=Paper=
{{Paper
|id=None
|storemode=property
|title=Attribute Dependencies in a Fuzzy Setting
|pdfUrl=https://ceur-ws.org/Vol-972/paper11.pdf
|volume=Vol-972
|dblpUrl=https://dblp.org/rec/conf/cla/Glodeanu12
}}
==Attribute Dependencies in a Fuzzy Setting==
https://ceur-ws.org/Vol-972/paper11.pdf
Attribute Dependencies in a Fuzzy Setting
Cynthia Vera Glodeanu
Technische Universität Dresden,
01062 Dresden, Germany
Cynthia_Vera.Glodeanu@mailbox.tu-dresden.de
Abstract. We present a new framework for modelling users preferences
in a fuzzy setting. Starting with a formal fuzzy context, the user en-
ters so-called attribute dependency formulas based on his priorities. The
method then yields the “interesting” formal concepts, that is, interesting
from the point of view of the user. Our approach is designed for com-
pounded attributes, i.e., attributes which include more than one trait.
In this paper, after studying some properties of the formulas, we start
investigating the computation of non-redundant bases for them. Such
bases are wishful for a better overview of the preferences.
Keywords: Formal Concept Analysis, fuzzy data, data reduction
1 Introduction
Attribute dependency formulas were introduced in [1] and further studied in
a series of papers, see for instance [2]. They were developed as a method of
controlling the size of crisp concept lattices. The most appealing aspect of this
method is that the reduction is done based on the user’s preferences, namely
he is allowed to define a sort of order on the attributes. In accordance with
these preferences, the user receives just the “interesting” concepts, “interesting”
from his point of view. In [1] such preferences were modelled in the language of
Formal Concept Analysis as follows: An attribute dependency formula (AD
formula) over a set M of attributes is A ⊑ B, where A, B ⊆ M . The meaning
of the formula is “the attributes from A are less important than the attributes
from B”. The AD formula A ⊑ B is true in N ⊆ M , written N |= A ⊑ B, if
if A ∩ N 6= ∅, then B ∩ N 6= ∅.
A formal concept (C, D) ∈ B(G, M, I) satisfies A ⊑ B if D |= A ⊑ B.
These formulas were the starting point of our work. However, we develop a
different kind of AD formulas, namely some that are appropriate for compounded
attributes, i.e., attributes which include more than one trait. For instance the
notion “wealth” is a compounded attribute consisting of “investment” and “flu-
ency”. A person who is wealthy has to have high values on both investment and
fluency. We develop such formulas for the fuzzy setting and automatically obtain
the crisp case by choosing L = {0, 1} for the residuated lattice.
Some proofs are omitted due to lack of space but can be found in [3].
c 2012 by the paper authors. CLA 2012, pp. 127–138. Copying permitted only for
private and academic purposes. Volume published and copyrighted by its editors.
Local Proceedings in ISBN 978–84–695–5252–0,
Universidad de Málaga (Dept. Matemática Aplicada), Spain.
128 Cynthia Vera Glodeanu
The paper is structured as follows: In Section 2 we give brief introductions to
Fuzzy Sets, Fuzzy Logic and Formal Fuzzy Concept Analysis. Section 3 contains
our new framework. Concluding remarks and further research topics are given
in the last section.
2 Preliminaries
2.1 Fuzzy Sets and Fuzzy Logics
In this subsection we present some basics about fuzzy sets and fuzzy logic. The
interested reader may find more details for instance in [4, 5].
A complete residuated lattice with (truth-stressing) hedge is an al-
gebra L := (L, ∧, ∨, ⊗, →,∗ , 0, 1) such that: (L, ∧, ∨, 0, 1) is a complete lattice;
(L, ⊗, 1) is a commutative monoid; 0 is the least and 1 the greatest element; the
adjointness property, i.e., a ⊗ b ≤ c ⇔ a ≤ b → c, holds for all a, b, c ∈ L.
