allow for constructors that are not considered in modal logics (e.g., the same-as constructor mentioned above). Second, some DL constructors that have been considered in modal logics, such as qualified number restrictions ( ≤ nR.C ), ( ≥ nR.C ) (Hollunder & Baader, 1991), which correspond to graded modalities (Van der Hoek & de Rijke, 1995), can easily be shown to be non-normal. Third, the transfer results for decidability are concerned with the stisfiability problem (with or without general inclusion axioms). ABoxes and the related inference problems are not considered. ABoxes can be simulated in modal logics allowing for so-called nominals, i.e., names for individuals, within formulae (Prior, 1967; Gargov
& Goranko, 1993; Areces, Blackburn, & Marx, 2000). However, as we will see below, the general transfer results do not apply to modal logics with nominals.

The purpose of this paper is to clarify for which DLs decidability of the component DLs transfers to their fusion. To this purpose, we introduceso-called abstract description systems
(ADSs), which can be seen as a common generalization of description and modal logics.

We define the fusion of ADSs, and state four theorems that say under which conditions decidability transfers from the component ADSs to their fusion. Two of these theorems are concerned with inference w.r.t. general concept inclusion axioms and two with inference without TBox axioms. In both cases, we first formulate and prove the results for the consistency problem of ABoxes (more precisely, the corresponding problem for ADSs) and then establish analogous results for the satisfiability problem of concepts.

From the DL point of view, the four theorems shown in this paper are concerned with the following four decision problems:
(i) decidability of consistency of ABoxes w.r.t. TBox axioms (Theorem 17);
(ii) decidability of satisfiability of concepts w.r.t. TBox axioms; (Corollary 22);
(iii) decidability of consistency of ABoxes without TBox axioms (Theorem 29); and (iv) decidability of satisfiability of concepts without TBox axioms (Corollary 34).

These theorems imply that decidability of the consistency problem and the satisfiability problem transfers to the fusion for most DLs considered in the literature. The main exceptions (which do not satisfy the prerequisites of the theorems) are
(a) DLs that are not propositionally closed, i.e., do not contain all Boolean connectives;

(b) DLs allowing for individuals (called nominals in modal logic) in concept descriptions; and
(c) DLs explicitly allowing for the universal role or for negation of roles.
Results from modal logic for problem (iv) usually require the component modal logics to be normal. Our Theorem 29 is less restrictive, and thus also applies to DLs allowing for constructors like qualified number restrictions.

## 2. Description Logics

Before defining abstract description systems in the next section, we introduce the main features of DLs that must be covered by this definition. To this purpose, we first introduce