=Paper=
{{Paper
|id=None
|storemode=property
|title=Attribute Exploration with Fuzzy Attributes and Background Knowledge
|pdfUrl=https://ceur-ws.org/Vol-1062/paper6.pdf
|volume=Vol-1062
|dblpUrl=https://dblp.org/rec/conf/cla/Glodeanu13
}}
==Attribute Exploration with Fuzzy Attributes and Background Knowledge==
https://ceur-ws.org/Vol-1062/paper6.pdf
Attribute exploration with fuzzy attributes and
background knowledge
Cynthia Vera Glodeanu
Technische Universität Dresden,
01062 Dresden, Germany
Cynthia-Vera.Glodeanu@tu-dresden.de
Abstract. Attribute exploration is a formal concept analytical tool for
knowledge discovery by interactive determination of the implications
holding between a given set of attributes. The corresponding algorithm
queries the user in an efficient way about the implications between the
attributes. The result of the exploration process is a representative set
of examples for the entire theory and a set of implications from which all
implications that hold between the considered attributes can be deduced.
The method was successfully applied in different real-life applications for
discrete data. In many instances, the user may know some implications
before the exploration starts. These are considered as background knowl-
edge and their usage shortens the exploration process. In this paper we
show that the handling of background information can be generalised to
the fuzzy setting.
Keywords: Knowledge discovery, Formal Concept Analysis, Fuzzy data
1 Introduction
Attribute exploration [1] allows the interactive determination of the implications
holding between the attributes of a given context. However, there are situations
when the object set of a context is too large (possibly infinite) or difficult to
enumerate. With the examples (possibly none) of our knowledge we build the
object set of the context step-by-step. The stem base of this context, that is a
minimal set of non-redundant implications from which all the implications of
the context follow, is built stepwise and we are asked whether the implications
of the base are true. If an implication holds, then it is added to the stem base. If
however, an implication does not hold, then we have to provide a counterexample.
While performing an attribute exploration we have to be able to distinguish
between true and false implications and to provide correct counterexamples for
false implications. This is a crucial point since the algorithm is naive and will
believe whatever we tell it. Once a decision was taken about the validity of an
implication, the choice cannot be reversed. Therefore, the counterexamples may
not contradict the so-far confirmed implications. The procedure ends when all
implications of the current stem base hold in general. This way we obtain an
object set which is representative for the entire theory, that may also be infinite.
c paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA
2013, pp. 69–80, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La
Rochelle, 2013. Copying permitted only for private and academic purposes.
70 Cynthia Vera Glodeanu
The exploration process can be shortened by taking some background knowledge
[2] into account that the user has at the beginning of the exploration.
As many data sets contain fuzzy data, it is a natural wish to generalise
attribute exploration to the fuzzy setting. In [3] we have already shown how
this can be done. It turned out that we have to make some restrictions on the
implications (using the globalisation as the hedge) in order to be able to perform
a successful attribute exploration. In this paper we generalise the fuzzy attribute
exploration to the case with background knowledge. The main work starts in
Section 3 where we show how non-redundant bases can be obtained while using
background knowledge. The theory for the exploration is developed in Section 4
including an appropriate algorithm. In Section 5 we use a real-world data set to
illustrative both exploration with and without background knowledge.
It should be mentioned that there is some overlap with results presented in
the authors PhD thesis [4], see Chapter 5.
2 Preliminaries
2.1 Crisp Attribute Exploration
We assume basic familiarities with Formal Concept Analysis [1].
In the introductory section we have already explained the principle of at-
tribute exploration. However, we have not yet presented its key to success. This
is, why we do not have to reconsider already confirmed implications after adding
new objects to the context.
Proposition 1. ([1]) Let K be a context and P1 , P2 , . . . , Pn be the first n pseudo-
intents of K with respect to the lectic order. If K is extended by an object g the
object intent g ↑ of which respects the implications Pi → Pi↓↑ , i ∈ {1, . . . , n},
then P1 , P2 , . . . , Pn are also the lectically first n pseudo-intents of the extended
context.
