Bag Semantics Query Containment: The CQ vs. UCQ Case and Other Stories

Query Containment Problem (QCP) is a fundamental decision problem in query processing and optimization. While QCP has for a long time been completely understood for the case of set semantics, decidability of QCP for conjunctive queries under multi-set semantics ($QCP_{\text{CQ}}^{\text{bag}}$) remains one of the most intriguing open problems in database theory. Certain effort has been put, in last 30 years, to solve this problem and some decidable special cases of $QCP_{\text{CQ}}^{\text{bag}}$ were identified, as well as some undecidable extensions, including $QCP_{\text{UCQ}}^{\text{bag}}$. In this paper we introduce a new technique which produces, for a given UCQ $Φ$, a CQ $ϕ$ such that the application of $ϕ$ to a database $D$ is, in some sense, an approximation of the application of $Φ$ to $D$. Using this technique we could analyze the status of $QCP^{\text{bag}}$ when one of the queries in question is a CQ and the other is a UCQ, and we reached conclusions which surprised us a little bit. We also tried to use this technique to translate the known undecidability proof for $QCP_{\text{UCQ}}^{\text{bag}}$ into a proof of undecidability of $QCP_{\text{CQ}}^{\text{bag}}$. And, as you are going to see, we got stopped just one infinitely small $\varepsilon$ before reaching this ultimate goal.