In this paper, we focus on the problem of determining whether two conjunctive (CQ) queries posed on relational data are combined-semantics equivalent [9]. We continue the tradition of [2,5,9] of studying this problem using the tool of containment between queries. We introduce a syntactic necessary and sufficient condition for equivalence of queries belonging to a large natural language of explicit-wave combined-semantics CQ queries; this language encompasses (but is not limited to) all set, bag, and bag-set queries, and appears to cover all combined-semantics CQ queries that are expressible in SQL. Our result solves in the positive the decidability problem of determining combined-semantics equivalence for pairs of explicit-wave CQ queries. That is, for an arbitrary pair of combined-semantics CQ queries, it is decidable (i) to determine whether each of the queries is explicit wave, and (ii) to determine, in case both queries are explicit wave, whether or not they are combined-semantics equivalent, by using our syntactic criterion. (The problem of determining equivalence for general combined-semantics CQ queries remains open. Even so, our syntactic sufficient containment condition could still be used to determine that two general CQ queries are combined-semantics equivalent.) Our equivalence test, as well as our general sufficient condition for containment of combined-semantics CQ queries, reduce correctly to the special cases reported in [2,5] for set, bag, and bag-set semantics. Our containment and equivalence conditions also properly generalize the results of [9], provided that the latter are restricted to the language of (combined-semantics) CQ queries.
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