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:

$\mathrm{\forall }{x}_1,\dots ,x_n.\varphi (x_1,\dots ,x_n)\to \psi (y_1,\dots ,y_m)$

where $\{y_1,\dots ,y_m\}\subseteq \{x_1,\dots ,x_n\}$, $\varphi$ is a conjunction of relational and equality atoms and $\psi$ is a non-empty conjunction of equality atoms. A relational atom has the form $R(w_1,\dots ,w_h)$ 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 $A(x,y)\wedge B(y,z,w)\wedge y=3\wedge z=w$, then it can be replaced with $A(x,3)\wedge B(3,z,z)$ (analogously replacing possible occurrences of the variables $y$ and $w$ in the head).

An equivalent definition is the following:

$\mathrm{\forall }{x}_1,\dots ,x_n.\varphi (x_1,\dots ,x_n)\to x_i=x_j$

where $i,j\in \{1,\dots ,n\}$. 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