The hedge ∗ is a unary operation on L satisfying V the following conditions:
i) a∗ ≤ a; ii) (a → b)∗ ≤ a∗ → b∗ ; iii) a∗∗ = a∗ ; iv) i∈I a∗i = ( i∈I ai )∗ ; for ev-
V
ery a, b, ai ∈ L (i ∈ I). Elements of L are called truth degrees, ⊗ and → are
(truth functions of) “fuzzy conjunction” and “fuzzy implication”. The hedge ∗
is a (truth function of) logical connective “very true”, see [4, 6]. The properties
(i)-(iv) have natural interpretations, i.e., (i) can be read as “if a is very true,
then a is true”, (ii) can be read as “if a → b is very true and if a is very true, then
b is very true”, etc. From the mathematical point of view, the hedge operator is
a special kernel operator controlling the size of the fuzzy concept lattice.
A common choice of L is a structure with L = [0, 1], ∧ and ∨ being minimum
and maximum, ⊗ being a left-continuous t-norm with the corresponding →. The
three most important pairs of adjoint operations on the unit interval are:
Lukasiewicz: a ⊗ b := max(0, a + b − 1) with a → b := min(1, 1 − a + b),
1, a ≤ b
Gödel: a ⊗ b := min(a, b) with a → b := ,
b, a b
1, a ≤ b
Product: a ⊗ b := ab with a → b := .
b/a, a b
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.
Let L be the structure of truth degrees. A fuzzy set (L-set) A in a universe
U is a mapping A : U → L, A(u) being interpreted as “the degree to which u
belongs to A”. We denote by u ∈ A the fact that A(u) = 1. If U = {u1 , . . . , un },
then A can be denoted by A = {a1/u1 , . . . , an/un } meaning that A(ui ) equals
ai for each i ∈ {1, . . . , n}. Let LU denote the collection of all L-sets in U . The
operations with L-sets are defined component-wise. For instance, the intersection
of L-sets A, B ∈ LU is an L-set A∩B in U s. t. (A∩B)(u) = A(u)∧B(u) for each
u ∈ U , etc. Binary fuzzy relations (L-relations) between X and Y can be thought
of as L-sets in the universe X × Y . For A, B ∈ LU , the subsethood degree,
Attribute Dependencies in a Fuzzy Setting 129
which
V generalises the classical subsethood relation ⊆, is defined as S(A, B) :=
u∈U (A(u) → B(u)). Therefore, S(A, B) represents a degree to which A is a
subset of B. In particular, we write A ⊆ B iff S(A, B) = 1.
Fuzzy closure operators were introduced in [7] and studied further by Be-
lohlavek at al., see for instance [8, 9]. The definition given in [7] mirrors more a
crisp thinking, representing a special case of the one given in [8]. Therefore, we
will use the latter.
Definition 1. Define on a set Y two mappings C, κ : LY → LY satisfying
A ⊆ C(A), κ(A) ⊆ A, (1)
S(A1 , A2 )∗ ≤ S(C(A1 ), C(A2 )), S(A1 , A2 )∗ ≤ S(κ(A1 ), κ(A2 )), (2)
C(A) = C(C(A)), κ(A) = κ(κ(A)), (3)
for every A, A1 , A2 ∈ LY . Then, C is called an L∗ -closure operator and κ an
L∗ -kernel operator on Y . A system S := {Aj ∈ LY | j ∈ J} is a L∗ -closure
system if for each A ∈ LU it holds that
\
(S(A, Aj )∗ → Aj ) ∈ S. (4)
j∈J
The system S is called an L∗ -kernel system if for each A ∈ LU it holds that
[
(S(A, Aj )∗ ⊗ Aj ) ∈ S. (5)
j∈J
For the globalisation, (2) becomes
A1 ⊆ A2 =⇒ C(A1 ) ⊆ C(A2 ), A1 ⊆ A2 =⇒ κ(A1 ) ⊆ κ(A2 ),
T S
and (4) and (5) become j∈J,A⊆Aj Aj and j∈J,A⊆Aj Aj , respectively.
Theorem 1. ([8, 9]) A system S which is closed under arbitrary intersections
is an L∗ -closure system iff for each a ∈ L and A ∈ S it holds that a∗ → A ∈ S.
A system S closed under arbitrary unions is an L∗ -kernel system iff for each
a ∈ L and A ∈ S it holds a∗ ⊗ A ∈ S.
2.2 Formal Fuzzy Concept Analysis
In the following we give brief introductions to Formal Fuzzy Concept Analysis
[10, 5, 11].
