Table 2: Fuzzy DL semantics for ALC.

$$\begin{array}{c}{{\bot^{\mathcal{I}}(x)}}\\ {{\top^{\mathcal{I}}(x)}}\\ {{(C\cap D)^{\mathcal{I}}(x)}}\\ {{(C\sqcup D)^{\mathcal{I}}(x)}}\\ {{(\neg C)^{\mathcal{I}}(x)}}\\ {{(\forall R.C)^{\mathcal{I}}(x)}}\\ {{(\exists R.C)^{\mathcal{I}}(x)}}\end{array}$$
I(x**) = 0**
⊤I(x**) = 1**
I(x) = C
I(x) = C
I(x) = ⊖C
$$\begin{array}{r l}{{=}}&{{0}}\\ {{=}}&{{1}}\\ {{=}}&{{C^{\mathcal{I}}(x)\otimes D^{\mathcal{I}}(x)}}\\ {{=}}&{{C^{\mathcal{I}}(x)\oplus D^{\mathcal{I}}(x)}}\\ {{=}}&{{\ominus C^{\mathcal{I}}(x)}}\\ {{=}}&{{\operatorname*{inf}_{y\in\Delta^{\mathcal{I}}}(R^{\mathcal{I}}(x,y)\Rightarrow C^{\mathcal{I}}(y))}}\\ {{=}}&{{\operatorname*{sup}_{y\in\Delta^{\mathcal{I}}}(R^{\mathcal{I}}(x,y)\otimes C^{\mathcal{I}}(y))}}\end{array}$$

1. to each atomic concept A a function AI: ∆I → [0, 1],
2. to each role R a function RI: ∆I × ∆I → [0, 1],
3. to each individual a **an element** a I ∈ ∆I.

The fuzzy interpretation function is extended to complex concepts as specified in the Table 2 (where x, y ∈ ∆I **are elements of the domain, and as usual we**
use ⊗, ⊕, ⊖ and ⇒ to denote respectively the t-norm, t-conorm, negation function and implication function of the corresponding family of fuzzy operators chosen). Hence, for every complex concept C **we get a function** C
I: ∆I −→ [0, 1].

Note that ha:C ≤ αi is equivalent to ha:¬C ≥ 1 − αi **under Lukasiewicz**
negation. Hence, in Lukasiewicz it is not necessary to explicitly consider fuzzy concept assertions of the form ha:C ≤ αi.

The (crisp) satisfiability of axioms in a fuzzy KB **is then defined by the**
following conditions:
1. I satisfies an axiom ha:C ≥ αi **in case that** C
I(a I) ≥ α, 2. I satisfies an axiom ha:C ≤ αi **in case that** C
I(a I) ≤ α, 3. I satisfies an axiom h(a, b):R ≥ αi **in case that** RI(a I, bI) ≥ α, 4. I satisfies an axiom hC ⊑ D ≥ αi **in case that (**C ⊑ D)
I ≥ α **where**
(C ⊑ D)
I **= inf**x∈∆I (C
I(x) ⇒ DI(x)).

It is interesting to point out that the satisfaction of a GCI of the form hC ⊑ D ≥ 1i is exactly the requirement that ∀x ∈ ∆I, CI(x) ≤ DI(x**) (i.e.,**
Zadeh's set inclusion); hence, in this particular case for the satisfaction it only matters the partial order and not the exact value of the implication ⇒.

As it is expected we will say that a fuzzy interpretation I **satisfies a KB** K
in case that it satisfies all axioms in K**. And it is said that a fuzzy KB** K is satisfiable iff there exist a fuzzy interpretation I **satisfying every axiom in** K.

As it is said in the introduction, here we mainly focus on witnessed models and compare them to finite models. This notion (see [11]) corresponds to the restriction to the DL language of the notion of witnessed model introduced, in