Incomplete information occurs when, due to insufficient knowledge about the state of the world, there is more than one candidate database to represent the current state of the world. In the database world, one can imagine the user keeping a set of relational databases (even an infinite set, if one imagines vigorously), knowing that one of these databases corresponds to the actual state of the world, but needing more information to know which database is the correct one. If the user wants to apply an ordinary relational update to this set of candidate databases, then the natural definition of the semantics of the update is to apply the update to each candidate database individually.

Though this imaginary scenario paints a clear picture of the semantics of ordinary updates when incomplete information is present, it is unsuitable for direct implementation in most applications, due to the prohibitive expense of storing multiple databases. A more compact representation of the candidate databases is required for the sake of efficiency. Our solution is to represent sets of databases as the models of simple relational theories in first-order logic, using an extension of Reiter's relational theories [Reiter 84]. Our relational theories are sufficiently powerful to represent in one theory any realistic set of relational databases all having the same schema and integrity constraints. Section 3.1 gives a formal description of the language and Section 3.2 of the structure of relational theories.