Definition 7. An abstract description system (ADS) is a pair ( L, M ), where L is an ADL
and M is a class of ADMs for L that is closed under isomorphic copies. 5 From the DL point of view, the choice of the class M defines the semantics of the concept and role constructors, and it allows us, e.g., to incorporate restrictions on role interpretations. In this sense, the ADS can be viewed as determining a (description) logic.

To be more concrete, in a DL interpretation the interpretation of the function symbols is determined by the interpretation of the role names. Thus one can, for example, restrict the class of models to ADMs that interpret a certain role as a transitive relation or as the composition of two other roles. Another restriction that can be realized by the choice of M is that nominals (corresponding to nullary function symbols) must be interpreted as singleton sets.

Let us now define reasoning problems for abstract description systems.  We will introduce sssees to the senses of the control of the control and the control consistency of ABoxes (with or without GCIs), and satisfiability of terms (with or without term assertions), which corresponds to satisfiability of concept descriptions (with or without GIS). Definition 8. Given an ADS (L,M), a finite set of assertions Γ is called satisfiable in
(L, M) iff there exists an ADM W E M and an assignment A for W such that (20, A)
satisfies all assertions in Γ . The term t is called satisfiable in ( L , M ) iff { a : t } is satisfiable in (L, M), where a is an arbitrary object variable.

- The satisfiability problem for ( L , M ) is concerned with the following question: given a finite set of object assertions Γ of L is Γ is satisfiable in ( L , M ).

- The relativized satisfiability problem for (L,M) is concerned with the following question: given a finite set of assertions Γ of L is Γ is satisfiable in ( L , M ).

- The term satisfiability problem for ( L, M ) is concerned with the following question:
given a term t of L, is t satisfiable in (L, M).

- The relativized term satisfiability problem for ( L , M ) is concerned with the following question: given a term t and a set of term assertions Γ of L , is { a : t } ∪ Γ satisfiable in (L, M).
In the next section, we will define the fusion of two ADSs, and show that (relativized)
stisfiability is decidable in the fusion if (relativized) satisfiability in the component ADSs is decidable. For these transfer results to hold, we must restrict ourselves to so-called local ADS. Definition 9. Given a family ( Up ) pEP of ADMs Up = ( Wp, F Wp, R Wp ) over pairwise disjoint domains W p , we say that 2D = (W, F 2D , R 2D ) is the disjoint union of (W p ) p EP iff
- W = UpEp Wp,