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Equivalcncc of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators min, max, count, count-distinct, and sum. Essentlally, this class contains all unnested SQL queries with the above aggregate operators, with a WHERE clause consisting of a conjunction of comparisons, and without a HAVING clause. The comparisons can be interpreted over either a dense order (c,g., over the rationals) or a discrete order (e.g., over the integers). Generally, however, different techniques and characterizations are needed in each of these two case-s. For queries with either max or min, equivalence is characterlzcd in terms of dominance mappings, which can be viewed aa a goncralization of containment mappings. For queries with the count-distinct operator, a sufficient condition for cquivalcncc is given in terms of equivalence of conjunctive qucrics under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism, This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable with polynomial spncc, Finally, it is shown that all the characterizations for min-, max-, count-, and sum-queries yield polynomial-time algorithms for linear queries, i.e., queries with no repeated prcdicatcs in their bodies.

Deciding equivalences among aggregate queries