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 CQ bag ) remains one of the most intriguing open problems in database theory. Certain effort has been put, in the last 30 years, to solve this problem and some decidable special cases of QCP CQ bag were identified, as well as some undecidable extensions, including QCP UCQ 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 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 UCQ bag into a proof of undecidability of QCP CQ bag . And, as you are going to see, we got stopped just one infinitely small ε before reaching this ultimate goal.