=Paper=
{{Paper
|id=None
|storemode=property
|title=Composition of L-Fuzzy contexts
|pdfUrl=https://ceur-ws.org/Vol-876/paper1.pdf
|volume=Vol-876
}}
==Composition of L-Fuzzy contexts==
https://ceur-ws.org/Vol-876/paper1.pdf
Composition of L-Fuzzy contexts
Cristina Alcalde1 , Ana Burusco2 , and Ramón Fuentes-González2
1
Dpt. Matemática Aplicada. Escuela Universitaria Politécnica
UPV/EHU. Plaza de Europa, 1
20018 - San Sebastián (Spain)
c.alcalde@ehu.es
2
Dpt. Automática y Computación. Universidad Pública de Navarra
Campus de Arrosadı́a
31006 - Pamplona (Spain)
{burusco,rfuentes}@unavarra.es
Abstract. In this work, we introduce and study the composition of two
L-fuzzy contexts that share the same attribute set. Besides studying its
properties, this composition allows to establish relations between the sets
of objects associated to both L-fuzzy contexts.
We also define, as a particular case, the composition of an L-fuzzy context
with itself.
In all the cases, we show some examples that illustrate the results.
Key words: Formal contexts theory, L-fuzzy contexts, Contexts asso-
ciated with a fuzzy implication operator
1 Introduction
In some situations we have information that relates two sets X and Z to the same
set Y and we want to know if these relations allow us to establish connections
between X and Z. In the present work we will try to deal with the study of this
problem using as tool the L-fuzzy Concepts Theory.
The Formal Concept Analysis developed by Wille ([13]) tries to extract some
information from a binary table that represents a formal context (X, Y, R) with
X and Y being two finite sets (of objects and attributes, respectively) and R ⊆
X × Y . This information is obtained by means of the formal concepts which are
pairs (A, B) with A ⊆ X, B ⊆ Y fulfilling A∗ = B and B ∗ = A (where ∗ is
the derivation operator which associates to each object set A the set B of the
attributes related to A, and vice versa). A is the extension and B the intension
of the concept.
The set of the concepts derived from a context (X, Y, R) is a complete lattice
and it is usually represented by a line diagram.
In some previous works ([4],[5]) we defined the L-fuzzy context (L, X, Y,
R), where L is a complete lattice, X and Y the sets of objects and attributes
respectively and R ∈ LX×Y an L-fuzzy relation between the objects and the
attributes, as an extension to the fuzzy case of the Wille’s formal contexts when
2 C. Alcalde et al.
the relation between the objects and the attributes that we want to study takes
values in a complete lattice L. When we work with these L-fuzzy contexts we
use the derivation operators 1 and 2 defined by: For every A ∈ LX , B ∈ LY
A1 (y) = inf {I(A(x), R(x, y))}, B2 (x) = inf {I(B(y), R(x, y))}
x∈X y∈Y
where I is a fuzzy implication operator defined in (L, ≤), I : L × L −→ L,
which is decreasing in its first argument, and, A1 represents, as a fuzzy set, the
attributes related to the objects of A and B2 the objects related to the attributes
of B.
The information of the context is visualized by means of the L-fuzzy concepts
which are pairs (A, A1 ) ∈ (LX , LY ) with A ∈ fix(ϕ) the set of fixed points of the
operator ϕ, being this one defined by the derivation operators 1 and 2 mentioned
above as ϕ(A) = (A1 )2 = A12 . These pairs, whose first and second components
are the extension and the intension respectively, represent, as a fuzzy set, the
set of objects that share some attributes.
The set L = {(A, A1 ) : A ∈ fix(ϕ)} with the order relation ≤ defined as:
(A, A1 ), (C, C1 ) ∈ L, (A, A1 ) ≤ (C, C1 ) if A ≤ C
(or equiv. C1 ≤ A1 ) is a complete lattice that is said to be the L-fuzzy concept
lattice ([4],[5]).
