A Formal Approach to Null Values in Database Relations

We study the problem of null values. By this we mean that an attribute is applicable but its value at present is unknown and also that an attribute is applicable but its value is arbitrary. We adopt the view that tuples denote statements of predicate logic about database relations. Then, a null value of the first kind, respectively second kind, corresponds to an existentially quantified variable, respectively universally quantified variable. For instance if r is a database relation without null values and X is a range declaration for r then the tuple (a, ∀,b, ∃) ∈ R is intended to mean “there exists an x ∈ X such that for all y ∈ X: (a,y,b,x) ∈ r”. We extend basic operations of the well-known relational algebra to relations with null values. Using formal notions of correctness and completeness (adapted from predicate logic) we show that our extensions are meaningful and natural. Furthermore we reexamine the generalized join within our framework. Finally we investigate the algebraic structure of the class of relations with null values under a partial ordering which can be interpreted as a kind of logical implication.