Delobel's theorem

The set of Delobel (Claude Delobel) provides an easy way to check if two fragments of a relationship in a database are a lossless representation of the output relation. A decomposition of relations is necessary in order to avoid the occurrence of anomalies .

Formal representation

The following important information is missing from this article or section:

The condition of freedom from loss nowhere refers to the original set of attributes U, so it does not depend on it and is therefore nonsensical. Obviously, there are still unspoken assumptions about what a decomposition actually is. You should then expressly state this. The same applies to F and F1, F2. In addition, there is no explanation of the or source for the APLUS algorithm, without which the easily possible check cannot or only not easily take place due to the lack of an alternative

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Let the relation and its decomposition with and be given . ${\displaystyle r:(U\mid F)}$ ${\displaystyle D:(\{r1:(A_{1}\mid F_{1}),r2:(A_{2}\mid F_{2})\})}$ ${\displaystyle U=A_{1}\cup A_{2}}$ ${\displaystyle F=F_{1}\cup F_{2}}$

We put: and with . ${\displaystyle A_{1}\ \cap \ A_{2}=B,A_{1}=AB}$ ${\displaystyle A_{2}\ =BC}$ ${\displaystyle A\ \cap \ B=A\ \cap \ C=B\ \cap \ C=\emptyset }$

D is lossless or ${\displaystyle \Longleftrightarrow (B\rightarrow A\in F^{+}}$ ${\displaystyle B\rightarrow C\in F^{+})}$

Now you only have to check the last condition, which is easily possible with the help of the APLUS algorithm .

example

The output relation is defined as having decompositions ${\displaystyle r:(a,b,c,d,e\mid a\rightarrow bcd,d\rightarrow bce,d\rightarrow e)}$

${\displaystyle r_{1}:(a,b,c,d\mid a\rightarrow bcd)}$and . ${\displaystyle r_{2}:(b,c,d,e\mid d\rightarrow bce,d\rightarrow e)}$

The attributes are distributed as follows:

amount	Attributes
B.	b, c, d
A.	a
C.	e

According to Delobel, it follows from this that the dismantling is loss-free if applies or . ${\displaystyle bcd\rightarrow a\in F^{+}}$ ${\displaystyle bcd\rightarrow e\in F^{+}}$

It follows immediately that also . ${\displaystyle d\rightarrow e\in F^{+}}$ ${\displaystyle bcd\rightarrow e\in F^{+}}$

See also

• Functional dependency
• Normalization of databases

swell

1. ↑ Solution slides (PDF; 661 kB)  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. of the AIFB Institute of the University of Karlsruhe for the lecture database systems from 2003