  The standard reasoning problem, concept satisfiability, in the basic
description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the
presence of unrestricted axioms. Several fragments of ALC, notably logics in
the FL, EL, and DL-Lite families, have an easier satisfiability problem;
sometimes it is even tractable. We classify the complexity of the standard
satisfiability problems for all possible Boolean and quantifier fragments of
ALC in the presence of general axioms.