A triple (G, M, I) is called a formal fuzzy context if I : G × M → L is
an L-relation between the sets G and M and L is the support set of some
residuated lattice. Elements from G and M are called objects and attributes,
respectively. The L-relation I assigns to each g ∈ G and each m ∈ M the truth
degree I(g, m) ∈ L to which the object g has the attribute m. For L-sets A ∈ LG
and B ∈ LM , the derivation operators are defined by
^ ^
A↑ (m) := (A(g)∗ → I(g, m)), B ↓ (g) := (B(m)∗ → I(g, m)) (6)
g∈G m∈M
130 Cynthia Vera Glodeanu
for g ∈ G and m ∈ M . Then, A↑ (m) is the truth degree of the statement “m
is shared by all objects from A”, and B ↓ (g) is the truth degree of “g has all
attributes from B”. The operators ↑ ,↓ form a so-called Galois connection with
hedges ([11]). A formal fuzzy concept (L-concept) is a tuple (A, B) with
A ∈ LG , B ∈ LM such that A↑ = B and B ↓ = A. Then, A is called the (fuzzy)
extent and B the (fuzzy) intent of (A, B). We denote the set of all L-concepts
of a given context (G, M, I) by B(G∗ , M ∗ , I). Concepts serve for classification.
Consequently, the super- and subconcept relation plays an important role. The
concept (A1 , B1 ) is a subconcept of (A2 , B2 ), written (A1 , B1 ) ≤ (A2 , B2 ), iff
A1 ⊆ A2 (or, equivalently, B1 ⊇ B2 ). Then, we call (A2 , B2 ) the superconcept
of (A1 , B1 ). The set of all L-concepts ordered by this concept order forms a
complete fuzzy lattice (with hedge), the so-called fuzzy concept lattice which
is denoted by B(G∗ , M ∗ , I) := (B(G∗ , M ∗ , I), ≤), see [11].
3 Fuzzy Attribute Dependencies
Now we are ready to present our new framework. Given a fuzzy formal context,
the user obtains the “interesting” concepts after entering a sort of order on
the groups of attributes, and fix the truth values for their relevance. Recall, we
designed this kind of AD formulas for compounded attributes. Thus, this notion
is not the fuzzy equivalent one of the formulas presented in Section 1. For a
straightforward fuzzification of those formulas see Remark 1.
Definition 2. A fuzzy attribute-dependency formula (fAD) over a set M
of attributes is an expression A ⊑ B, where A, B ∈ LM are L-sets of attributes.
A ⊑ B is true in an L-set N ∈ LM for α, β ∈ L \ {0} and α ≤ β, written
N |=α,β A ⊑ B, if the following condition is satisfied:
if S(A, N ) ≥ α, then S(B, N ) ≥ β. (7)
For an fAD formula or a set T of fAD formulas, the values α and β are called
the thresholds of A ⊑ B and T . An L-concept (C, D) ∈ B(G, M, I) satisfies
A ⊑ B if D |=α,β A ⊑ B.
For notational simplicity we will sometimes omit α and β from |=α,β provided
they are clear from the setting.
The set of all formal concepts from B(G, M, I) that satisfy a given set T of
fAD formulas is denoted by BT (G, M, I). We call BT (G, M, I) together with the
restricted concept order the fuzzy concept lattice of (G, M, I) constrained
by T and denote it by BT (G, M, I).
The fAD formulas permit a two-sided modelling of the extracted L-concepts.
On the one hand, α and β provide the thresholds to which an intent has to
contain all elements of A and B. On the other hand, the truth degrees of the
elements contained in A and B fix the thresholds to which we want the attributes
to be contained in the intent of a concept satisfying the fAD formula. We will
illustrate this fact in the forthcoming example.
Attribute Dependencies in a Fuzzy Setting 131
In applications it is particularly useful to associate to the truth values of
a residuated lattice L a Likert scale L. This allows the user to have a better
understanding of the truth values. For instance let L = {0, 0.25, 0.5, 0.75, 1} be
the support set of some residuated lattice with the associated Likert scale L =
{not important, less important, important, very important, most important},
i.e, 0 =not important, 0.25 =less important, etc.
