Equivalences Among Aggregate Queries with Negation
  Query equivalence is investigated for disjunctive aggregate queries with negated subgoals, constants and comparisons. A full characterization of equivalence is given for the aggregation functions count, max, sum, prod, toptwo and parity. A related problem is that of determining, for a given natural number N, whether two given queries are equivalent over all databases with at most N constants. We call this problem bounded equivalence. A complete characterization of decidability of bounded equivalence is given. In particular, it is shown that this problem is decidable for all the above aggregation functions as well as for count distinct and average. For quasilinear queries (i.e., queries where predicates that occur positively are not repeated) it is shown that equivalence can be decided in polynomial time for the aggregation functions count, max, sum, parity, prod, toptwo and average. A similar result holds for count distinct provided that a few additional conditions hold. The results are couched in terms of abstract characteristics of aggregation functions, and new proof techniques are used. Finally, the results above also imply that equivalence, under bag-set semantics, is decidable for non-aggregate queries with negation. 
Sara Cohen, Werner Nutt, Yehoshua Sagiv