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We investigate the complexity of the typability problem for the relational algebra. This problem consists of deciding, for a given relational algebra expression, whether there exists an assignment of types to variables occurring in the expression such that the expression is well-typed under the assignment. We obtain that the problem is NP-complete in general. In particular, we show that the problem becomes NP-hard due to (1) the cartesian product operator, (2) the selection operator on arbitrary sets of typed predicates, (3) the selection operator on “well-behaved” sets of typed predicates together with join and projection or renaming. However, the problem is in P when (1) we only allow union, difference, join and selection on “well-behaved” sets of typed predicates, or (2) we allow all operators except cartesian product, where the set of selection predicates can mention at most one base type. Most of these results follow from a close connection of the typability problem to non-uniform constraint satisfaction.

On the complexity of deciding typability in the relational algebra