  Finite chase, or alternatively chase termination, is an important condition
to ensure the decidability of existential rule languages. In the past few
years, a number of rule languages with finite chase have been studied. In this
work, we propose a novel approach for classifying the rule languages with
finite chase. Using this approach, a family of decidable rule languages, which
extend the existing languages with the finite chase property, are naturally
defined. We then study the complexity of these languages. Although all of them
are tractable for data complexity, we show that their combined complexity can
be arbitrarily high. Furthermore, we prove that all the rule languages with
finite chase that extend the weakly acyclic language are of the same
expressiveness as the weakly acyclic one, while rule languages with higher
combined complexity are in general more succinct than those with lower combined
complexity.
