Definition 5. Let L be an ADL as in Definition 4. An abstract description model (ADM)
for L is of the form

$$\mathfrak{M}=\left\langle W,{\mathcal{F}}^{\mathfrak{M}}=\{f^{\mathfrak{M}}\mid f\in{\mathcal{F}}\},{\mathcal{R}}^{\mathfrak{M}}=\{R^{\mathfrak{M}}\mid R\in{\mathcal{R}}\}\right\rangle,$$

where W is a nonempty set, the f 20 are functions mapping every sequence 〈 X 1 , . . , X n , 〉
of subsets of W to a subset of W t and the R 2 p are binary relations on W .

Since ADMs do not interpret variables, we need an assignment that assigns a subset of W to each set variable, before we can evaluate terms in an ADM. To evaluate object assertions, we need an additional assignment that assigns an element of W to each object variable.

Definition 6. Let L be an ADL and M = (W, FM, RM) be an ADM for L. An assignment for 20 is a pair A = (A1, A2) such that A1 is a mapping from the set of set variables V into 2W, and A 2 is an injective 4 mapping from the set of object variables X into W. Let W be an ADM and A = ( A 1 , A 2 ) be an assignment for M. With each L -term t , we inductively associate a value t 2 , A in 2 W as follows:
- x 20,A := A 1 (x) for all variables x E V,

* $(\neg t)^{\mathfrak{M},\mathcal{A}}:=W\setminus(t)^{\mathfrak{M},\mathcal{A}}$, $(t_{1}\wedge t_{2})^{\mathfrak{M},\mathcal{A}}:=t_{1}^{\mathfrak{M},\mathcal{A}}\cap t_{2}^{\mathfrak{M},\mathcal{A}}$, $(t_{1}\lor t_{2})^{\mathfrak{M},\mathcal{A}}:=t_{1}^{\mathfrak{M},\mathcal{A}}\cup t_{2}^{\mathfrak{M},\mathcal{A}}$, * $f(t_{1},\ldots,t_{n_{f}})^{\mathfrak{M},\mathcal{A}}:=f^{\mathfrak{M}}(t_{1}^{\mathfrak{M},\mathcal{A}},\ldots,t_{n_{f}}^{\mathfrak{M},\mathcal{A}})$.  
If x 1 , . . , x n are the set variables occurring in t, then we often write t 2 ( X 1 , . . . , X n ) as shorthand for t 20 , A , where A is an assignment with x ; A = X i for 1 < i < n .

The truth-relation = between (20, A) and assertions is defined as follows:
- ( W, A ) |= R(a, b ) iff A 2 (a)R 2WA 2 (b),
- (20, A) |= a : t iff A2(a) E t20,A,
- (20, A) |= t 1 In this case we say that the assertion is satisfied in ( 20 , A ). If, for an ADM 20 and a set of assertions Γ there exists an assignment A for 20 such that each assertion in Γ is satisfied in (20, A), then 20 is a model for F
There are two differences between ADMs and DL interpretations. First, in a DL interpretation, the interpretation of the role names fixes the interpretation of the function symbols corresponding to concept constructors that involve roles (like value restrictions, nmber restrictions, etc.). The interpretation of the concept names corresponds to an assignment. Thus, a DL model is an ADM together with an assignment, whereas an ADM alone corresponds to what is called frame in modal logics. Second, in DL the roles used in concept constructors may, of course, also occur in role assertions. In contrast, the definition of ADMs per se does not enforce any connection between the interpretation of the function symbols and the interpretation of the relation symbols. Such connections can, however, be enforced by restricting the attention to a subclass of all possible ADMs for the ADL.

4. This corresponds to the unique name assumption.