La sostenibilidad

4.5.4 A different Perspective for MVDs

M VDs have been defined as expressions

X -

Y where

X, Y E Sub(N).

Alternatively, we can view MVDs as expressions

X -

Y where

X, Y E

eN and (eN, �'

U,

n, -'-eN , JN) is the Brouwerian algebra of closed subsets of the PO-space on the join-irreducible elements J N

of

_Sub(N).

A set

_r

�

dom(N) satisfies

the MVD

_{X -}

y

on eN, denoted by

_{Fr X -}

Y, if and only if there is

a t E r

with

1r�(t)

=

1r�(t1)

for all

B E XUY

and

1rg(t)

=

1rg(t2)

for all

C E X U

(JN-'-cNY) whenever

_7r1(tl) = _7r1(t2)

for all

_{A E X}

holds for any

_{t1 , t2 E r.}

This view can again be justified in the following sense. Lemma 3.9 shows that for all

r

�

dom(N)

we have

Fr X

- y for

X, y E

eN if and only if

Fr u X - u

y in terms of Definition 4. 1 . The minimal axiomatisation from Theorem 4.44 reads then as follows. The following inference rules

x - Y, x - z x - Y u z

are minimal, sound and complete for the implication of MVDs in the presence of records and lists.

4.6 Related and Future Work

MVDs have been studied very well in relational databases. The next goal is the proposal of a nested list normal form for nested attributes with respect to the class of MVDs and the class of FDs and MVDs, to semantically justify this proposal and generalise the decomposition approach. Research papers that may be used as guidelines are [103, 133, 289, 290] . The proposal of such a normal form can be found in Section 6.2 of this thesis. It is desirable to improve the running time of Algorithm 4.4. 1 for deciding the implication of FDs and MVDs. Substantial research on that subject has again been done for relational databases and the papers [98, 1 18, 135, 152, 173, 223, 239, 277] may give some more information. The paper [2 1 6] proposes algorithms how to obtain

_{reduced MVDs}

and

_{minimal covers}

of sets of MVDs for relational databases. The concept of a

pure set of FDs and MVDs

was introduced in [154] . An MVD

X

----» Y of a set

E

of FDs and MVDs on a relation schema R is called pure iff it is non-trivial and neither

X

-t Y nor

X

-t (R - Y) are in

E+.

A related definition aimed at factoring out MVDs which cannot be derived from FDs appears in the concept of an

envelope set

due to [301, 302] in a work on desirable 4NF decompositions. So-called

conflict-free MVDs

are introduced in [247] . MVDs of this class have the property that they allow a unique 4NF dependency preserving database schema. Moreover it is stated that non conflict-free sets of dependencies are inadequately specified. It is interesting to study these different notions in the context of complex object types.

Multi-valued dependencies have been the subject of

data mining.

In [242] two algo­ rithms for the discovery of multi-valued dependencies from relations are presented. The top-down algorithm enumerates the hypotheses from the most general to more specific hy­ potheses which are checked on the input relation. The bottom-up algorithm first computes the invalid multi-valued dependencies. Starting with the most general dependencies, the algorithm iteratively refines the set of dependencies to conform with each particular invalid dependency. The implementation of the algorithms is analysed and some empirical results are presented. A different approach is proposed in [300] .

Recent papers that study multi-valued dependencies in the context of XML are [286, 287] . The work in [286] introduces MVDs in XML (XMVDs) and justifies the definition by showing that for a general class of mappings from relations to XML, a relation satisfies an MVD if and only if the corresponding XML document satisfies the corresponding XMVD. As this justification of XMVDs already suggests, XMVDs provide semantics for XML documents that are exported or imported from relational databases. Therefore, XMVDs do not cover multi-valued dependencies among complex objects such as lists. The definition of XMVDs is again based on the notion of a path. The work in [287] proposes an extension of the well-known fourth normal form ( 4NF) from relational data bases to XML in order to syntactically describe semantically well-designed XML documents with respect to XMVDs as studied in [286] .

A conceptual treatment of MVDs is introduced in [266] . It is proposed that entity­ relationship modelling techniques enable a more natural and intuitive way of handling MVDs. Based on the concept of

competing MVDs

it is proven in which case a unique entity-relationship schema representation exists. If MVDs are competing, then either one

4.6. RELATED AND FUTURE WORK Sebastian Link of the competing schemata is chosen or an approximation which combines the competing schemata can be used.

For more comments on future work see Section 6.2. Let us finally look at a further example of MVDs among complex objects. Suppose we store nucleotide sequences together with certain genes that occur in it, i.e. sequences of amino acids, and together with a certain base and the sequence of positions in which that base appears in the original nucleotide sequence. We may use the nested attribute

Genes(Sequence[N ucleotide] , Gene[ Amino-Acid] , Occurs(Base, Position[N umber] ) ) . There might b e several genes encoded within the nucleotide sequence, and there are dif­ ferent bases together with a certain sequence of positions in which they occur. The set of genes, however, is independent from the set of bases and the corresponding sequence of occurrences. We therefore have the following MVDs

Genes(Sequence[Nucleotide] ) ----* Genes(Gene[Amino-Acid] ) and Genes(Sequence[Nucleotide]) ----* Genes(Occurs(Base, Position[Number]) ) . Moreover there are the FDs

Genes(Sequence[Nucleotide] ,Occurs(Base)) -+ Genes(Occurs(Position[Number] ) ) and

Genes (Sequence[N ucleotide J, Occurs (Position [Number] ) ) -+ Genes( Occurs (Base) ) . It appears that the chance of MVDs occurring among complex objects is as good as the chance of MVDs occurring among fiat data. The techniques provided in this chapter may therefore help to cover more application domains.

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