Join dependency

In database theory, a join dependency is a constraint on the set of legal relations over a database scheme. A table ${\displaystyle T}$ is subject to a join dependency if ${\displaystyle T}$ can always be recreated by joining multiple tables each having a subset of the attributes of ${\displaystyle T}$. If one of the tables in the join has all the attributes of the table ${\displaystyle T}$, the join dependency is called trivial.

The join dependency plays an important role in the fifth normal form (5NF), also known as project-join normal form, because it can be proven that if a scheme ${\displaystyle R}$ is decomposed in tables ${\displaystyle R_{1}}$ to ${\displaystyle R_{n}}$, the decomposition will be a lossless-join decomposition if the legal relations on ${\displaystyle R}$ are restricted to a join dependency on ${\displaystyle R}$ called ${\displaystyle *(R_{1},R_{2},\ldots ,R_{n})}$.

Another way to describe a join dependency is to say that the relationships in the join dependency are independent of each other.

Unlike in the case of functional dependencies, there is no sound and complete axiomatization for join dependencies,^[1] though axiomatization exist for more expressive dependency languages such as full typed dependencies.^{[2]: Chapter 8} However, implication of join dependencies is decidable.^{[2]: Theorem 8.4.12}