On the other hand, given A ∈ LX , (or B ∈ LY ) we can obtain the derived
L-fuzzy concept applying the defined derivation operators. In the case of the use
of a residuated implication operator (as it holds in this work), the associated
L-fuzzy concept is (A12 , A1 ) (or (B2 , B21 )).
Other extensions of the Formal Concept Analysis to the fuzzy area are in
[14], [12], [3], [8], [10], [11] and [6].
2 Composed formal contexts
The composition of formal contexts allows to establish relations between the
elements of two sets of objects that share the same attribute set.
Definition 1. Let (X, Y, R1) and (Z, Y, R2) be two formal contexts, the com-
posed formal context is defined as the context (X, Z, R1 ? R2), where ∀(x, z) ∈
X × Z:
(
1 if R2(z, y) = 1, ∀y such that R1(x, y) = 1
R1 ? R2(x, z) =
0 in other case
That is, the object x is related to z in the composed context if z shares all the
attributes of x in the original contexts.
Proposition 1. The relation of the composed context, R1 ? R2, can also be
defined as:
R1 ? R2(x, z) = min{max{R10 (x, y), R2(z, y)}} ∀(x, z) ∈ X × Z
y∈Y
Composition of L-Fuzzy contexts 3
0
where R10 is the negation of the relation R1, that is, R10 (x, y) = (R1(x, y)) ∀(x, y) ∈
X ×Y.
This property will be helpful in the following sections.
Remark 1. Given the formal contexts (X, Y, R1) and (Z, Y, R2), the relation of
the composed context R1 ? R2 is not necessarily the opposed of the relation
R2 ? R1, that is, in general,
There exists (x, z) ∈ X × Z such that R1 ? R2(x, z) 6= R2 ? R1(z, x)
Example 1. Let us consider the formal contexts (X, Y, R1) and (Z, Y, R2), where
X = {x1 , x2 , x3 }, Y = {y1 , y2 , y3 , y4 , y5 }, Z = {z1 , z2 , z3 , z4 }, and the respective
relations are the following ones:
R2 y1 y2 y3 y4 y5
R1 y1 y2 y3 y4 y5
z1 1 1 0 1 0
x1 0 1 1 0 1
z2 0 1 0 0 1
x2 1 1 0 1 0
z3 1 1 0 1 1
x3 0 0 1 0 1
z4 0 1 1 1 1
If we calculate the composition of the contexts defined above in the two
possible orders, then the obtained relations are:
R2 ? R1 x1 x2 x3
R1 ? R2 z1 z2 z3 z4
z1 0 1 0
x1 0 0 0 1
z2 1 0 0
x2 1 0 1 0
z3 0 0 0
x3 0 0 0 1
z4 0 0 0
and, as can be seen, (R1 ? R2)op 6= R2 ? R1.
This property will be helpful in the following sections.
2.1 Particular case: when a formal context is composed with itself
Let us analyze a particular case where some interesting results are obtained.
Proposition 2. Let (X, Y, R) be a formal context. If (X, Y, R) is composed with
itself, then the obtained context is (X, X, R ? R) where the sets of objects and
attributes are coincident and the relation R ? R is a binary relation defined on
X as follows:
R ? R(x1 , x2 ) = min{max{R0 (x1 , y), R(x2 , y)}} ∀(x1 , x2 ) ∈ X × X
y∈Y
4 C. Alcalde et al.
Remark 2. The object x1 is related to attribute x2 in the composed context, if
in the original context the object x2 has at least the same attributes than the
object x1 .
Example 2. Returning to the formal context (X, Y, R) that we studied in the
previous example, where the relation R was:
R y1 y2 y3 y4 y5
x1 0 1 1 0 1
x2 1 1 0 1 0
x3 0 0 1 0 1
The composition of this context with itself is the context (X, X, R ? R), and
relation is given by the table:
R ? R x1 x2 x3
x1 1 0 0
x2 0 1 0
x3 1 0 1
Proposition 3. The relation R ? R obtained by the composition of the formal
context (X, Y, R) with itself is a preorder relation defined on the object set X.
