Space Usage in Functional Query Languages

We consider evaluation strategies for database queries expressed in three functional query languages: the complex value algebra, the simply typed lambda calculus, and method schemas. Each of these query languages derives its expressive power from a different primitive: the complex value algebra from the powerset operator, the simply typed lambda calculus from list iteration, and method schemas from recursion. We show that “natural” evaluation strategies for these primitives may lead to very inefficient space usage, but that with some simple optimizations many queries can be evaluated with little or no space overhead. In particular, we show. (1) In the complex value algebra, all expressions with set nesting depth at most 2 can be evaluated in PSPACE, and this set of expressions is sufficient to express all queries in the polynomial hierarchy; (2) In the simply typed lambda calculus with equality and constants, all query terms of order at most 5 (where “query term” is a syntactic condition on types) can be evaluated in PSPACE, and this set of terms expresses exactly the PSPACE queries; (3) There exists a set of second-order method schemas (with no simple syntactic characterization) that can be evaluated in PSPACE, and this set of schemas is sufficient to express all PSPACE queries.