Given this semantics, we can now formally define the relevant inference problems.

Definition 3 (Inferences). Let C and D be concept descriptions, i an individual name, T a TBox, and A an ABox. We say that C subsumes D relative to the TBox T (D ET C)
iff D X ⊂ C X for all models I of T . The concept description C is satisfiable relative to the TBox T iff there exists a model I of T such that C T ̸ = ∅ . The individual i is an instance of C in the ABox A relative to the TBox T iff i T E CI for all models of A relative to T . The ABox A is consistent relative to the TBox T iff there exists a model of A relative to T.

These three inferences can also be considered without reference to a TBox: C subsumes D ( C is satisfiable) iff C subsumes D ( C is satisfiable) relative to the empty TBox, and i is an instance of C in A (A is consistent) iff i is an instance of C in A (A is consistent)
relative to the empty TBox.

We restrict our attention to DLs that are propositionally closed (i.e., allow for the Boolean operators conjunction, disjunction, and negation).  Consequently, subsumption can be reduced to (un)satisfiability since C E T D iff C n ¬ D is unsatisfiable relative to T . Conversely, (un)satisfiability can be reduced to subsumption since C is unsatisfiable relative to T iff C E T L . For this reason, it is irrelevant whether we consider the subsumption or the stisfiability problem in our results concerning the transfer of decidability of these problems from component DLs to their fusion (informally called transfer results in the following).

Similarly, the instance problem can be reduced to the (in)consistency problem and vice versa: i is an instance of C in A relative to T iff A U {¬C(i)} is inconsistent relative to T ; and A is inconsistent relative to T iff i is an instance of ⊥ in A relative to T , where i is an arbitrary individual name. Consequently, it is irrelevant whether we consider the instance problem or the consistency problem in our transfer results.

Finally, the satisfiability problem can be reduced to the consistency problem: C is satisfiable relative to T iff the ABox {C(i)} is consistent relative to T, where i is an arbitrary individual name. However, the converse need not be true. It should be obvious that this implies that a transfer result for the satisfiability problem does not yield the corresponding transfer result for the consistency problem: from decidability of the consistency problem for the component DLs we can only deduce decidability of the satisfiability problem in their fusion. What might be less obvious is that a transfer result for the consistency problem need not imply the corresponding transfer result for the satisfiability problem: if the satisfiability problems in the component DLs are decidable, then the transfer result for the consistency problem can just not be applied (since the prerequisite of this transfer result, namely, decidability of the consistency problem in the component DLs, need not be satisfied). However, we will show that the method used to show the transfer result for the consistency problem also applies to the satisfiability problem.

## 2.1  More Expressive Dls

There are several possibilities for extending ALC in order to obtain a more expressive DL.

The three most prominent are adding additional concept constructors, adding role constructors, and formulating restrictions on role interpretations. In addition to giving examples for such extensions, we also introduce a naming scheme for the obtained DLs. Additional concept constructors are indicated by appending caligraphic letters to the language name, role constructors by symbols in superscript, and restrictions on roles by letters in subscript.