  Description logics are knowledge representation languages that have been
designed to strike a balance between expressivity and computational
tractability. Many different description logics have been developed, and
numerous computational problems for these logics have been studied for their
computational complexity. However, essentially all complexity analyses of
reasoning problems for description logics use the one-dimensional framework of
classical complexity theory. The multi-dimensional framework of parameterized
complexity theory is able to provide a much more detailed image of the
complexity of reasoning problems.
  In this paper we argue that the framework of parameterized complexity has a
lot to offer for the complexity analysis of description logic reasoning
problems---when one takes a progressive and forward-looking view on
parameterized complexity tools. We substantiate our argument by means of three
case studies. The first case study is about the problem of concept
satisfiability for the logic ALC with respect to nearly acyclic TBoxes. The
second case study concerns concept satisfiability for ALC concepts
parameterized by the number of occurrences of union operators and the number of
occurrences of full existential quantification. The third case study offers a
critical look at data complexity results from a parameterized complexity point
of view. These three case studies are representative for the wide range of uses
for parameterized complexity methods for description logic problems.