Attribute exploration was successfully applied in both theoretical and prac-
tical research domains. On the one hand it facilitated the discovery of implica-
tions between properties of mathematical structures, see for example [5–7]. On
the other hand it was also used in real-life scenarios, for instance in chemistry
[8], information systems [9], etc.
In case the user knows some implications between the attributes in advance,
attribute exploration with background knowledge [10, 2] can be applied. Using
background knowledge in the exploration will considerably reduce the time of the
process as the user has to answer less questions and provide less counterexamples.
2.2 Formal Fuzzy Concept Analysis
Due to lack of space we omit the introduction of some basic notions from fuzzy
logic, fuzzy sets and Fuzzy Formal Concept Analysis. We assume that the reader
is familiar with these notions and refer to standard literature. For fuzzy theory
Attribute exploration with fuzzy attributes and background knowledge 71
we refer to [11, 12], in particular, notions like residuated lattices with hedges
and L-sets can be found for instance in [13]. For Fuzzy Formal Concept Analysis
see [14, 15] but also [16, 17]. The theory about Fuzzy Formal Concept Analysis
with hedges can be found in [13]. In this section we only present some notions
concerning lectic order, attribute implications and the computation of their non-
redundant bases.
We start with the fuzzy lectic order [18] which is defined as follows: Let
L = {l0 < l1 < · · · < ln = 1} be the support set of some linearly ordered residu-
ated lattice and M = {1, 2, . . . , m} the attribute set of an L-context (G, M, I).
For (x, i), (y, j) ∈ M × L, we write
(x, li ) ≤ (y, lj ) :⇐⇒ (x < y) or (x = y and li ≥ lj ).
We define B ⊕ (x, i) := ((B ∩ {1, 2, . . . , x − 1}) ∪ {li/x})↓↑ for B ∈ LM and
(x, i) ∈ M × L. Furthermore, for B, C ∈ LM , we define B <(x,i) C by
B ∩ {1, . . . , x − 1} = C ∩ {1, . . . , x − 1} and B(x) < C(x) = li . (1)
We say that B is lectically smaller than C, written B < C, if B <(x,i) C for
some (x, i) satisfying (1). As in the crisp case, we have that B + := B ⊕ (x, i) is
the least intent which is greater than a given B with respect to < and (x, i) is
the greatest with B <(x,i) B ⊕ (x, i) (for details we refer to [18]).
Fuzzy Implications and Non-redundant Bases. Fuzzy attribute implica-
tions were studied in a series of papers by Bělohlávek and Vychodil [19, 20].
We denote by S(A, B) the truth value of “the L-set A is a subset of the L-set
B”. Futher, (−)∗ denotes the hedge of a residuated lattice L, i.e., (−)∗ : L → L
is a map satisfying a∗ ≤ a, (a → b)∗ ≤ a∗ → b∗ , a∗∗ = a∗ and 1∗ = 1 for every
a, b ∈ L. Typical examples for the hedge are the identity, i.e., a∗ := a for all
a ∈ L, and the globalisation, i.e., a∗ := 0 for all a ∈ L \ {1} and a∗ := 1 if and
only if a = 1.
A fuzzy attribute implication (over the attribute set M ) is an expression
A ⇒ B, where A, B ∈ LM . The verbal meaning of A ⇒ B is: “if it is (very) true
that an object has all attributes from A, then it also has all attributes from B”.
The notions “being very true”, “to have an attribute”, and the logical connective
“if-then” are determined by the chosen L. For an L-set N ∈ LM of attributes,
the degree ||A ⇒ B||N ∈ L to which A ⇒ B is valid in N is defined as
||A ⇒ B||N := S(A, N )∗ → S(B, N ).
