  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.
