86
A. Kerber and R. Bruggemann
and we evaluate A ∈ L A implies B ∈ L A in E by:
τ
∗ (A ⇒ B) =
o∈O
τ
∗
A
(o), B
(o)
.
A ⇒ B holds in E if and only if τ * (A ⇒ B) = 1, i.e., iff A ’ ⊆ L B ’ . Defining
pseudo-contents (Ganter and Wille 1996), by
P == P
" and for each pseudo − content Q ⊂ L P : Q
"
⊆ L P,
we get the Duquenne/Guigues-basis (Duquenne 1987) which implies every attribute
implication following from E,
P =
P ⇒
P
"
\P
| P pseudo − content
.
1.4 Example
Adding substructures, Cl-, F-, Br-, I-atoms, and using simplified binary parameters
nODP ∗ , nGWP ∗ , nALT ∗ , ... , we obtain for an arbitrary subset of refrigerants (see
for the complete set (Restrepo 2008)) in order not to get too huge outputs:
E nODP ∗ nGWP ∗ nALT ∗ nC Cl F Br I ether CO 2 NH 3
1
1
0
0
0
1
1 0
0 0
0
0
2
0
1
0
0
1
1 0
0 0
0
0
6
0
0
0
1
1
1 0
0 0
0
0
7
0
0
0
1
1
1 0
0 0
0
0
8
0
1
1
0
0
1 0
0 0
0
0
16 0
0
0
1
0
0 0
0 0
0
0
21 0
0
0
0
0
0 0
0 0
1
0
22 1
0
0
0
1
1 1
0 0
0
0
23 0
1
1
1
0
1 0
0 0
0
0
29 0
1
1
1
0
1 0
0 1
0
0
32 0
0
0
0
1
0 0
0 0
0
0
33 1
0
0
1
1
1 0
0 0
0
0
35 1
0
1
1
1
1 0
0 0
0
0
36 0
0
0
0
0
1 0
1 0
0
0
37 0
0
0
1
0
0 0
0 1
0
0
38 0
0
0
0
0
0 0
0 0
0
1
39 0
0
0
1
0
1 0
0 1
0
0
40 0
0
0
1
0
1 0
0 1
0
0
The Duquenne/Guigues basis of it yields all what follows, it can be obtained
online, using CONEXP–1.3 (Yevtushenko 2000). We find the implications