X X X XL(B)
X X X XL(A, B)
X X X XIt can be seen that a
�
E� . However, a can be inferred from E using the transitivityrule. 0
Lemma 4.22.
The implication rule is independent from the set
91 ={ reflexivity axiom, ex
tension rule, transitivity rule, Brouwerian complement rule, pseudo-transitivity rule, mixed
pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
LetN
=A
, E = 0 and a =A - A.
The closure of E under 91 is represented by the following tables.A
It can be seen that a
�
E�. However, a can be inferred from E using first the reflexivityaxiom to infer
A ---+ A,
and subsequently the implication rule. 0Lemma 4.23.
The Brouwerian complement rule is independent from the set
91{reflexivity axiom, extension rule, transitivity rule, implication rule, pseudo-transitivity
rule, mixed pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
Suppose we interpretX
-Y
as"X
functionally determinesY" ,
and consider the set of FDs on a nested attributeN.
Under this interpretation, implication rule, pseudo transitivity rule, mixed pseudo-transitivity rule, multi-valued join rule and mixed meet rule are all still valid, but the Brouwerian complement rule is not. Hence, it cannot be logicallyimplied by the set given. 0
Lemma 4.24.
The pseudo-transitivity rule is independent from the set
91 ={ reflexivity ax
iom, extension rule, transitivity rule, implication rule, Brouwerian complement rule, mixed
pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
4.2.
MINIMALITY Sebastian LinkProof.
LetN
=L(A, B, C),
E ={.A ----* L (A), L(A) ----* L(B)}
and a- =.A
----*L(B).
Theclosure of E under 91: is represented by the following tables
and
�
.A
B,
.A
XL(A)
X XL(B)
X XL(C)
X XL(A, B)
X X X XL(A, C)
X X X XL(B, C)
X X X XL(A, B, C)
X X X X X X X X ----*.A
B,
.A
X • • XL(A)
X X 0 0 0 0 X XL (B)
X X X XL(C)
X X X XL(A, B)
X X X X X X X XL(A, C)
X X X X X X X XL(B, C)
X X X X X X X XL(A, B, C)
X X X X X X X XA filled circle • in line
X
and columnY
indicates thatX ----* Y
follows from the given MVD.A ----* L(A) ,
and a o in lineX
and columnY
indicates thatX ----* Y
follows from the given MVDL(A)
----*L(B) .
This shows thatA ----* L(B)
tJ. E�, but a- can be derived usingthe pseudo-transitivity rule. D
Lemma 4.25.
The mixed pseudo-transitivity rule is independent from the set
91: ={ reflexivity axiom, extension rule, transitivity rule, implication rule, Brouwerian comple
ment rule, pseudo-transitivity rule, multi-valued join rule, mixed meet rule} .
Proof.
LetN
=L(A, B),
E ={.A ----* L (A), L(A)
--+L(B)}
and a- =.A � L(B).
Theclosure of E under 91: is represented by the following tables.
�
----*
.A
X.A
X X X XL(A)
X X X XL(A)
X X X XL(B)
X XL(B)
X X X XIt can be seen that a � .E� . However, a can be inferred from E using the mixed pseudo-
transitivity rule. 0
In order to show the independence of the multi-valued join rule, we make use of the fact that non-maximal basis attributes cannot be represented as the Brouwerian complement of any subattribute.
Lemma 4.26.
The multi-valued join rule is independent from the set
9t ={ reflexivity
axiom, extension rule, transitivity rule, implication rule, Brouwerian complement rule,
pseudo-transitivity rule, mixed pseudo-transitivity rule, mixed meet rule} .
Proof.
LetN
=L(A, K[B]),
.E ={A __,. L(A) , A __,. L(K[A] ) }
and a =A
__,.L(A, K[A]).
The closure of E under 9t is represented by the following tablesand ---+
A
A
X 0L(A)
X X 0 0L(K[A])
X XL(K[B])
X X XL(A, K[A])
X X X XL(A, K[B])
X X X X X XA
A
X • 0 • XL(A)
X X 0 X 0 XL(K[A] )
X • X • XL(K[B])
X X X X XL(A, K[A])
X X X X X XL(A, K[B])
X X X X X XA filled circle • in line
X
and columnY
indicates thatX
__,.Y
follows from the givenMVD
A
__,.L(A) ,
whereas a circle o in lineX
and columnY
indicates thatX
---+Y
orX
__,.Y
follows from the given MVDA
__,.L(K[B]) .
One can see thatA __,. L(A, K[A])
�E�, but a can be derived using the multi-valued join rule. 0
Lemma 4.27.
The mixed meet rule is independent from the set
9t ={reflexivity axiom,
extension rule, transitivity rule, implication rule, Brouwerian complement rule, pseudo
transitivity rule, mixed pseudo-transitivity rule, multi-valued join rule} .
Proof.
LetN
=L[K(A, B)],
.E= {A __,. L[K(A) ] }
and a =A __,. L[A] .
We use a instead ofA
---+L[A]
for technical reasons. The closure of .E under 9t is represented by the following tables4.3.
