2. **Expansion** of K′T
: every defined concept (i.e. the first argument of a concept **definition)**
which occurs in the defining term of a concept definition (i.e. the second **argument** of a concept definition) is substituted by its defining term. This process is iterated **until**
there remain only undefined concepts in the second arguments of concept **definitions.**
This yields a **terminology** K′′T
.

3. Expansion of KA: every primitive concept occurring in KA which is **defined** in K′′T
is substituted by its defining **term** in K′′T
. This **yields** K′A.
The transformation has the nice property that K |= α iff K′A |= α
′

, **where** α
′
is **obtained** by replacing every primitive concept occurring in α, which is **defined** in K′′T
, with its **defining**
term in K′′T
. While this allows us to restrict our attention to purely assertional KBs **only,**
it is worth noting that the expansion process can be exponential (Nebel, **1988).**
From (10)–(13), it follows that the above problems can be reduced to the **satisfiability**
problem. There exists a well known technique based on constraint propagation solving **this**
problem (Schmidt-Schauß & Smolka, 1991; Buchheit, Donini, & Schaerf, 1993b; **Donini**
et al., **1994).**
We conclude with an **example.**
Example 1 Consider the following **terminology** KT .

| SportKind       | <·   | ⊤                                                                                                            |
|-----------------|------|--------------------------------------------------------------------------------------------------------------|
|                 |      |                                                                                                              |
| SportTool       | <·   | ⊤                                                                                                            |
|                 |      |                                                                                                              |
| IndividualSport | <·   | SportKind                                                                                                    |
| TeamSport       | <·   | SportKind                                                                                                    |
| Basketball      | <·   | SportTool                                                                                                    |
| TennisRacket    | <·   | SportTool                                                                                                    |
| Basket          | : =  | SportKind⊓ (∃KindOfSport.⊤)⊓ (∀KindOfSport.TeamSport)⊓ (∃HasSportTool.⊤)⊓ (∀HasSportTool.Basketball)         |
| Tennis          | : =  | SportKind⊓ (∃KindOfSport.⊤)⊓ (∀KindOfSport.IndividualSport)⊓ (∃HasSportTool.⊤)⊓ (∀HasSportTool.TennisRacket) |

Suppose that there are two video sequences v1, v2, which are about basket and tennis, respectively. We may represent the semantic content of them **through**

$K_{\nu1}=\{\nu1:$Video $\Gamma$ 3About. Basket$\}$  $K_{\nu2}=\{\nu2:$Video $\Gamma$ 3About. Tennis$\}$.  
Consider K = KT ∪Kv1 ∪Kv2. If we are interested in retrieving videos about sport, we may query K through the query **concept** Q = Video ⊓ ∃About.SportKind and the answer **will** be the list containing both v1 and v2, as K |= v1:Q and K |= v2:Q **hold.**