Example 1. Consider the fuzzy context given in Figure 1. It represents the eval-
good team good organi- adaptive confi- computer
player sational skills towards new dential skills
e: analytical thinking
b: not discriminative
c: time management
k: word processing
d: problem solver
a: collaborative
f: environment
g: assignment
i: judgement
j: discretion
l: database
h: priority
1 0 0.5 0.5 1 1 0 0 0.5 0.5 0.5 1 1
2 1 1 1 1 1 1 1 1 0.5 0.5 0 0
3 0.5 0.5 0.5 1 1 0 1 1 1 1 0.5 0.5
4 1 0.5 1 1 1 0 1 1 0.5 0.5 1 1
5 0 0.5 0 0.5 0.5 0 0 0.5 0.5 0.5 0 0
6 1 1 0.5 1 1 1 1 1 0.5 0.5 0.5 0.5
7 0 0.5 0 0 0.5 0 0 0.5 0 0.5 0 0
Fig. 1. Fuzzy context about employees
uation of the employees of a small business regarding some qualities. Here each
quality is compounded of two or more traits. For instance, an employee is a “good
team player” if he/she is collaborative and not discriminative. The context has
44 L-concepts with the Gödel logic which are far too many to be analysed by a
busy manager. The manager however knows how good or bad the employees do
their jobs and he is interested more in their collaboration than their organisa-
tional skills and more in their adaptivity than in their confidentiality. Therefore,
he chooses the following two fAD formulas
{0.5/c, d, e} ⊑ {a, b} and {0.5/i, 0.5/j} ⊑ {f, g, h}, (8)
with α = 0.5 and β = 1. Then, he obtains 11 L-concepts which are given in
Figure 2.
The manager realises that the company does neither send its employees to
business trips nor to other companies and the employees should know their
priorities. Therefore, he changes the second fAD formula into
{0.5/i, 0.5/j} ⊑ {g, 0.5/h}. (9)
132 Cynthia Vera Glodeanu
Extent Intent
1 2 3 4 5 6 7 a b c d e f g h i j k l
1 0 0 0 0 0 0.5 0 1 1 1 1 1 1 1 1 1 1 1 1
2 0 0.5 0 0 0 0.5 0 1 1 1 1 1 1 1 1 1 1 0 0
3 0 1 0 0 0 0.5 0 1 1 1 1 1 1 1 1 0.5 0.5 0 0
4 0 0 0 0 0 1 0 1 1 0.5 1 1 1 1 1 0.5 0.5 0.5 0.5
5 0 1 0 0 0 1 0 1 1 0.5 1 1 1 1 1 0.5 0.5 0 0
6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0 1 0 0 1 0 0 1 0 1 0 0
7 0.5 1 0.5 0.5 0.5 1 0.5 0 1 0 0 1 0 0 1 0 0.5 0 0
8 0.5 0.5 1 0.5 0.5 0.5 0.5 0 0.5 0 0 1 0 0 1 0 1 0 0
9 0.5 1 1 1 0.5 1 0.5 0 0.5 0 0 1 0 0 1 0 0.5 0 0
10 1 1 1 1 0.5 1 0.5 0 0.5 0 0 1 0 0 0.5 0 0.5 0 0
11 1 1 1 1 1 1 1 0 0.5 0 0 0.5 0 0 0.5 0 0.5 0 0
Fig. 2. Concepts satisfying (8)
Obviously the concepts from the first fAD formulas are a subset of the concepts
from the second fAD formulas. The second couple of formulas yields 16 concepts,
namely those from Figure 2 and 3.
Extent Intent
1 2 3 4 5 6 7 a b c d e f g h i j k l
12 0 0 0.5 0.5 0 0.5 0 1 1 1 1 1 0 1 1 1 1 1 1
13 0 0.5 0.5 0.5 0 0.5 0 1 1 1 1 1 0 1 1 1 1 0 0
14 0 1 0.5 0.5 0 0.5 0 1 1 1 1 1 0 1 1 0.5 0.5 0 0
15 0 0 0.5 0.5 0 1 0 1 1 0.5 1 1 0 1 1 0.5 0.5 0.5 0.5
16 0 1 0.5 0.5 0 1 0 1 1 0.5 1 1 0 1 1 0.5 0.5 0 0
Fig. 3. Concepts satisfying (9) and the first fAD from (8)
We already have the framework of how to select interesting concepts based
on the preferences of the user. Let us further investigate some properties of the
fAD formulas.
Proposition 1. Let T be a set W of fAD formulas. Then, BT (G, M, I) is a com-
plete fuzzy lattice, which is a -sublattice of B(G, M, I).
