2. if L provides unqualified number restrictions, then, for every n E N and every role description R E R , function symbols f> n R and f< n R of arity 0, 3. if L provides qualified number restrictions, then, for every n E N and every role R E R,
unary function symbols f> n R and f< n R ;
4. if L provides nominals, then, for every I E No, a function symbol f, of arity 0, 5.  if L provides feature agreement and disagreement, then, for every pair of feature chains
(u 1 ,u 2 ), two function symbols fu 1 ,u 2 and fu 1 ,tu 2 of arity 0.
For an L -concept description C , let t C denote the representation of C as an L -term, which is defined in the obvious way: concept names A are translated into set variables XA,
the concept constructors ¬, ¬, and ¶ are mapped to ¬, ¬, and V, respectively, and all other concept constructors are translated to the corresponding function symbols.  Obviously, both the sets of function and relation symbols of L may be infinite.

An example of the translation of concept descriptions into terms of an ADL was already given above: the ALCN -concept description A Π ∀ ( R 1 Π R 2 ). ¬ ( B Π ( ≥ 2 R 1 )) corresponds to the term xA A fv(RinR2)(~(xB A f(>2Ri)).

We now define the set of abstract description models M corresponding to the DL L .

For every L -frame, M contains a corresponding ADM.

Definition 12 (Corresponding ADM). Let & = (A&, &) be a frame. The corresponding abstract description model 20 = ( W, F 20 , R 20 ) has domain W := A 3 . The relation symbols of L are just the role descriptions of L , and thus they are interpreted in the frame F . For each relation symbol R E R we can hence define R 201 := R 8 ,
To define F W , we need to define f 2 U for every nullary function symbol f in L , and f 2 1 ( X ) for every unary function symbol f in L and every X C A 1 . Let A be an arbitrary concept name. For each X c A s , let I X be the interpretation based on & mapping the concept name A to X and every other concept name to 0.6 To define f 20 , we make a case distinction according to the type of f:

$$1.\;\;f_{\,\exists R}^{\mathfrak{M}}(X):=(1)$$

1. f H R ( X ) := ( B R A ) T × ,    f H R ( X ) := ( A R A ) T × ,

en R  = (≥ n R) 

$\left({X\over2}\right)\,\cdot-\,\left({1\over2}\right)$

(X):=(>nR.A) Ix, f

$\mathbf{a}=\mathbf{b}$. 
$$\mathbf{\Pi}$$
$\uparrow$
$\mathbf{a}=\lambda\hat{\mathbf{e}}$
(X):=(<nRA) Ix 4. fW := II
= ( u 1 ↓ u 2 ) Z 0 , 
The class of ADMs M thus obtained from a DL L is obviously closed under isomorphic copies since this also holds for the set of L -frames (independently of which DL L we consider). Hence, the tuple S = ( L , M ) corresponding to a DL L is indeed an ADS.

As an example, let us view the DL ALCN n as an ADS. The ADL L corresponding to ALCN n has already been discussed. Thus, we concentrate on the class of ADMs M induced 6. Taking the empty set here is arbitrary.