Polymorphic Type Inference for the Relational Algebra

Abstract

We give a polymorphic account of the relational algebra. We introduce a formalism of ''type formulas'' specifically tuned for relational algebra expressions, and present an algorithm that computes the ''principal'' type for a given expression. The principal type of an expression is a formula that specifies, in a clear and concise manner, all assignments of types (sets of attributes) to relation names, under which a given relational algebra expression is well-typed, as well as the output type that expression will have under each of these assignments. Topics discussed include complexity and polymorphic expressive power.

Towards a Type Discipline for Answer Set Programming

“…Finally we mention the extensive research about polymorphic type inference in relation algebra and (object-oriented) databases, see [7,4] for examples. However, this is only loosely related to our efforts, as there the aim is to extend a ML-like type system to capture the principal type of a program involving database operations or more in general of an expression in relational algebra, where a type is seen as a set of attribute names for a given schema.…”

On the complexity of deciding typability in the relational algebra

“…The type system of Van den Bussche and Waller [7] considers an expression ill-defined if the schema of some subexpression is required to have an attribute present and absent at the same time. Practical query languages also consider an expression ill-defined if attributes are not used consistently with regard to their base types.…”

“…The corresponding SQL expression would be considered incorrect however, since an attribute cannot have base type bool and int at the same time. In this section, we therefore augment the type system of Van den Bussche and Waller [7] with the ability to deal with types on the attribute value level.…”

“…In particular, well-defined expressions Research Assistant of the Fund for Scientific Research -Flanders (Belgium) are those expressions that can be assigned a type in the system. Recently, Van den Bussche and Waller [7] considered a type system for the relational algebra. This system consists of a set of type rules allowing to derive the output type of a program given types for its relation variables.…”

“…For general motivations for studying type inference for the relational algebra, we refer to Van den Bussche and Waller [7]. In this paper, we will focus on typability: deciding whether a given expression can ever be welldefined; that is, does there exist a type assignment for which the expression is well-typed?…”