Proof. Clearly, BT (G, M, I) ⊆ B(G, M, I) and B(G, M, I) with the restricted
concept order, is a partially ordered subset of B(G, M, I). Further note that
BT (G, M, I) is bounded from below because the least L-concept of B(G, M, I)
is (M ↓ , M ), concept which is compatible with every fAD formula. Now, we have
to show that BT (G, M, I) is closed under arbitrary suprema in B(G, M, I). To
this end let (Aj , Bj ) ∈ BT (G, M, I) (j ∈ J) be L-concepts. For any fAD formula
A ⊑ B ∈ T , we have Bj |= A ⊑ B for every j ∈ J. Now, if there is j ∈ J such
that Bj (a) < α for some a ∈ A, then ∩j∈J Bj (a) < α and we are done because
Attribute Dependencies in a Fuzzy Setting 133
then ∩j∈J Bj |= A ⊑ B. Contrary, if for all j ∈ J and all a ∈ A we have
Bj (a) ≥ α, then ∩j∈J Bj (a) ≥ α for all a ∈ A. Since Bj |= A ⊑ B holds for
all j ∈ J, we also have that Bj (b) ≥ β for all j ∈ J and b ∈ B. Due to the
same argument as before, it holds ∩j∈J Bj (b) ≥ β for all b ∈ B and hence we
have ∩j∈J Bj |= A ⊑ B, showing that BT (G, M, I) is closed under arbitrary
suprema. ⊔
⊓
In general BT (G, M, I) is not closed under arbitrary infima in B(G, M, I).
The fAD formulas are entered by the user. Thus, the chance that these for-
mulas are redundant is very high. Wishful thinking suggests to have a set of
non-redundant formulas because these are then easier to follow and to modify.
Therefore, in the following we will develop methods for removing such redun-
dancies.
Definition 3. An L-set N ∈ LM is a model of a set T of fAD formulas if we
have N |= A ⊑ B for each A ⊑ B ∈ T . Let Mod(T ) denote the set of all models
of T , i.e.,
Mod(T ) := {N ∈ LY | N |= A ⊑ B for each A ⊑ B ∈ T }.
An fAD formula A ⊑ B follows semantically from T , written T |= A ⊑ B, if
for each N ∈ Mod(T ), we have N |= A ⊑ B.
Lemma 1. i) N |= A ⊑ {l1/y1 , . . . , ln/yn } iff N |= A ⊑ {li/yi } holds for each
i ∈ {1, . . . , n}.
ii) For each set T of fAD formulas and each fAD formula ϕ, we have T |= ϕ iff
⌊T ⌋ |= ϕ, where ⌊T ⌋ := {A ⊑ {l/y} | A ⊑ B ∈ T and B(y) = l}.
Proof. i) If S(A, N ) < α, then we are done. Therefore, suppose S(A, N ) ≥ α.
By the definition of S, we have S({l1/y1 , . . . , ln/yn }, N ) ≥ β if and only if we
have N ({li/yi }) ≥ β for all i ∈ {1, . . . , n} and thus N |= A ⊑ {li/yi } for all
i ∈ {1, . . . , n}.
ii) Follows by i), omitted due to lack of space. ⊔
⊓
Thanks to Lemma 1 we may merge fAD formulas with the same left-hand
side into a single fAD formula. The new formula is true in a model if and only
if all its component fAD formulas are true in that model. This lemma allows us
also to test semantic entailment in fAD formulas A ⊑ {l/y} rather than on the
whole A ⊑ B.
In the following we will study the connection between the models of fAD
formulas and L∗ -closure systems. It will turn out that any L∗ -closure system
can be described by a set of fAD formulas.
Proposition 2. Let T be a set of fAD formulas. Then, Mod(T ) is an L∗ -closure
system with ∗ being the globalisation.