Proof. As a consequence of the definition, it is immediate to prove that:
1. The relation R ? R is reflexive.
2. The relation R ? R is transitive.
t
u
Remark 3. It is a simple verification to see that:
– The relation R ? R is not, in general, a symmetric relation. To be symmetric
it is necessary that whenever an object x2 in the original context (X, Y, R)
has all the attributes of another object x1 , both objects have the same set
of attributes.
– The relation R ? R is not antisymmetric either. Therefore, R ? R is not, in
general, an order relation.
3 Extension to the L-fuzzy context case
The expression given in proposition 1 can be generalized to the fuzzy case sub-
stituting the maximum operator by a t-conorm S and taking a strong negation
0
. In this way, we can define the compositions of two L-fuzzy contexts as follows:
Composition of L-Fuzzy contexts 5
Definition 2. Let (L, X, Y, R1) and (L, Z, Y, R2) be two L-fuzzy contexts, we
define the composed L-fuzzy context (L, X, Z, R1 ? R2), where:
R1 ? R2(x, z) = inf {S(R10 (x, y), R2(z, y))} ∀(x, z) ∈ X × Z
y∈Y
with S being a t-conorm defined in the lattice L.
If we remind the definition of a fuzzy S-implication, the previous one can be
expressed in this way:
Definition 3. Let (L, X, Y, R1) and (L, Z, Y, R2) be two L-fuzzy contexts, and
I an S-implication operator. We define the composed L-fuzzy context (L, X, Z,
R1 ? R2), where:
R1 ? R2(x, z) = inf {I(R1(x, y), R2(z, y))} ∀(x, z) ∈ X × Z
y∈Y
We can generalize this definition to any fuzzy implication as we will see next.
3.1 Composition of L-fuzzy contexts associated with an implication
operator
Definition 4. Let (L, X, Y, R1) and (L, Z, Y, R2) be two L-fuzzy contexts, and
let I be a fuzzy implication operator, we define the composed L-fuzzy context
associated with the implication I as the L-fuzzy context (L, X, Z, R1 ?I R2),
where:
R1 ?I R2(x, z) = inf {I(R1(x, y), R2(z, y))} ∀(x, z) ∈ X × Z
y∈Y
Remark 4. If we remind the definition of the triangle subproduct operator / given
by [9], one of the standard operators in the fuzzy relation theory which has been
previously used in diverse works [1, 2], we can see that the composed relation
defined here can be written as:
R1 ?I R2 = R1 / (R2)op
As can be observed, also in this case a similar result to the crisp case is obtained.
Proposition 4. Let (L, X, Y, R1) and (L, Z, Y, R2) be two L-fuzzy contexts. Then,
the relation of the composed L-fuzzy context (L, X, Z, R1?I R2) is not, in general,
the opposite of the relation of the composed L-fuzzy context (L, Z, X, R2 ?I R1).
(R1 ?I R2)op 6= R2 ?I R1
That is, if we change the order of the composition, the obtained relation between
the elements of X and Z is different.
6 C. Alcalde et al.
Proof. Given two L-fuzzy contexts (L, X, Y, R1) and (L, Z, Y, R2), and a fuzzy
implication operator I, the relation of the composed L-fuzzy context (L, X, Z,
R1 ?I R2) is:
R1 ?I R2(x, z) = inf {I(R1(x, y), R2(z, y))} ∀(x, z) ∈ X × Z
y∈Y
On the other hand, the relation of the composed L-fuzzy context (L, Z, X,
R2 ?I R1) is defined as:
R2 ?I R1(z, x) = inf {I(R2(z, y), R1(x, y))} ∀(z, x) ∈ Z × X
y∈Y
As, in general, given a fuzzy implication I(a, b) 6= I(b, a), then these relations
are not opposed. t
u
Example 3. We have a company of temporary work in which we want to ana-
lyze the suitability of some candidates to obtain some offered employments. The
company knows the requirements of knowledge to occupy each one of the posi-
tions, represented by means of the L-fuzzy context (L, X, Y, R1), where the set of
objects X is the set of employments, the attributes Y the necessary knowledge,
and the relation among them appears in Table 1 with values in the chain L={0,
0.1, 0.2, . . . , 1}.
