Having defined subsumption, equivalence of concept descriptions is defined in the usual way: C ≡ D if and only if C ⊆ D and D ⊆ C . T-equivalence C ≡ t D is specified analogously.

As already mentioned in the introduction, the main difference between partial and total attributes with respect to subsumption is that u ↓ v ⊆ i u ◦ w ↓ v ◦ w holds for all attribute chains u, v, w, whereas it is not necessarily the case that u ↓ v ∈ u ◦ w ↓ v ◦ w, Finally, before introducing the lcs operation formally and concluding this section, we comment on the expressive power of CLASSIC¯, since (syntactically) CLASSIC¯ lacks some common constructors. Although CLASSIC¯, as introduced here, does not contain the bottom concept ⊥ explicitly, it can be expressed by, e.g., (≥ 1 r ) ∩ (≤ 0 r ). We will use ⊥ as an abbreviation for inconsistent concept descriptions.    Furthermore, primitive negation, i.e.,
nnegation of concept names, can be simulated by number restrictions. For a concept name E one can replace every occurrence of E by ( ≥ 1 r E ) and the negation ¬ E of E by ( ≤ 0 r E )
where rE is a new role name. Finally, for an attribute a the following equivalences hold:
(> n a) ≡ ⊥ for n ≥ 2; (> 1 a) ≡ (a ↓ a); (> 0 a) ≡ T; (≤ n a) ≡ T for n ≥ 1; and
( < 0 a )  ≡ ( Va. ). These show that we do not lose any expressive power by not allowing for number restrictions on attributes. Still, full CLASSIC is somewhat more expressive than CLASSIC¯. This is mainly due to the introduction of individuals (also called nominals) in CLASSIC. For the sake of completeness we give the syntax of the full CLASSIC language. 3 This requires a further set, O , representing the set of individual names.  Then we can define two additional concept constructors
- {e1,...,em}, for individuals ei E O (enumeration as in {Fall, Summer, Spring}) - p : e for a role or attribute p , and an individual e ( fills as in currentSeason : Summer ).

In a technical report, Küsters and Borgida (1999) extend some of the results presented in this work to full CLASSIC, in the case when individuals have a non-standard semantics.

The least common subsumer of a set of concept descriptions is the most specific concept subsuming all concept descriptions of the set:
Definition 4 The concept description D is the least common subsumer (lcs) of the concept descriptions C 1 , . . , C n (lcs(C 1 , . . , C n ) for short) iff i ) C i E D for all i = 1 , . . , n and ii)
for every D' with that property D ⊆ D'. Analogously, we define lcst (C 1 , . . , C n ) using ⊆ t instead of E
Note that the lcs of concept descriptions may not exist, but if it does, by definition it is uniquely determined up to equivalence. In this sense, we may refer to the lcs.

In the following two sections, attributes are always interpreted as partial functions; only in Section 5 do we consider total attributes.

## 3. Characterizing Subsumption In Classic-

In this section we modify the characterization of t-subsumption for CLASSIC, as proposed by Borgida and Patel-Schneider (1994), to handle the case of partial attributes. We do