Proof. Let {Nj ∈ Mod(T ) | j ∈ J}. T We will be showing that Mod(T ) is closed
under arbitrary intersection, i.e., j∈J Nj is a model of T . For any fAD formula
134 Cynthia Vera Glodeanu
A ⊑ B ∈ T we have Nj |= A ⊑ B for every j ∈ J. Now, if there is j ∈ J such that
Nj (a) < α for some a ∈ A, then ∩j∈J Nj (a) < α and we are done because then
∩j∈J Nj |= A ⊑ B. Contrary, if for all j ∈ J and all a ∈ A we have Nj (a) ≥ α,
then ∩j∈J Nj (a) ≥ α for all a ∈ A. Since Nj |= A ⊑ B for all j ∈ J holds, we
also have that Nj (b) ≥ β for all j ∈ J and b ∈ B. Due to the same argument
as before, ∩j∈J Nj (b) ≥ β for all b ∈ B and hence we have ∩j∈J Nj |= A ⊑ B,
showing that Mod(T ) is a closed under arbitrary intersection.
Mod(T ) is a closed under arbitrary intersection and due to Theorem 1 we
just have to show that for any N ∈ Mod(T ) and any a ∈ L, a∗ → N is a model
of T . However, this condition only holds if ∗ is the globalisation because, then
we have
1, a = 0,
S(A, a∗ → N ) = a∗ → S(A, N ) =
S(A, N ), a = 1,
i.e., a∗ → N trivially satisfies any fAD formula if a = 0 or we do not gain
anything new to N in the case that a = 1. ⊔
⊓
Remark 1. One may argue that due to Proposition 2 the fAD formulas are not
strong enough. However, we consider that this is not the case. In the crisp set-
ting, the AD formulas form a kernel system, and due to the connection between
AD formulas and attribute implications (for details see [1]) one may efficiently
compute a non-redundant base of formulas. For instance, if we generalise in a
straight-forward way the AD formulas from [1] to the fuzzy setting, then we also
obtain just crisp like kernel systems. This can be done as follows: Define a fAD
formula A ⊑ B, where A, B ∈ LM . Further, A ⊑ B is true in an L-set N ∈ LM
for α, β ∈ L \ {0} and α ≤ β, written N |=α,β A ⊑ B, if the following condition
is satisfied:
if A ∩ N α-true, then B ∩ N β-true,
where an L-set X ∩ Y ∈ LM is α-true if there is at least one attribute m ∈ M
such that (X ∩ Y )(m) ≥ α, where α ∈ L. We need the thresholds in order to
ensure that the obtained concepts are indeed relevant. Now, the models of such
formulas form a crisp like kernel system which can be shown in an analogous
way to Proposition 2.
Lemma 2. For any L∗ -closure system S in M there is a set T of fAD formulas
over M such that S = Mod(T ).
Proof. Define a set T of fAD formulas by T := {A ⊑ CS (A) | A ∈ LM }, where
CS (A) is the closure of A given by the L∗ -closure operator CS . Further, choose
α = β = 1. Let N ∈ S, i.e., N = CS (N ). We have to show that N is a model
of T , thus let N |= A ⊑ CS (A) for every A ⊑ CS (A) ∈ T . If S(A, N ) < 1, then
N |= A ⊑ CS (A) and we are done. Now take S(A, N ) ≥ 1, i.e., A ⊆ N . Since
CS is a closure operator we have CS (A) ⊆ CS (N ) = N , hence S(CS (A), N ) ≥ 1,
i.e., N |= A ⊑ CS (A). Thus, N is a model of T and we have the first inclusion,
namely S ⊆ Mod(T ). For the converse inclusion let N ∈ Mod(T ). We have
if S(N, N ) ≥ 1, then S(CS (N ), N ) ≥ 1,
Attribute Dependencies in a Fuzzy Setting 135
where S(N, N ) ≥ 1 obviously holds and since N is a model of T we must
also have S(CS (N ), N ) ≥ 1 yielding that N = CS (N ), i.e., N ∈ S and hence
Mod(T ) ⊆ S. ⊔ ⊓
According to Proposition 2, Mod(T ) is an L∗ -closure system, and there-
fore there must exist an L∗ -closure operator CMod(T ) : LM → LM such that
N = CMod(T ) (N ) if and only if N ∈ Mod(T ). Hence, by definition CMod(T ) (N )
is the least model in Mod(T ) which contains N . This definition of the L∗ -closure
operator does not provide a useful method to compute the closure of a given N .
First, because one has to iterate over all models in Mod(T ) and second, such a
iteration may be impossible if L is infinite because then Mod(T ) is infinite.
