Hilbert's tenth problem
Computability with low-dimensional dynamical systems
Theoretical Computer Science
Selected papers of the 9th annual ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
Reachability and connectivity queries in constraint databases
PODS '00 Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Fixed-point query languages for linear constraint databases
PODS '00 Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Linear approximation of planar spatial databases using transitive-closure logic
PODS '00 Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Deciding stability and mortality of piecewise affine dynamical systems
Theoretical Computer Science
The stability of saturated linear dynamical systems is undecidable
Journal of Computer and System Sciences
Constraint Databases
Operational Semantics for Fixed-Point Logics on Constraint Databases
LPAR '01 Proceedings of the Artificial Intelligence on Logic for Programming
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Query Languages for Constraint Databases: First-Order Logic, Fixed-Points, and Convex Hulls
ICDT '01 Proceedings of the 8th International Conference on Database Theory
DBPL '01 Revised Papers from the 8th International Workshop on Database Programming Languages
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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 over the reals. 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 two-dimensional plane, viewed as a binary relation over the reals, is decidable, and even expressible in first-order logic over the reals. Based on this result, we identify a particular transitive-closure logic for which termination of query evaluation is decidable and which is more expressive than first-order logic over the reals. Furthermore, we can define a guarded fragment in which exactly the terminating queries of this language are expressible.