If N is the L-set of all attributes of an object g, then ||A ⇒ B||N is the truth
degree to which A ⇒ B holds for g. For N ⊆ LM , the degreeV ||A ⇒ B||N ∈ L to
which A ⇒ B holds in N is defined by ||A ⇒ B||N := N ∈N ||A ⇒ B||N . For
an L-context (G, M, I), let Ig ∈ LM (g ∈ G) be an L-set of attributes such that
Ig (m) = I(g, m) for each m ∈ M . Clearly, Ig corresponds to the row labelled
g in (G, M, I). We define the degree ||A ⇒ B||(G,M,I) ∈ L to which A ⇒ B
holds in (each row of) K = (G, M, I) by ||A ⇒ B||K := ||A ⇒ B||N , where
72 Cynthia Vera Glodeanu
N := {Ig | g ∈ G}. Denote by Int(G∗ , M, I) the set of all intents of B(G∗ , M, I).
The degree ||A ⇒ B||B(G∗ ,M,I) ∈ L to which A ⇒ B holds in (the intents of)
B(G∗ , M, I) is defined by
||A ⇒ B||B(G∗ ,M,I) := ||A ⇒ B||Int(G∗ ,M,I) . (2)
Lemma 1. ([20]) For each fuzzy attribute implication A ⇒ B it holds that
||A ⇒ B||(G,M,I) = ||A ⇒ B||B(G∗ ,M,I) = S(B, A↓↑ ).
Due to the large number of implications in a formal context, one is interested
in the stem base of the implications. Neither the existence nor the uniqueness of
the stem base for a given L-context are guaranteed in general [20].
Let T be a set of fuzzy attribute implications. An L-set of attributes N ∈ LM
is called a model of T if ||A ⇒ B||N = 1 for each A ⇒ B ∈ T . The set of all
models of T is denoted by Mod(T ) := {N ∈ LM | N is a model of T }. The
degree ||A ⇒ B||T ∈ L to which A ⇒ B semantically follows from T is
defined by ||A ⇒ B||T := ||A ⇒ B||Mod(T ) . T is called complete in (G, M, I) if
||A ⇒ B||T = ||A ⇒ B||(G,M,I) for each A ⇒ B. If T is complete and no proper
subset of T is complete, then T is called a non-redundant basis.
Theorem 1. ([20]) T is complete iff Mod(T ) = Int(G∗ , M, I).
As in the crisp case the stem base of a given L-context can be obtained
through the pseudo-intents. P ⊆ LM is called a system of pseudo-intents if
for each P ∈ LM we have:
P ∈ P ⇐⇒ (P 6= P ↓↑ and ||Q ⇒ Q↓↑ ||P = 1 for each Q ∈ P with Q 6= P ).
Theorem 2. ([20]) T := {P ⇒ P ↓↑ | P ∈ P} is complete and non-redundant.
If (−)∗ is the globalisation, then T is unique and minimal.
3 Adding background knowledge to the stem base
The user may know some implications between attributes in advance. We will call
such kind of implications background implications. In this section we will focus
on finding a minimal list of implications, which together with the background
implications will describe the structure of the concept lattice.
The theory about background knowledge for the crisp case was developed in
[2] and a more general form of it in [10]. The results from [2] for implication
bases with background knowledge follow by some slight modifications of the
results about implication bases without background knowledge presented in [1].
The same applies for the fuzzy variant of this method. Hence, if we choose the
empty set as the background knowledge, we obtain the results from [19, 20].
We start by investigating the stem bases of L-contexts relative to a set of
background implications. Afterwards we show how some notions and results for
fuzzy implications and their stem bases change for our new setting.
The attribute sets of L-contexts and the residuated lattices L will be consid-
ered finite. Further, L is assumed to be linearly ordered.
