Chase Termination for Guarded Existential Rules

The chase procedure is considered as one of the most fundamen- tal algorithmic tools in database theory. It has been successfully applied to different database problems such as data exchange, and query answering and containment under constraints, to name a few. One of the central problems regarding the chase procedure is all- instance termination, that is, given a set of tuple-generating depen- dencies (TGDs) (a.k.a. existential rules), decide whether the chase under that set terminates, for every input database. It is well-known that this problem is undecidable, no matter which version of the chase we consider. The crucial question that comes up is whether existing restricted classes of TGDs, proposed in different contexts such as ontological query answering, make the above problem de- cidable. In this work, we focus our attention on the oblivious and the semi-oblivious versions of the chase procedure, and we give a positive answer for classes of TGDs that are based on the notion of guardedness. To the best of our knowledge, this is the first work that establishes positive results about the (semi-)oblivious chase termination problem. In particular, we first concentrate on the class of linear TGDs, and we syntactically characterize, via rich- and weak-acyclicity, its fragments that guarantee the termination of the oblivious and the semi-oblivious chase, respectively. Those syn- tactic characterizations, apart from being interesting in their own right, allow us to pinpoint the complexity of the problem, which is PSPACE-complete in general, and NL-complete if we focus on predicates of bounded arity, for both the oblivious and the semi- oblivious chase. We then proceed with the more general classes of guarded and weakly-guarded TGDs. Although we do not provide syntactic characterizations for its relevant fragments, as for linear TGDs, we show that the problem under consideration remains de- cidable. In fact, we show that it is 2EXPTIME-complete in general, and EXPTIME-complete if we focus on predicates of bounded arity, for both the oblivious and the semi-oblivious chase. Finally, we investigate the expressive power of the query languages obtained from our analysis, and we show that they are equally expressive with standard database query languages. Nevertheless, we have strong indications that they are more succinct.