BROUWERIAN-COMPLEMENT RULE Sebastian Link-+
A
XL[A]
X XL[K(A)]
X X XL[K(B)]
X X XL[K(A, B)]
X X X X X andA
X • • XL[A]
X X • • XL[K(A)]
X X X X XL[K(B)]
X X X X XL[K(A, B)]
X X X X Xas before. A filled circle • in line
X
and columnY
indicates thatX
-Y
follows from the given MVDA
--*L[K(A)].
One can see thatA -+ L[A]
� E�, but it can be derived using the mixed meet rule. Furthermore a � E�, but a can be derived by applying theimplication rule to
A -+ L[A].
0The combination of the previous lemmata gives the following result.
Theorem 4.28.
Reflexivity axiom, extension rule, transitivity rule, implication rule,
Brouwerian complement rule, pseudo-transitivity rule, mixed pseudo-transitivity rule,
multi-valued join rule and mixed meet rule form a minimal, sound and complete set of
inference rules for the implication of FDs and MVDs in the presence of records and lists.
0 Theorem
4.28
is somewhat surprising. We have seen that in the presence of lists the multi-valued join rule is independent from the rest of the rules in Theorem4.28.
For relational databases, however, it was proven in[204]
that the multi-valued join rule is logically implied by a corresponding subset of the rules above. The fact that this is not the case in the presence of lists results from the existence of some subattributes which are not the Brouwerian complement of any other subattributes, e.g.L [A]
is not the Brouwerian complement of any subattribute ofL[A].
4 . 3 B rouwerian- Complement Rule
The Brouwerian complement rule is the analogue of the complementation rule for MVDs in relational databases. There are a few papers
[32, 46, 204]
which point at the significance of the complementation rule. In fact, it is the only rule that does not have a direct analogue in the axiomatisation of FDs since it is the only rule that takes into account the contextof the dependencies, that is, the underlying relation schema R. The rest of the inference rules apply independently of whatever relation schema the attributes are embedded in.
The situation is again slightly different in the presence of lists. Here, the Brouwerian complement rule is not the only context-sensitive rule. The mixed meet rule depends on the underlying nested attribute as well. That is, the FD
X
--7 Y n Y� can be inferred fromthe MVD
X ---*
Y.EXAMPLE
4 . 8 .
Let Y =L(A, K[A]),
N1 =L(A, K[A])
and N2 =L(A, K[B]).
In thesecases, Y�1 =
L(A, A)
and Y�2 =L(A, K[B]).
It follows that Y nN1 Y�1 =L(A, A) ,
butY nN2 Y�2 =
L(A, K[A]).
0We delay a detailed study of the mixed meet rule to future research, and focus for the remainder of this section on the role of the Brouwerian complement rule. It is interesting to study whether one can obtain a (minimal) sound and complete set of inference rules that does not include the Brouwerian complement rule. A further reason for this might appear in the future when the algorithmic synthesis of nested attributes will be studied. This will cause at least as many problems as in relational databases
[29, 204].
Given a nested attribute N we introduce the N-axiom
A ---* N
which is sound by the definition of MVDs. It is a very weak form of the Brouwerian complement rule, and corre-sponds to the so-called R-axiom 0
---*
R for a relation schema R in relational databases (see[46]) .
We will show in this section that the Brouwerian complement rule in the minimal set of inference rules from Theorem4.28
can be replaced by the N-axiom and still maintain minimality.First of all, we will show that N-axiom and Brouwerian complement rule are equivalent in the presence of reflexivity axiom, implication rule and pseudo-transitivity rule.
Lemma 4.29.
The Brouwerian complement rule is not independent from { reflexivity ax
iom,
N-axiom, implication rule, pseudo-transitivity} . Furthermore, the
N-axiom is not
independent from {reflexivity axiom, Brouwerian complement rule, implication rule} .
Proof.
The derivation tree for the first statement is given byy --+ A>.::;Y
Y ---* A
A ---* N
X
-Y Y - NX ---*
N....:.... Y "-v--'=Ye
and the proof for the second statement is given by
A --+ A>.::;>.
A ---* A
A ---* N
4.3.
BROUWERIAN-COMPLEMENT RULE Sebastian Link While all elements of Dt = { reflexivity axiom, extension rule, transitivity rule, implication rule, pseudo-transitivity rule, mixed pseudo-transitivity rule, multi-valued join rule, mixed meet rule} are still sound when the MVDs are interpreted as FDs, the N-axiom is not. This shows the independence of the N-axiom from m. above.
Let Dt = {reflexivity axiom, extension rule, transitivity rule, implication rule, N-axiom,
pseudo-transitivity rule, mixed pseudo-transitivity rule, multi-valued join rule, mixed meet rule } . It follows from Lemma
4.29
and the lemmata from the previous section that R is independent from Dt - { R}, ifR
E {extension rule, transitivity rule, mixed pseudotransitivity rule, multi-valued join rule, mixed meet rule} . We deal with the remaining cases in the following lemma.
Lemma 4.30.