Dedicated to the memory of Paris C. Kanellakis The expressive power of first-order query languages with several classes of equality and inequality constraints is studied in this paper. We settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals. Furthermore, noting that relational queries exhibit several forms of genericity, we establish a number of collapse results of the following form: The class of generic boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus (with or without an order relation). We prove such results for both the natural and active-domain semantics. As a consequence, the relational calculus augmented with polynomial inequalities expresses the same classes of generic boolean queries under both the natural and active-domain semantics. In the course of proving these results for the active-domain semantics, we establish Ramsey-type theorems saying that any query involving certain kinds of constraints coincides with a constraintfree query on databases whose elements come from a certain infinite subset of the domain. To
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