Restricted Chase Termination: You Want More than Fairness

The chase is a fundamental algorithm with ubiquitous uses in database theory. Given a database and a set of existential rules (aka tuple-generating dependencies), it iteratively extends the database to ensure that the rules are satisfied in a most general way. This process may not terminate, and a major problem is to decide whether it does. This problem has been studied for a large number of chase variants, which differ by the conditions under which a rule is applied to extend the database. Surprisingly, the complexity of the universal termination of the restricted (aka standard) chase is not fully understood. We close this gap by placing universal restricted chase termination in the analytical hierarchy. This higher hardness is due to the fairness condition, and we propose an alternative condition to reduce the hardness of universal termination.