Equality-generating dependency

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

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

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

Actually, one can remove all equality atoms from the body of the dependency without loss of generality.^[1] For instance, if the body consists in the conjunction ${\displaystyle A(x,y)\land B(y,z,w)\land y=3\land z=w}$ , then it can be replaced with ${\displaystyle A(x,3)\land B(3,z,z)}$ (analogously replacing possible occurrences of the variables ${\displaystyle y}$ and ${\displaystyle w}$ in the head).

An equivalent definition is the following:^[2]

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

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

• 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