On the first-order expressibility of computing certain answers to conjunctive queries over uncertain databases

A natural way for capturing uncertainty in the relational data model is by having relations that violate their primary key constraint, that is, relations in which distinct tuples agree on the primary key. A repair (or possible world) of a database is then obtained by selecting a maximal number of tuples without ever selecting two distinct tuples that have the same primary key value. For a Boolean query q, CERTAINTY(q) is the problem that takes as input a database db and asks whether q evaluates to true on every repair of db. We are interested in determining queries q for which CERTAINTY(q) is first-order expressible (and hence in the low complexity class AC0). For queries q in the class of conjunctive queries without self-join, we provide a necessary syntactic condition for first-order expressibility of CERTAINTY(q). For acyclic queries, this necessary condition is also a sufficient condition. So we obtain a decision procedure for first-order expressibility of CERTAINTY(q) when q is acyclic and without self-join. We also show that if CERTAINTY(q) is first-order expressible, its first-order definition, commonly called (certain) first-order rewriting, can be constructed in a rather straightforward way.