Similarly to (fuzzy) attribute implications we proceed as follows: For any set
T of fAD formulas and for any L-set N ∈ LM of attributes, we define an L-set
N T ∈ LM of attributes as follows:
[
N T := N ∪ {β ⊗ B | A ⊑ B ∈ T, S(A, N ) ≥ α}. (10)
Further, we define an L-set N Tn ∈ LY of attributes for each non-negative integer
by
N if n = 0,
N Tn := (11)
(N Tn−1 )T if n ≥ 1.
We define an operator clT : LM → LM by
∞
[
clT (N ) := N Tn . (12)
n=0
Proposition 3. For each N ∈ Mod(T ) we have clT (N ) = N .
Proof. Omitted due to lack of space. ⊔
⊓
∗
The next lemma shows that the L -closure operator defined on the models
of T coincides with the clT -operator defined in (12).
Lemma 3. Let T be a set of fAD formulas over M . Further let both M and
L be finite. Then, clT is an L∗ -closure operator such that for each N ∈ LM ,
CMod(T ) (N ) = clT (N ).
Proof. CMod(T ) is an L∗ -closure operator, therefore it suffices to check that
CMod(T ) and clT coincide. To this end let N ∈ LM be an L-set of attributes. By
the definition of clT we have N ⊆ clT (N ). We still have to show that clT (N )
belongs to Mod(T ) and that clT (N ) is the least model containing N . First of all
note that the finiteness of L and M imply that LM is finite and that there exists
a non-negative integer k such that clT (N ) = N Tk , where N Tk is given by (11).
We still have to show that clT (N ) ∈ Mod(T ), i.e., for any A ⊑ B ∈ T we have
clT (N ) |= A ⊑ B. Suppose that S(A, clT (N )) ≥ α. Then, clT (N ) = N ∪ {β ⊗ B}.
Obviously, S(B, N ∪{β⊗B}) ≥ β, proving that clT (N ) ∈ Mod(T ) which contains
N . For any X ∈ Mod(T ) such that N ⊆ X we have to show that clT (N ) ⊆ X.
This easily follows by the properties of closure operators and by Proposition 3.
In fact, we have that clT (N ) ⊆ clT (X) = X. ⊔
⊓
136 Cynthia Vera Glodeanu
Based on the previous result we present an algorithm for the computation of
the closure CMod(T ) (N ) of a fuzzy attribute set N ∈ LM provided L and M are
finite.
Algorithm 1: Closure( N, T )
1 repeat
2 take A ⊑ B ∈ T such that S(A, N ) ≥ α and S(B, N ) < β;
3 set N to N ∪ {β ⊗ B};
4 until forall A ⊑ B ∈ T , (S(A, N ) < α) or (S(A, N ) ≥ α and S(B, N ) ≥ β);
5 return N
If we choose more restrictive values for α and β, we have the following con-
nection between fuzzy implications and fAD formulas:
Lemma 4. Let Imp(T ) := {A ⇒ B | for all A ⊑ B ∈ T } and T be a set of
fAD formulas. If we choose α = β = 1, then the following holds
Mod(Imp(T )) ⊆ Mod(T ).
Proof. Omitted due to lack of space. ⊔
⊓
Definition 4. Two sets T1 and T2 of fAD formulas are called equivalent, writ-
ten T1 ≡ T2 , if for each ϕ1 ∈ T1 and ϕ2 ∈ T2 we have T1 |= ϕ2 and T2 |= ϕ1 .
Lemma 5. Let T1 and T2 be sets of fAD formulas. Then, the following are
equivalent:
i) Mod(T1 ) = Mod(T2 ),
ii) For any fAD formula ϕ we have T1 |= ϕ ⇐⇒ T2 |= ϕ,
iii) T1 ≡ T2 .
Proof. Omitted due to lack of space. ⊔
⊓
Now we are prepared to introduce non-redundant bases.
Definition 5. A set T1 of fAD formulas is called a non-redundant base of T
if T ≡ T1 and there is no T2 ⊂ T1 with T2 ≡ T . A set T1 of fAD formulas is
called a minimal base of T if T ≡ T1 and for each T2 such that T ≡ T2 , we
have |T1 | ≤ |T2 |.
Obviously, if T1 is a minimal base of T , then T1 is a non-redundant base of T .
The converse implication is not true in general.
For a given set T of fAD formulas we may compute a non-redundant base as
follows: First note that if T1 := T \ {A ⊑ B} and T1 |= A ⊑ B, then T ≡ T1 .
