## Paper: First-Order Queries on Finite Structures Over the Reals (at LICS 1995)

**Jan Paredaens Jan Van den Bussche Dirk Van Gucht**

### Abstract

We investigate properties of finite relational structures over the reals expressed by first-order sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, this corresponds to Boolean real polynomial constraint queries on finite structures. The fact that quantifiers range over all reals seems crucial; however, we observe that each sentence in the first-order theory of the reals can be evaluated by letting each quantifier 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 uniformly on all finite structures by a sentence of which the quantifiers range only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinary query evaluation as usual in finite model theory and databases. Moreover, if only ``generic'' queries are taken into consideration, we show that this can be reduced even further by proving that such queries can be expressed by sentences using as polynomial inequalities only those of the simple form x < y .

### BibTeX

@InProceedings{ParedaensVandenBuss-FirstOrderQuerieson, author = {Jan Paredaens and Jan Van den Bussche and Dirk Van Gucht}, title = {First-Order Queries on Finite Structures Over the Reals}, booktitle = {Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science (LICS 1995)}, year = {1995}, month = {June}, pages = {79--87}, location = {San Diego, CA, USA}, publisher = {IEEE Computer Society Press} }