On the Logic of SQL Nulls

The logic of nulls in databases has been subject of investigation since their introduction in Codd’s Relational Model, which is the foundation of the SQL standard. In the logic based approaches to modelling relational databases proposed so far, nulls are considered as representing unknown values. Such existential semantics fails to capture the behaviour of the SQL standard. We show that, according to Codd’s Relational Model, a SQL null value represents a non-existing value; as a consequence no indeterminacy is introduced by SQL null values. In this paper we introduce an extension of first-order logic accounting for predicates with missing arguments. We show that the domain independent fragment of this logic is equivalent to Codd’s relational algebra with SQL nulls. Moreover, we prove a faithful encoding of the logic into standard first-order logic, so that we can employ classical deduction machinery. 1 Relational Databases and SQL Null Values Consider a database instance with null values over the relational schema {R/2}, and a SQL query asking for the tuples in R being equal to themselves: R : 1 2 a b b N SELECT * FROM R WHERE R.1 = R.1 AND R.2 = R.2 ; ⇒ 1 | 2 ---+--a | b (1 row)