Functional Database Query Languages as Typed Lambda Calculi of Fixed Order

We present functional database query languages expressing the FO- and PTIME-queries. This framework is a functional analogue of the logical languages of first-order and fixpoint formulas over finite structures, and its formulas consist of: atomic constants of order 0, equality among these constants, variables, application, lambda and let abstraction; all typable in 4 functionality order. In this framework, proposed in [25] for arbitrary functionality order, typed lambda terms are used for input-output databases and for query program syntax, and reduction is used for query program semantics. We define two families of languages: TLI = i or simply-typed list iteration of order i + 3 with equality and MLI = i or ML-typed list iteration of order i + 3 with equality; we use i + 3 since our list representation of input-output databases requires at least order 3. We show that, over list-represented databases, both TLI = 0 and MLI = 0 exactly express the FO-queries and both TLI = 1 and ..