FIRST-ORDER QUERIES ON FINITE STRUCTURES OVER THE REALS

We investigate properties of finite relational structures over the rea ls expressed by first-order sentences whose predicates are the relatio ns of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint progr amming terminology, this corresponds to Boolean real polynomial constr aint queries on finite structures. The fact that quantifiers range ove r all reals seems crucial; however, we observe that each sentence in t he first-order theory of the reals can be evaluated by letting each qu antifier range over only a finite set of real numbers without changing its truth value. Inspired by this observation, we then show that when all polynomials used are linear, each query can be expressed uniforml y on all finite structures by a sentence of which the quantifiers rang e only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinar y query evaluation as usual in finite model theory and databases. More over, if only ''generic'' queries are taken into consideration, we sho w that this can be reduced even further by proving that such queries c an be expressed by sentences using as polynomial inequalities only tho se of the simple form chi < y.