  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
parameterized class of the finite restricted chase, called
$k$-$\mathsf{safe}(\Phi)$. 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. Under consideration in Theory and Practice of Logic Programming
(TPLP).
