Definition
The implication problem is to decide whether or not a given set of constraints logically implies another constraint. With any constraint (dependency) language \({\cal L}\) there are two implication problems associated, which do not coincide in general.
In the traditional logic framework, an instance of a schema R is a logical structure that is either finite or infinite, referred to as an unrestricted instance of R. A set Σ of constraints over R implies without restriction a constraint ϕ, denoted by Σ ⊧unr ϕ, if for each unrestricted instance I of R that satisfies Σ, I also satisfies ϕ. The unrestricted implication problem for \({\cal L}\) is to determine, given a set Σ of constraints in \({\cal L}\) and another constraint ϕ in \({\cal L}\), whether or not Σ ⊧unr ϕ.
In the context of databases, only finite instances are considered and implication analysis lies within finite model theory. A set Σ of constraints over R finitely implies a constraint ϕ, denoted by Σ ⊧fin...
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Fan, W. (2009). Implication of Constraints. In: LIU, L., ÖZSU, M.T. (eds) Encyclopedia of Database Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-39940-9_1238
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