Uniform Quanti er Elimination and Constraint Query

In this paper we introduce a variant of the quantiier elimination problem for the rst order theory of real closed elds. Instead of considering a single quantiied formula, we consider a uniform sequence of such formulas and eliminate quantiiers to obtain another uniform sequence. Our immediate motivation comes from a problem in the theory of constraint databases with real polynomial constraints. Using the uniform quantiier elimination algorithm, we give an algorithm for converting a query with natural domain semantics to an equivalent one with active domain semantics. A non-constructive version of this result was proved in 2]. Very recently, a constructive proof was also given independently in 5]. In the last part of the paper, using some elementary tools from rst order logic and results described above, we prove that parity is not expressible in the constraint query language over the reals. This was proved by Benedikt et al 2]. However unlike ours, their proof uses diicult techniques from non-standard analysis and model theory of ordered structures.