  The subsumption problem with respect to terminologies in the description
logic ALC is EXPTIME-complete. We investigate the computational complexity of
fragments of this problem by means of allowed Boolean operators. Hereto we make
use of the notion of clones in the context of Post's lattice. Furthermore we
consider all four possible quantifier combinations for each fragment
parameterized by a clone. We will see that depending on what quantifiers are
available the classification will be either tripartite or a quartering.
