Value addiction

A polyhydric dependence ( English multivalued dependency (MVD) ) dependence describes a set of attributes of a set of attributes . ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \beta }$ ${\displaystyle \alpha }$

Definition and explanation

In the following represent all attributes (columns) of the tuple (row) . A multi-valued dependency between attributes of a relation exists if the following applies: ${\displaystyle t[\alpha ]}$ ${\displaystyle \alpha }$ ${\displaystyle t}$ ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle R}$

For two tuples and with, there are always two further tuples in every permissible instance of and with: ${\displaystyle t_{1}}$ ${\displaystyle t_{2}}$ ${\displaystyle t_{1}[\alpha ]=t_{2}[\alpha ]}$ ${\displaystyle R}$ ${\displaystyle t_{3}}$ ${\displaystyle t_{4}}$

${\displaystyle {\begin{matrix}t_{1}[\alpha ]=t_{2}[\alpha ]=t_{3}[\alpha ]=t_{4}[\alpha ]\\t_{1}[\beta ]=t_{3}[\beta ]\\t_{2}[\beta ]=t_{4}[\beta ]\\t_{1}[R\setminus (\alpha \cup \beta )]=t_{4}[R\setminus (\alpha \cup \beta )]\\t_{2}[R\setminus (\alpha \cup \beta )]=t_{3}[R\setminus (\alpha \cup \beta )]\end{matrix}}}$

This clearly shows:

${\displaystyle {\begin{matrix}{\text{Tupel}}&\alpha &\beta &R\setminus (\alpha \cup \beta )\\t_{1}&a_{1}..a_{n}&b_{1}..b_{m}&d_{1}..d_{k}\\t_{2}&a_{1}..a_{n}&c_{1}..c_{m}&e_{1}..e_{k}\\t_{3}&a_{1}..a_{n}&b_{1}..b_{m}&e_{1}..e_{k}\\t_{4}&a_{1}..a_{n}&c_{1}..c_{m}&d_{1}..d_{k}\end{matrix}}}$

Multi-valued dependencies are trivial if or . ${\displaystyle \beta \subseteq \alpha }$ ${\displaystyle \alpha \cup \beta =R}$

Shell formation

In connection with the normalization of databases, the set of all dependencies implied by multivalued dependencies is often required. The starting point is the set consisting of functional dependencies and multi-valued dependencies . The aim is to determine the envelope . Analogous to the Armstrong axioms for expanding the functional dependencies, the following axioms are applied here: ${\displaystyle D}$ ${\displaystyle FD}$ ${\displaystyle MVD}$ ${\displaystyle D^{+}}$

1. Reflexivity , expansion and transitivity for functional dependencies
2. Repetition : If so , then too ${\displaystyle \alpha \rightarrow \beta }$ ${\displaystyle \alpha \twoheadrightarrow \beta }$
3. Complement : for everyone also exists ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \alpha \twoheadrightarrow R\setminus \{\alpha \cup \beta \}}$
4. Multi- valued extension : apply and be as well , then also applies ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \gamma \subseteq R}$ ${\displaystyle \delta \subseteq \gamma }$ ${\displaystyle \alpha \gamma \twoheadrightarrow \beta \delta }$
5. Multi-valued transitivity : if and , then also applies ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \beta \twoheadrightarrow \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \gamma \setminus \beta }$
6. Merger : Applies , and there is a with , and then applies ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \gamma \subseteq \beta }$ ${\displaystyle \delta }$ ${\displaystyle \delta \subseteq R}$ ${\displaystyle \gamma \cap \delta =\varnothing }$ ${\displaystyle \delta \rightarrow \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \gamma }$

A few more derived rules help here as well:

1. Multi-valued union : if and , then also applies ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \alpha \twoheadrightarrow \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \beta \gamma }$
2. Average : If and , then also applies ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \alpha \twoheadrightarrow \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \beta \cap \gamma }$
3. Difference : If and , then also applies or ${\displaystyle \alpha \twoheadrightarrow \beta }$ ${\displaystyle \alpha \twoheadrightarrow \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \beta \setminus \gamma }$ ${\displaystyle \alpha \twoheadrightarrow \gamma \setminus \beta }$