Embedded dependency

An embedded dependency (ED) is a sentence in first-order logic of the form:

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

where ${\displaystyle \{z_{1},\ldots ,z_{k}\}=\{y_{1},\ldots ,y_{m}\}\setminus \{x_{1},\ldots ,x_{n}\}}$ and ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are conjunctions of relational and equality atoms.[1] 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.[2] 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). Analogously, one can replace existential variables occurring in the head if they appear in some equality atom.[2]

In literature there are many common restrictions on embedded dependencies, among with:[1][3]

• full (or universal) dependencies, which are the ones without existentially-quantified variables (${\displaystyle \exists z_{i}}$)
• tuple-generating dependencies (TGDs)
• equality-generating dependencies (EGDs)
• single-head (or 1-head) dependencies, which have only one atom in the head
• unirelational dependencies, in which only one relation symbol occurs

When all atoms in ${\displaystyle \psi }$ are equalities, the ED is an EGD and, when all atoms in ${\displaystyle \psi }$ are relational, the ED is a TGD. Every ED is equivalent to an EGD and a TGD.

A common extension of embedded dependencies are disjunctive embedded dependencies (DED),[4] which can be defined as follows:

${\displaystyle \forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \bigvee _{i=1}^{\ell }\exists z_{1}^{i},\ldots ,z_{k}^{i}.\psi (y_{1}^{i},\ldots ,y_{m}^{i})}$

where ${\displaystyle \{z_{1}^{i},\ldots ,z_{k}^{i}\}=\{y_{1}^{i},\ldots ,y_{m}^{i}\}\setminus \{x_{1},\ldots ,x_{n}\}}$ and ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are conjunctions of relational and equality atoms.

Disjunctive embedded dependencies are more expressive than simple embedded dependencies, because DEDs in general can not be simulated using one or more EDs. An even more expressive constraint is the disjunctive embedded dependency with inequalities (indicated with DED${\displaystyle ^{\neq }}$), in which every ${\displaystyle \psi }$ may contain also inequality atoms.[4]

All the restriction above can be applied also to disjunctive embedded dependencies. Beside them, DEDs can also be seen as a generalization of disjunctive tuple-generating dependencies and disjunctive equality-generating dependencies.

• Kanellakis, Paris C. (1990). "Elements of Relational Database Theory". Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics. Amsterdam: Elsevier. pp. 1073–1156. doi:10.1016/b978-0-444-88074-1.50022-6. ISBN 978-0-444-88074-1.
• Abiteboul, Serge; Hull, Richard B.; Vianu, Victor (1995). Foundations of Databases. Addison-Wesley. ISBN 0-201-53771-0.
• Deutsch, Alin (2009). "FOL Modeling of Integrity Constraints (Dependencies)". Encyclopedia of Database Systems. Boston, MA: Springer US. pp. 1155–1161. doi:10.1007/978-0-387-39940-9_980. ISBN 978-0-387-39940-9.