Comprehending nulls

The Nested Relational Calculus (NRC) has been an influential high-level query language, providing power and flexibility while still allowing translation to standard SQL queries. It has also been used as a basis for language-integrated query in programming languages such as F#, Scala, and Links. However, SQL's treatment of incomplete information, using nulls and three-valued logic, is not compatible with `standard' NRC based on two-valued logic. Nulls are widely used in practice for incomplete data, but the question of how to accommodate SQL-style nulls and incomplete information in NRC, or integrate such queries into a typed programming language, appears not to have been studied thoroughly. In this paper we consider two approaches: an explicit approach in which option types are used to represent (possibly) nullable primitive types, and an implicit approach in which types are treated as possibly-null by default. We give translations relating the implicit and explicit approaches, discuss handling nulls in language integration, and sketch extensions of normalization and conservativity results.