Testing bag-containment of conjunctive queries

Abstract.

Under the bag-theoretic semantics relations are bags of tuples, that is, a tuple may have any number of duplicates. Under this semantics, a conjunctive query $Q$ is bag-contained in a conjunctive query $Q^{\prime }$, denoted $Q\leq _bQ^{\prime }$, if for all databases ${\cal D}$, $Q({\cal D})$, the result of applying $Q$ to ${\cal D}$, is a subbag of $Q^{\prime }({\cal D)}$. It is not known whether testing $Q\leq _bQ^{\prime }$ is decidable. In this paper we prove that $Q\leq _bQ^{\prime }$ can be tested on a finite set of canonical databases built from the body of $Q$. Using that result we give a procedure that decides the bag-containment problem of conjunctive queries in a large number of cases.