Deciding Termination of Query Evaluation in Transitive-Closure Logics for Constraint Databases

We study extensions of first-order logic over the reals with different types of transitive-closure operators as query languages for constraint databases that can be described by Boolean combinations of polynomial inequalities. We are in particular interested in deciding the termination of the evaluation of queries expressible in these transitive-closure logics. It turns out that termination is undecidable in general. However, we show that the termination of the transitive closure of a continuous function graph in the twodimensional plane is decidable, and even expressible in first-order logic over the reals. Based on this result, we identify a particular transitiveclosure logic for which termination of query evaluation is decidable and which is more expressive than first-order logic. Furthermore, we can define a guarded fragment in which exactly the terminating queries of this language are expressible.