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.
All these fragments restrict the use of Boolean operators in one way or another.
We look at systematic and more general restrictions
of the Boolean operators and establish the complexity of the concept satisfiability problem
in the presence of axioms. We separate tractable from intractable cases.