  In this paper we take closer look at recent developments for the chase
procedure, and provide additional results. Our analysis allows us create a
taxonomy of the chase variations and the properties they satisfy. Two of the
most central problems regarding the chase is termination, and discovery of
restricted classes of sets of dependencies that guarantee termination of the
chase. The search for the restricted classes has been motivated by a fairly
recent result that shows that it is undecidable to determine whether the chase
with a given dependency set will terminate on a given instance. There is a
small dissonance here, since the quest has been for classes of sets of
dependencies guaranteeing termination of the chase on all instances, even
though the latter problem was not known to be undecidable. We resolve the
dissonance in this paper by showing that determining whether the chase with a
given set of dependencies terminates on all instances is coRE-complete. For the
hardness proof we use a reduction from word rewriting systems, thereby also
showing the close connection between the chase and word rewriting. The same
reduction also gives us the aforementioned instance-dependent RE-completeness
result as a byproduct. For one of the restricted classes guaranteeing
termination on all instances, the stratified sets dependencies, we provide new
complexity results for the problem of testing whether a given set of
dependencies belongs to it. These results rectify some previous claims that
have occurred in the literature.