Attribute exploration with fuzzy attributes and background knowledge 73
Definition 1. Let K be a finite L-context and let L be a set of background
attribute implications. A set B of fuzzy attribute implications of K is called L-
complete if every implication of K is entailed by L ∪ B. We call B, L-non-
redundant if no implication A ⇒ B from B is entailed by (B \ {A ⇒ B}) ∪ L.
If B is both L-complete and L-non-redundant, it is called an L-base.
Note that if we have L = ∅ in the above definition, then all L-notions are
actually the notions introduced for sets of fuzzy implications. This remark holds
also for the other notions introduced in this section.
For any set L of background attribute implications and any attribute L-set
X ∈ LM we define an L-set X L ∈ LM and an L-set X Ln ∈ LM for each
non-negative integer n by
[
X L := X ∪ {B ⊗ S(A, X)∗ | A ⇒ B ∈ L},
Ln X, n = 0,
X :=
(X Ln−1 )L , n ≥ 1.
An operator L on these sets is defined by
∞
[
L(X) := X Ln . (3)
n=0
From [20] we know that L(−) is an L∗ -closure operator for a finite set M of
attributes and a finite residuated lattice L.
Definition 2. For an L-context (G, M, I), a subset P ⊆ LM is called a system
of L-pseudo-intents of (G, M, I) if for each P ∈ LM the following holds
P ∈ P ⇐⇒ (P = L(P ) 6= P ↓↑ and ||Q ⇒ Q↓↑ ||P = 1 for each Q ∈ P : Q 6= P ).
As in the case without background knowledge, the usage of the globalisation
simplifies the definition of the system of L-pseudo-intents. We have that P ⊆ LM
is a system of pseudo-intents if
P ∈ P ⇐⇒ (P = L(P ) 6= P ↓↑ and Q↓↑ ⊆ P for each Q ∈ P with Q & P ).
Theorem 3. The set BL := {P ⇒ P ↓↑ | P is an L-pseudo-intent} is an L-base
of K. We call it the L-Duquenne-Guigues-base or the L-stem base.
Proof. First note that all implications from BL are implications of (G, M, I). We
start by showing that BL is complete, i.e., ||A ⇒ B||BL ∪L = ||A ⇒ B||(G,M,I)
for every fuzzy implication A ⇒ B. By Equation (2) follows ||A ⇒ B||(G,M,I) =
||A ⇒ B||Int(G∗ ,M,I) . Hence, it is sufficient to prove that ||A ⇒ B||BL ∪L =
||A ⇒ B||Int(G∗ ,M,I) for every fuzzy attribute implication A ⇒ B. For any L-set
N ∈ LM , N ⇒ L(N ) is entailed by L, therefore we have N = L(N ).
Each intent N ∈ Int(G∗ , M, I) is a model of BL . Now let N ∈ Mod(BL ) and
assume that N 6= N ↓↑ , i.e., N is not an intent. Since N ∈ Mod(BL ) we have
74 Cynthia Vera Glodeanu
||Q ⇒ Q↓↑ ||N = 1 for every L-pseudo-intent Q ∈ P. By definition, N is an
L-pseudo-intent and hence N ⇒ N ↓↑ belongs to BL . But now, we have
||N ⇒ N ↓↑ ||N = S(N, N )∗ → S(N ↓↑ , N ) = 1∗ → S(N ↓↑ , N ) = S(N ↓↑ , N ) 6= 1,
which is a contradiction because N does not respect this implication.
To finish the proof, we still have to show that BL is L-non-redundant. To
this end let P ⇒ P ↓↑ ∈ BL . We show that this implication is not entailed
by L := (BL \ {P ⇒ P ↓↑ }) ∪ L. As P = L(P ), it is obviously a model of
L. We have ||Q ⇒ Q||P = 1 for every L-pseudo-intent Q ∈ P different from
P since P is an L-pseudo-intent. Therefore, P ∈ Mod(L). We also have that
||P ⇒ P ↓↑ ||P = S(P ↓↑ , P ) 6= 1 and thus P is not a model of BL ∪ L. Hence, we
have ||P ⇒ P ↓↑ ||(G,M,I) = ||P ⇒ P ↓↑ ||BL ∪L 6= ||P ⇒ P ↓↑ ||L , showing that L is
not complete and thus BL ∪ L is non-redundant. t
u
In general we write P ⇒ P ↓↑ \ {m ∈ M | P (m) = P ↓↑ (m)} instead of P ⇒ P ↓↑ .
