  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 family, 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.
