Equality-generating dependency

In relational database theory, an equality-generating dependency (EGD) is a certain kind of constraint on data. It is a subclass of the class of embedded dependencies (ED).

An algorithm known as the chase takes as input an instance that may or may not satisfy a set of EGDs (or, more generally, a set of EDs), and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EGDs.

An important subclass of equality-generating dependencies are functional dependencies.

Definition

An equality-generating dependency is a sentence in first-order logic of the form:

$\forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \psi (y_{1},\ldots ,y_{m})$

where $\{y_{1},\ldots ,y_{m}\}\subseteq \{x_{1},\ldots ,x_{n}\}$, $\phi$ is a conjunction of relational and equality atoms and $\psi$ is a non-empty conjunction of equality atoms. A relational atom has the form $R(w_{1},\ldots ,w_{h})$ and an equality atom has the form $w_{i}=w_{j}$, where each of the terms $w,...,w_{h},w_{i},w_{j}$ are variables or constants.

Actually, one can remove all equality atoms from the body of the dependency without loss of generality. For instance, if the body consists in the conjunction $A(x,y)\land B(y,z,w)\land y=3\land z=w$, then it can be replaced with $A(x,3)\land B(3,z,z)$ (analogously replacing possible occurrences of the variables $y$ and $w$ in the head).

An equivalent definition is the following:

$\forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow x_{i}=x_{j}$

where $i,j\in \{1,\ldots ,n\}$. Indeed, generating a conjunction of equalities is equivalent to have multiple dependencies which generate only one equality.

Further reading

• Abiteboul, Serge; Hull, Richard B.; Vianu, Victor (1995). Foundations of Databases. Addison-Wesley. ISBN 0-201-53771-0.
• Alin Deutsch, FOL Modeling of Integrity Constraints, https://web.archive.org/web/20140912044956/http://db.ucsd.edu/pubsFileFolder/305.pdf