Note that computing the stem-base and closing the implications from it with
respect to the operator L(−) will yield a different set of implications than the
L-stem-base. Let us take a look at the following example.
Example 1. Consider the L-context given in Figure 1. In order to ensure that its
stem-base and L-stem-base exist, we use the globalisation. Further, we use the
Gödel logic. The stem-base is displayed in the left column of Figure 1. For the
background implications L := {b ⇒ a, d ⇒ a, {a, c} ⇒ b} we obtain the L-stem-
base displayed in the middle column of the figure. If we close the pseudo-intents
stem base L-stem base L(P ) ⇒ P ↓↑
0.5 0.5 0.5
a b c d /b ⇒ a, /b ⇒ a, /b ⇒ a,
0.5 0.5 0.5
/a ⇒ a, /a ⇒ a, /a ⇒ a,
x 1 0.5 0 0
d ⇒ a, b, c, c, 0.5/d ⇒ a, b, d, a, d ⇒ b, c,
y 1 0 0 0
c, 0.5/d ⇒ a, b, d, a, 0.5/d ⇒ b, c, d, c, 0.5/d ⇒ a, b, d,
z 0 0 1 0
b ⇒ a, c, d, a, d ⇒ b, c, a, b ⇒ c, d,
t 0 0 0 0.5
a, 0.5/d ⇒ b, c, d, a, b ⇒ c, d. a, 0.5/d ⇒ b, c, d,
a, c ⇒ b, d. a, b, c ⇒ d.
Fig. 1. An L-context and its different stem bases.
of the stem-base with respect to the operator L(−), we obtain implications of
the form L(P ) ⇒ P ↓↑ , which are displayed in the right column of the figure. As
one easily sees, the latter set of implications and the L-stem-base are different.
The set {L(P ) ⇒ P ↓↑ | P is a pseudo-intent with L(P ) 6= P ↓↑ } contains an
additional implication, namely {a, b, c} ⇒ d.
For developing our theory about fuzzy attribute exploration with background
knowledge, the following results are useful. First, the set of all intents and all
Attribute exploration with fuzzy attributes and background knowledge 75
L-pseudo-intents is an L∗ -closure system, as stated below. Due to lack of space
we omit the proofs of the following two lemmas.
Lemma 2. Let (G, M, I) be an L-context, let L be a set of fuzzy implications of
(G, M, I). Further, let P and Q be intents or L-pseudo-intents such that
S(P, Q)∗ ≤ S(P ↓↑ , P ∩ Q) and S(Q, P )∗ ≤ S(Q↓↑ , P ∩ Q).
Then, P ∩ Q is an intent.
Note that if we choose the globalisation for (−)∗ , then P ∩ Q is an intent
provided that P and Q are (L-pseudo-)intents with P * Q and Q * P .
Now we are interested in the closure of an L-set with respect to the impli-
cations of the L-base BL . Therefore, we first define for each L-set X ∈ LM and
• •
each non-negative integer n the L-sets X L , X Ln ∈ LM as follows:
• [
X L := X ∪ {B ⊗ S(A, X)∗ | A ⇒ B ∈ BL , A 6= X},
L• X, n = 0,
X :=
n • •
(X Ln−1 )L , n ≥ 1.
Further, we define an operator L• (−) on these sets by
∞
[ •
L• (X) := X Ln . (4)
n=0
Lemma 3. If (−)∗ is the globalisation, then L• given by (4) is an L∗ -closure
operator and {L• (X) | X ∈ LM } coincides with the set of all L-pseudo-intens
and intents of (G, M, I).