Table 1. The requirements of knowledge to obtain each one of the employments.
R1 computer science accounting mechanics cooking
domestic helper 0.1 0.3 0.1 1
waiter 0 0.4 0 0.7
accountant 0.9 1 0 0
car salesman 0.5 0.7 0.9 0
On the other hand, we have the knowledge of some candidates for these
positions, represented by the L-fuzzy context (L, Z, Y, R2) in which the objects
are the different candidates to occupy the jobs, the attributes the necessary
knowledge and the relation among them is given by Table 2.
A candidate will be suitable to obtain a job if he owns all the knowledge
required in this position. Therefore, to analyze what candidate is adapted for
each job, we would use the composed L-fuzzy context (L, X, Z, R1 ? R2). The
relation of this composed context, calculated using the Lukasiewicz implication
operator, is the represented in Table 3.
To obtain the information of this L-fuzzy context we will use the ordinary
tools of the L-fuzzy Concept Theory to analyze the associated L-fuzzy concepts.
Thus, for example, if we want to find the best candidate to occupy the job of
waiter, we take the set:
Composition of L-Fuzzy contexts 7
Table 2. Knowledge of the candidates.
R2 computer science accounting mechanics cooking
C1 0.5 0.8 0.3 0.6
C2 0.2 0.5 0.1 1
C3 0 0.2 0 0.3
C4 0.9 0.4 0.1 0.5
C5 0.7 0.5 0.2 0.1
Table 3. Suitability of each candidate for each position.
R1 ? R2 C1 C2 C3 C4 C5
domestic helper 0.6 1 0.3 0.5 0.1
waiter 0.9 1 0.6 0.8 0.4
accountant 0.6 0.3 0.1 0.4 0.5
car salesman 0.4 0.2 0.5 0.2 0.3
{domestic helper/0, waiter/1, accountant/0, car salesman/0}
and we obtain the derived L-fuzzy concept, whose intension is:
{C1 /0.9, C2 /1, C3 /0.6, C4 /0.8, C5 /0.4}
If we look at the attributes with the highest membership degree, we can
deduce that the most suitable candidate for the job of waiter is C2 , followed by
C1 and C4 .
If, for instance, we want to find the best person to be accountant in a restau-
rant that also could work as a waiter, we take the set
{domestic helper/0, waiter/1, accountant/1, car salesman/0}
and the derived L-fuzzy concept is
{C1 /0.6, C2 /0.3, C3 /0.1, C4 /0.4, C5 /0.4}
where we can see that the most suitable candidate is C2 .
On the other hand, if our interest is to analyze which of the jobs is the
most suitable for each candidate, we do the composition in the contrary order,
obtaining the L-fuzzy context (L, Z, X, R2 ? R1), where the composed relation
is represented in Table 4.
We can see in this example that both compositions are different: A candidate
can be the best to occupy a concrete job, but that job need not be the most
appropriate for this candidate.
8 C. Alcalde et al.
Table 4. Suitability of each employment for each candidate.
R2 ? R1 domestic helper waiter accountant car salesman
C1 0.5 0.5 0.4 0.4
C2 0.8 0.7 0 0
C3 0.3 0.2 0.2 0.7
C4 0.2 0.1 0.5 0.5
C5 0.4 0.3 0.8 0.8
The following result will be of interest to study the L-fuzzy concepts associ-
ated to the objects of the composed L-fuzzy context.
