We present a decision procedure for a quantified fragment of

set theory involving ordered pairs and some operators to manipulate them. When our decision procedure is applied to formulae in this fragment whose quantifier prefixes have length bounded by a fixed constant, it runs in nondeterministic polynomial-time.

Related to this fragment, we also introduce a description logic which

provides an unusually large set of constructs, such as, for instance,

Boolean constructs among roles. The set-theoretic nature of the description logics semantics yields a straightforward reduction of the

knowledge base consistency problem to the satisfiability problem for formulae of our fragment with quantifier prefixes of length at most 2, from which the NP-completeness of reasoning in this novel description logic follows. Finally, we extend this reduction to cope also with SWRL rules.