Universal (and Existential) Nulls

Incomplete Information research is quite mature when it comes to so called existential nulls, where an existential null is a value stored in the database, representing an unknown object. For some reason universal nulls, that is, values representing all possible objects, have received almost no attention. We remedy the situation in this paper, by showing that a suitable finite representation mechanism, called Star Cylinders, handling universal nulls can be developed based on the Cylindric Set Algebra of Henkin, Monk and Tarski. We provide a finitary version of the cylindric set algebra, called Cylindric Star Algebra, and show that our star-cylinders are closed under this algebra. Moreover, we show that any First Order Relational Calculus query over databases containing universal nulls can be translated into an equivalent expression in our cylindric star-algebra, and vice versa. All cylindric star-algebra expressions can be evaluated in time polynomial in the size of the database.

The representation mechanism is then extended to Naive Star Cylinders, which are star-cylinders allowing existential nulls in addition to universal nulls. For positive queries (with universal quantification), the well known naive evaluation technique can still be applied on the existential nulls, thereby allowing polynomial time evaluation of certain answers on databases containing both universal and existential nulls. If precise answers are required, certain answer evaluation with universal and existential nulls remains in coNP. Note that the problem is coNP-hard, already for positive existential queries and databases with only existential nulls. If inequalities ¬(x_i ≈ x_j) are allowed, reasoning over existential databases is known to be ∏ 2 p -complete, and it remains in ∏ 2 p when universal nulls and full first order queries are allowed.