having exacily two children can be represented by the concept description

## Woman ∩ ( ≤ 2Child ) ∩ ( ≥ 2Child ).

The knowledge base of a DL system consists of a terminological component (TBox) and an assertional component (ABox). In its simplest form, the TBox consists of concept definitions, which assign names (abbreviations) to complex descriptions.  More general TBox formalisms allow for so-called general concept inclusion axioms (GCIs) between complex descriptions. For example, the concept inclusion

## Human   (≥ 3Child)

states that people having at least three ildren are entitled to a tax break. The ABox formalism consists of concept assertions (stating that an individual belongs to a concept)
and role assertions (stating that two individuals are related by a role). For example, the assertions Woman(MARY), child(MARY, TOM), Human(TOM) state that Mary is a woman, who has a child, Tom, who is a human.

DL systems provide their users with various inference capabilities that allow them to deduce implicit knowledge from the explicitly represented knowledge. For instance, the subsumption problem is concerned with subconcept-superconcept relationships: C is subsumed by D ( C E D ) if, and only if, all instances of C are also instances of D , i.e., the first description is always interpreted as a subset of the second description. For example, the concept description Woman obviously subsumes the concept description Woman Π ∀ child.Woman. The concept description C is satisfiable iff it is non-contradictory, i.e., it can be interpreted by a nonempty set. In DLs allowing for conjunction and negation of concepts, subsumption can be reduced to (un)satisfiability: C E D iff C Π ¬ D is unsatisfiable. The instance checking problem consists of deciding whether a given individual is an instance of a given concept.  For example, w.r.t. the assertions from above, MARY is an instance of the concept description Woman □ ∃ child . Human. The ABox A is consistent iff it is non-contradictory, i.e., it has a model. In DLs allowing for negation of concepts, the instance problem can be reduced to (in )consistency of ABoxes:  is an instance of C w.r.t. the ABox A iff AU{¬C(i)}
is inconsistent.

In order to ensure a reasonable and predictable behavior of a DL system, reasoning in the DL employed by the system should at least be decidable, and preferably of low complexity. Consequently, the expressive power of the DL in question must be restricted in an appropriate way. If the imposed restrictions are too severe, however, then the important notions of the application domain can no longer be expressed. Investigating this trade-off between the expressivity of DLs and the complexity of their inference problems has thus been one of the most important issues in DL research (see, e.g., Levesque & Brachman, 1987; Nebel, 1988; Schmidt-Schauß, 1989; Schmidt-Schauß & Smolka, 1991; Nebel, 1990; Donini, Lenzerini, Nardi, & Nutt, 1991, 1997; Donini, Hollunder, Lenzerini, Spaccamela, Nardi, & Nutt, 1992; Schaerf, 1993; Donini, Lenzerini, Nardi, & Schaerf, 1994; De Giacomo & Lenzerini, 1994a, 1994b, 1995; Calvanese, De Giacomo, & Lenzerini, 1999; Lutz, 1999; Horrocks, Sattler, & Tobies, 2000).

This paper investigates an approach for extending the expressivity of DLs that (in many cases) guarantees that reasoning remains decidable: the fusion of DLs. In order to explain