Safe constraint queries

We cxlend some of the classical characterization theorems of the relational theory particularly those related to query safety to the context where database elements come with fixed interpreted structure, and where formulae over elements of that structure can be used in queries. We show that the addition of common interpreted functions such as real addition and multiplication to the relational calculus preserves important characterization theorems of the relational calculus, and also preserves certain combinatorial properlies of queries. Our main result of the first kind is that there is a syntactic characterization of the collection of safe queries over the relational calculus supplemented by a wide class of interpreted functions a class that includes addition, multiplication, and exponcntiation and that this characterization gives us an interpreted analog of the concept of range-restricted query from the uninterpreted setting. Furthermore, our range-restricted queries are particularly intuitive for the relational calculus with real arithmetic, and give a natural syntax for safe queries in the presence of polynomial functions, We use these characterizations to show that safety is decidable for Boolean combinations of conjuncLive queries for a large class of interpreted structures. WC show a dichotomy theorem that sets a polynomial bound on the growth of the output of a query that might refer to addition, multiplication and exponentiation. We apply the above results for finite databases to get results on finitely representable databases. We obtain syntactic characterizations of the queries on finitely rcprcscntablc databases that preserve certain geometric conditions, such as being convex polytopes, polyhedra, and compact semi-linear sets. The latter corresponds to many spatial applications. We show how to give an cffectivc syntax to safe queries, and prove decidability of preservation properties for conjunctive queries.