We may then remove fAD formulas A ⊑ B from T step-by-step until there is
no T1 ⊂ T such that T1 ≡ T . The computation of a non-redundant base with
this method is quite laborious. In what follows, we present another connection
between fuzzy attribute implications and fAD formulas which will considerably
simplify this task.
Attribute Dependencies in a Fuzzy Setting 137
Lemma 6. Let T be a set of fAD formulas. We have Mod(T ) = Mod(Imp(T ∗ )),
where
Imp(T ∗ ) := {α ⊗ A ⇒ β ⊗ B | ∀A ⊑ B ∈ T, α, β thresholds of T }. (13)
Proof. First note that an L-set N ∈ LM is a model of an attribute implication
A ⇒ B in a fuzzy setting if ||A ⇒ B||N := S(A, N )∗ → S(B, N ) = 1.
Let N ∈ Mod(T ) and A ⊑ B ∈ T . We have two cases: i) S(A, N ) ≥ α
and S(B, N ) ≥ β both hold. Consider its first part. Then, for every attribute
m ∈ M , we have A(m) → N (m) ≥ α which by the adjointness property gives
us α ⊗ A(m) ≤ N (m) and therefore S(α ⊗ A, N ) = 1 and thus S(β ⊗ B, N ) = 1.
Hence, α ⊗ A ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 1∗ → 1 = 1.
ii) We have S(A, N ) < α which is equivalent to S(α ⊗ A, N ) < 1. Therefore,
||α ⊗ B ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 0 → S(β ⊗ B, N ) = 1.
Cases i) and ii) show that N is a model of Imp(T ∗ ).
For the converse inclusion let N ∈ Mod(Imp(T ∗ )). Then, we have
||α ⊗ A ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 1 (14)
for any fuzzy implication A ⇒ B ∈ Imp(T ∗ ). Equation 14 holds if and only if one
of the following situations appears: i) (S(α ⊗ A, N )∗ = 1 and S(β ⊗ B, N ) = 1)
⇐⇒ (S(A, N ) ≥ 1 and S(B, N ) ≥ 1). ii) S(α ⊗ A, N )∗ = 0 is equivalent to
S(α ⊗ A, N ) < 1 which further is equivalent to S(A, N ) < α. The two cases
prove N |=α,β A ⊑ B. ⊔
⊓
Thus, a fAD formula A ⊑ B with thresholds α, β is satisfied if and only if the
corresponding implication from Imp(T ∗ ), where ∗ is the globalisation, holds with
truth value 1. With this link between fAD formulas and fuzzy attribute impli-
cations we may easily compute a minimal base for any set T of fAD formulas.
First, we build the set Imp(T ∗ ) associated to T as given in (13). For this set
we compute a minimal base of attribute implications BT ∗ . Finally, from BT ∗ we
obtain a minimal base of fAD formulas for T by
BT := {A ⊑ B \ A | A ⇒ B ∈ BT ∗ },
where
A :=
_
{C ∈ LM | α ⊗ C = α ⊗ A}, B :=
_
{D ∈ LM | α ⊗ D = α ⊗ B}.
Note that, unlike the crisp case, in the fuzzy setting a formal context does
not have to have a unique stem base (see [12]). The uniqueness is ensured just
in the case of the globalisation.
With the possibility of computing a non-redundant base, the user may review
his choices and alter them conveniently.
4 Conclusion
We have presented two generalisations of crisp attribute dependency formulas
into the fuzzy setting. Both variants allow the user to define a sort of order on
138 Cynthia Vera Glodeanu
the attributes. According to the entered constraints the user sees just a part of
the concept lattice, namely the one containing the relevant concepts for him/her.
Due to lack of space, the straightforward generalisation was just briefly presented
in Remark 1.
The second approach was designed for compound attributes, such which in-
corporate more than one quality or specification. This time we required from the
“interesting” concepts that if they contain all less important attributes with a
threshold α, then they should also contain all more important attributes with
a threshold β, were the thresholds are truth degrees such that α ≤ β. For such
formulas, besides showing some of their properties, we focused mainly on the
computation of their non-redundant bases.
Future work will focus on applying the method on real-world data and eval-
uating the outcomes by experts. Another research topic is the exploration of
fAD formulas, where the user may alter the choices made without starting from
scratch each time.
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