Algebraic query optimisation for database programming languages

Abstract.

A major challenge still facing the designers and implementors of database programming languages (DBPLs) is that of query optimisation. We investigate algebraic query optimisation techniques for DBPLs in the context of a purely declarative functional language that supports sets as first-class objects. Since the language is computationally complete issues such as non-termination of expressions and construction of infinite data structures can be investigated, whilst its declarative nature allows the issue of side effects to be avoided and a richer set of equivalences to be developed. The language has a well-defined semantics which permits us to reason formally about the properties of expressions, such as their equivalence with other expressions and their termination. The support of a set bulk data type enables much prior work on the optimisation of relational languages to be utilised.

In the paper we first give the syntax of our archetypal DBPL and briefly discuss its semantics. We then define a small but powerful algebra of operators over the set data type, provide some key equivalences for expressions in these operators, and list transformation principles for optimising expressions. Along the way, we identify some caveats to well-known equivalences for non-deductive database languages. We next extend our language with two higher level constructs commonly found in functional DBPLs: set comprehensions and functions with known inverses. Some key equivalences for these constructs are provided, as are transformation principles for expressions in them. Finally, we investigate extending our equivalences for the set operators to the analogous operators over bags. Although developed and formally proved in the context of a functional language, our findings are directly applicable to other DBPLs of similar expressiveness.