Relational Algebra and Calculus with SQL Null Values

TL;DR

This paper provides a logical characterization of SQL with null values by extending relational algebra and proposing two domain relational calculi, establishing their equivalence and extending Codd's theorem.

Contribution

It introduces a formal framework capturing SQL null semantics through extended algebra and calculi, extending Codd's theorem to null-inclusive SQL.

Findings

Relational algebra extended with nulls captures SQL fragment.

Two domain relational calculi with nulls are equivalent to extended algebra.

Codd's theorem is extended to include null values in relational models.

Abstract

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. We show a logical characterisation of a first-order fragment of SQL with null values, by first focussing on a simple extension with null values of standard relational algebra, which captures exactly the SQL fragment, and then proposing two different domain relational calculi, in which the null value is a term of the language but it does not appear as an element of the semantic interpretation domain of the logics. In one calculus, a relation can be seen as a set of partial tuples, while in the other (equivalent) calculus, a relation is horizontally decomposed as a set of relations each one holding regular total tuples. We extend Codd's theorem by proving the equivalence of the relational algebra with both domain relational calculi in presence of SQL null values.