Remark 1. Note that for a general hedge, L• (−) does not need be an L∗ -closure
operator. For instance, choose the Goguen structure and the identity for the
hedge (−)∗ . Further, let L := {0.3/y ⇒ y}. Then,
•
L• ({0.2/y})(y) ≥ ({0.2/y})L (y) = {0.2/y} ∪ {y ⊗ (0.3 → 0.2)}
= {0.2/y} ∪ {0.(66)/y} = {0.(66)/y},
and L• ({0.3/y})(y) = {0.3/y}. Hence, L• (−) does not satisfy the monotony prop-
erty, because we have {0.2/y} ⊆ {0.3/y} but L• ({0.2/y}) * L• ({0.3/y}).
4 Attribute exploration with background knowledge
Particularly appealing is the usage of background knowledge in the exploration
process. This proves itself to be very useful and time saving for the user. He/she
will have to answer less questions, as the algorithm does not start from scratch.
Due to Remark 1 and the fact that we are only able to perform a successful
attribute exploration if the chosen hedge is the globalisation, we will consider
76 Cynthia Vera Glodeanu
only this hedge in this section. In order to arrive at the exploration with back-
ground knowledge we will present the lectic order, the “key proposition” and an
appropriate algorithm for attribute exploration in this setting.
The lectic order is defined analogously as in Section 2, see (1). The only
difference lies in the definition of “⊕”. This time we are using the L∗ -closure
•
operator (−)L instead of (−)↓↑ .
•
Theorem 4. The lectically first intent or L-pseudo-intent is ∅L . For a given
L-set A ∈ LM , the lectically next intent or L-pseudo-intent is given by the L-set
A⊕(m, l), where (m, l) ∈ M ×L is the greatest tuple such that A <(m,l) A⊕(m, l).
The lectically last intent or L-pseudo-intent is M .
Now we are prepared to present the main proposition regarding attribute
exploration with background knowledge in a fuzzy setting.
Proposition 2. Let L be a finite, linearly ordered residuated lattice with glob-
alisation. Further, let P be the unique system of L-pseudo-intents of a finite
L-context K with P1 , . . . , Pn ∈ P being the first n L-pseudo-intents in P with
respect to the lectic order. If K is extended by an object g, the object intent g ↑
of which respects the implications from L ∪ {Pi ⇒ Pi↓↑ | i ∈ {1, . . . , n}}, then
P1 , . . . , Pn remain the lectically first n L-pseudo-intents of the extended context.
Proof. Let K = (H, M, J) be the initial context and let (G, M, I) be the extended
context, namely G = H ∪ {g} and J = I ∩ (H × M ). To put it briefly, since
g I is a model of Pi ⇒ PiJJ for all i ∈ {1, . . . , n} we have that PiJJ = PiII . By
the definition of L-pseudo-intents and the fact that every L-pseudo-intent Q of
(H, M, J) with Q ⊂ Pi is lectically smaller than Pi , we have that P1 , . . . , Pn are
the lectically first n L-pseudo-intents of (G, M, I).
We now have the key to a successful attribute exploration with background
knowledge in the fuzzy setting, at least when we use the globalisation. With this
result we are able to generalise the attribute exploration algorithm as presented
by Algorithm 1. Its input is the L-context K, the residuated lattice L and the set
of background implications L. The first intent or L-pseudo-intent is the empty
set. If it is an intent, add it to the set of intents of the context. Otherwise, ask
the expert whether the implication is true in general. If so, add this implication
to the L-stem base, otherwise ask for a counterexample and add it to the context
(line 2 − 11). Until A is different from M , repeat the following steps: Search for
the largest attribute i in M with its largest value l such that A(i) < l. For this
attribute compute its closure with respect to the L• (−)-closure operator given
by (4) and check whether the result is the lectically next intent or L-pseudo-
intent (line 12 − 16). Thereby, A & i := A ∩ {1, . . . , i − 1}. In lines 17 − 25 we
repeat the same procedure as in lines 2 − 11.
