  When reasoning in description, modal or temporal logics it is often useful to
consider axioms representing universal truths in the domain of discourse.
Reasoning with respect to an arbitrary set of axioms is hard, even for
relatively inexpressive logics, and it is essential to deal with such axioms in
an efficient manner if implemented systems are to be effective in real
applications. This is particularly relevant to Description Logics, where
subsumption reasoning with respect to a terminology is a fundamental problem.
Two optimisation techniques that have proved to be particularly effective in
dealing with terminologies are lazy unfolding and absorption. In this paper we
seek to improve our theoretical understanding of these important techniques. We
define a formal framework that allows the techniques to be precisely described,
establish conditions under which they can be safely applied, and prove that,
provided these conditions are respected, subsumption testing algorithms will
still function correctly. These results are used to show that the procedures
used in the FaCT system are correct and, moreover, to show how efficiency can
be significantly improved, while still retaining the guarantee of correctness,
by relaxing the safety conditions for absorption.
