The chase procedure for existential rules is an indispensable tool for several database applications, where its termination guarantees the decidability of these tasks. Most previous studies have focused on the skolem chase variant and its termination analysis. It is known that the restricted chase variant is a more powerful tool in termination analysis provided a database is given. But all-instance termination presents a challenge since the critical database and similar techniques do not work. In this paper, we develop a novel technique to characterize the activeness of all possible cycles of a certain length for the restricted chase, which leads to the formulation of a framework of parameterized classes of the finite restricted chase, called $k$-$\mathsf{safe}(\Phi)$ rule sets. This approach applies to any class of finite skolem chase identified with a condition of acyclicity. More generally, we show that the approach can be applied to the hierarchy of bounded rule sets previously only defined for the skolem chase. Experiments on a collection of ontologies from the web show the applicability of the proposed methods on real-world ontologies.

We present trichotomy results characterizing the complexity of reasoning with disjunctive logic programs. To this end, we introduce a certain definition schema for classes of programs based on a set of allowed arities of rules. We show that each such class of programs has a finite representation, and for each of the classes definable in the schema, we characterize the complexity of the existence of an answer set problem. Next, we derive similar characterizations of the complexity of skeptical and credulous reasoning with disjunctive logic programs. Such results are of potential interest. On the one hand, they reveal some reasons responsible for the hardness of computing answer sets. On the other hand, they identify classes of problem instances, for which the problem is “easy” (in P) or “easier than in general” (in NP). We obtain similar results for the complexity of reasoning with disjunctive programs under the supported-model semantics.