The algorithm generates interactively the L-stem base of the L-context. We
enumerate the intents and pseudo-intents in the lectic order. Due to Proposi-
tion 2 we are allowed to extend the context by objects whose object intents
respect the already confirmed implications. This way, the L-pseudo-intents al-
ready contained in the stem base do not change. Hence, the algorithm is sound
and correct.
Attribute exploration with fuzzy attributes and background knowledge 77
Algorithm 1: FuzzyExploration(K, L, L)
1 L := ∅; A := ∅;
↓↑
2 if A = A then
3 add A to Int(K)
4 else
5 Ask expert whether A ⇒ A↓↑ is valid;
6 if yes then
7 add A ⇒ A↓↑ to L
8 else
9 Ask for counterexample g and add it to K
10 end
11 end
12 while A 6= M do
13 for i = n, . . . , 1 and l = max L, . . . , min L with A(i) < l do
14 B := L• (A);
15 if A & i = B & i and A(i) < B(i) then
16 A := B;
17 if A = A↓↑ then
18 add A to Int(K)
19 else
20 Ask expert whether A ⇒ A↓↑ is valid;
21 if yes then
22 add A ⇒ A↓↑ to L
23 else
24 Ask for counterexample g and add it to K
25 end
26 end
27 end
28 end
29 end
5 Illustrative example
For our illustrative example we will consider the data from Figure 2. The ob-
jects are different universities from Germany and the attributes are indicators
rating these institutions. Study situation overall: M.Sc. students were asked
about their overall rating of their studies. IT-infrastructure: the availability of
subject-specific software, PC pools and wifi were taken into account. Courses
offered: amount of courses and the interdisciplinary references were relevant.
Possibility of studying: timing of the courses and content of the curriculum
were decisive. Passage to M.Sc.: clearly formulated admission requirements
and assistance of the students with the organisational issues were relevant.
Suppose we want to explore the implications between the attributes from
the L-context from Figure 2. We also know some examples, namely the TUs.
These will be the objects of the L-context we start with. Further, we heard
78 Cynthia Vera Glodeanu
from others that the implications {a, b} ⇒ c, d ⇒ {0.5/a, e}, {a, e} ⇒ {c, d} and
a ⇒ {0.5/b, 0.5/c, 0.5/e} hold. These will be considered the set of background im-
plications L. The exploration process is displayed in the left column of Figure 3.
The first L-pseudo-intent is ∅ and we ask the expert whether ∅ ⇒ ∅↓↑ holds.
This is not the case and a counterexample is Uni Bochum. For implications that
hold, for instance in step no. 3, the expert answers just “yes” and the implication
is added to the L-base. Afterwards, the validity of the implication induced by
the next L-pseudo-intent is asked, and so on. The algorithm continues until we
reach M as an intent or L-pseudo-intent. In our case, however, the algorithm
stops at step no. 10. This is due to the fact that the implications induced by the
L-pseudo-intents after {b, 0.5/d} are confirmed by the implications from the back-
ground knowledge. The result of the exploration is an extended context, namely
study situation IT- infra- courses possibility of passage to
overall structure offered studying M.Sc.
