Subclasses of Presburger arithmetic and the polynomial-time hierarchy
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We consider the data complexity of various logics on two important classes of constraint databases: dense order and linear constraint databases. For dense order databases, we present a general result allowing us to lift results on logics capturing complexity classes from the class of finite ordered databases to dense order constraint databases. Considering linear constraints, we show that there is a significant gap between the data complexity of first-order queries on linear constraint databases over the real and the natural numbers. This is done by proving that for arbitrary high levels of the Presburger arithmetic there are complete first-order queries on databases over (N,