Minimal Cover

Set of dependencies $F$ is minimal if it satisfies the following conditions:

1. Every right-hand side of a dependency in $F$ is a single attribute
2. For no $X\to A$ is the set $F-\{X\to A\}$ equivalent to $F$
3. For no $X\to A$ and $Z\subset X$ is the set $F-\{X\to A\}\bigcup \{Z\to A\}$ equivalent to $F$

Theorem

For every set of dependencies $F$ there is an equivalent minimal set of dependencies called a minimal cover

Computation

Given a set of function dependencies apply each step until it no longer succeeds in updating $F$

1. Replace $X\to YZ$ with the pair $X\to Y$ and $X\to Z$
2. Replace $X\to A$ from $F$ if $A\in compute{X}^{+}(X,F-\{X\to A\})$
3. Remove $A$ from left-hand side of $X\to B$ in $F$ if $B$ is in $compute{X}^{+}(X-\{A\},F)$
4. Replace $X\to Y$, $X\to Z$ in $F$ by $X\to YZ$