a b c d e
TU Braunschweig 0.5 0 0.5 0.5 0
TU Chemnitz 0.5 1 0.5 0.5 0.5
TU Clausthal 1 1 1 1 1
TU Darmstadt 0.5 0.5 0.5 0.5 0.5
TU Ilmenau 0.5 1 0.5 0.5 1
TU Kaiserslautern 1 0.5 0.5 0.5 0
Uni Bielefeld 0.5 0 0.5 1 1
Uni Bochum 0 0.5 0.5 0.5 1
Uni Duisburg 0.5 0.5 0 0.5 0.5
Uni Erlangen 0.5 0.5 0.5 0.5 0
Uni Heidelberg 0.5 1 0.5 1 1
Uni Koblenz 0.5 0.5 0.5 0.5 0.5
Uni Saarbrücken 1 0.5 1 1 1
Fig. 2. The data is an extract from the data published in the journal Zeit Campus in
January 2013. The whole data can be found under https://bit.ly/ZCinsr-informatik.
that contains our initial examples and the counterexamples we (or the expert)
has entered. We also obtain an L-base consisting of the background implica-
tions L and the implications we have confirmed during the exploration process.
In the right column of Figure 3 the exploration without background knowledge
is displayed. One immediately sees that there are 4 additional steps. The first
difference between the explorations appears in step no. 8. Without using back-
ground knowledge we have to answer the implication from the right column,
while this implication is already confirmed if we use background knowledge. The
exploration without background knowledge yields the same extended context
whereas the stem base contains the implications we have confirmed during the
process.
Attribute exploration with fuzzy attributes and background knowledge 79
no. expl. w. background know. simple expl.
0.5 0.5 0.5
1 Q: {} ⇒ { /a, /c, /d} Q: {} ⇒ {0.5/a, 0.5/c, 0.5/d}
E: no, ex. Uni Bochum E: no, ex. Uni Bochum
2 Q: {} ⇒ {0.5/c, 0.5/d} Q: {} ⇒ {0.5/c, 0.5/d}
E: no, ex. Uni Duisburg E: no, ex. Uni Duisburg
3 Q: {} ⇒ 0.5/d Q: {} ⇒ 0.5/d
E: yes E: yes
0.5 0.5 0.5
4 Q: { /d, /e} ⇒ /b Q: {0.5/d, 0.5/e} ⇒ 0.5/b
E: no, ex. Uni Bielefeld E: no, ex. Uni Bielefeld
5 Q: {0.5/b, 0.5/d} ⇒ 0.5/e Q: {0.5/b, 0.5/d} ⇒ 0.5/e
E: no, ex. Uni Erlangen E: no, ex. Uni Erlangen
0.5 0.5
6 Q: { /d, e} ⇒ /c Q: {0.5/d, e} ⇒ 0.5/c
E: yes E: yes
7 Q: {0.5/a, 0.5/b, 0.5/c, 0.5/d, e} ⇒ b Q: {0.5/a, 0.5/b, 0.5/c, 0.5/d, e} ⇒ b
E: no, ex. Uni Saarbrücken E: no, ex. Uni Saarbrücken
8 Q: {0.5/a, 0.5/b, 0.5/c, d, e} ⇒ {a, c} Q: d ⇒ {0.5/a, 0.5/c, e}
E: no, ex. Uni Heidelberg E: yes
0.5 0.5 0.5
9 Q: {c, /d} ⇒ {a, /b, d, /e} Q: {0.5/a, 0.5/b, 0.5/c, d, e} ⇒ {a, c}
E: yes E: no, ex. Uni Heidelberg
10 Q: {b, 0.5/d} ⇒ {0.5/a, 0.5/c} Q: {c, 0.5/d} ⇒ {a, 0.5/b, d, 0.5/e}
E: yes E: yes
exploration stopped.
11 Q: {b, 0.5/d} ⇒ {0.5/a, 0.5/c, 0.5/e}
E: yes
12 Q: {a, 0.5/d} ⇒ {0.5/b, 0.5/c, 0.5/e}
E: yes
13 Q: {a, 0.5/b, 0.5/c, 0.5/d, e} ⇒ {c, d}
E: yes
14 Q: {a, b0.5/c, 0.5/d, 0.5/e} ⇒ {c, d, e}
E: yes
exploration stopped.
Fig. 3. Exploration with and without background knowledge.
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