Before to proceed with the proposition, we are going to introduce a new
notation: If the subscripts point out the derivation operators and the superscripts
the L-fuzzy contexts where they are applied, then A1? is the derived set from A
obtained in the composed L-fuzzy context, A11 is the derived set obtained in the
L-fuzzy context (L, X, Y, R1), and (A11 )22 the derived set of the last one in the
L-fuzzy context (L, Z, Y, R2)
Proposition 5. If the implication operator I is residuated and we consider the
set: (
1 if x = xi
A(x) =
0 in other case
then, the intension of the L-fuzzy concept obtained in the composed L-fuzzy con-
text (L, X, Z, R1 ?I R2) from the set A, is equal to the extension of the L-fuzzy
concept obtained in (L, Z, Y, R2) from the intension of the L-fuzzy concept ob-
tained in (L, X, Y, R1) from A. That is, we obtain the same fuzzy set Z applying
the derivation operators twice (once in each one of the contexts that make up the
composition), or once in the composed context.
Moreover, it is verify that:
∀z ∈ Z, A1? (z) = (A11 )22 (z) = R1 ?I R2(xi , z)
That is, the membership degrees obtained are the values of the row of R1 ?I R2
that corresponds to the object xi .
(
1 if x = xi
Proof. Let be A(x) = , the intension of the L-fuzzy concept
0 in other case
obtained from A in the context (L, X, Y, R1) is the L-fuzzy subset of Y :
A11 (y) = inf {I(A(x), R1(x, y))}, ∀y ∈ Y.
x∈X
As the implication I is residuated, ∀a ∈ L it is verified that I(0, a) = 1 and
I(1, a) = a, thus,
A11 (y) = R1(xi , y), ∀y ∈ Y.
Composition of L-Fuzzy contexts 9
Taking now the set A11 , we obtain the derived L-fuzzy concept in the L-fuzzy
context (L, Z, Y, R2), the extension of which is:
(A11 )22 (z) = inf {I(A11 (y), R2(z, y))} =
y∈Y
= inf {I(R1(xi , y), R2(z, y))} = R1 ?I R2(xi , z), ∀z ∈ Z.
y∈Y
On the other hand, the intension of the obtained L-fuzzy concept in the
composed L-fuzzy context from A is:
A1? (z) = inf {I(A(x), R1 ?I R2(x, y))} = R1 ?I R2(xi , z), ∀z ∈ Z.
x∈X
t
u
Example 4. If we come back to example 3, we have analyzed which candidate is
the most suitable for the job of waiter.
To do this, in the L-fuzzy context (L, X, Z, R1 ? R2) (see Table 3) we have
taken the set
A = {domestic helper/0, waiter/1, accountant/0, car salesman/0}
and we have calculated the closed L-fuzzy concept, where the fuzzy intension is:
A1? = {C1 /0.9, C2 /1, C3 /0.6, C4 /0.8, C5 /0.4}
And here, if we look at those attributes whose membership degrees stand out
from the others, we deduce that the most suitable candidates to be good waiters
were, C2 , C1 and C4 , in this order.
The same result is obtained if we take the L-fuzzy context (L, X, Y, R1) (see
Table1) and we calculate the L-fuzzy concept from A, which intension is:
A11 = {computer science/0, accounting/0.4, mechanics/0, cooking/0.7}
And, from this fuzzy set we obtain in the L-fuzzy context (L, Z, Y, R2) (see
Table2) the derived L-fuzzy concept the extension of which is:
(A11 )22 = {C1 /0.9, C2 /1, C3 /0.6, C4 /0.8, C5 /0.4}
As can be seen, the result is the same that the obtained in the composed L-fuzzy
context.
3.2 Composition of an L-fuzzy context with itself
The composition of an L-fuzzy context (L, X, Y, R) with itself will allow us to
set up some relationships between the elements of the object set X.
Proposition 6. If I is a residuated implication associated with a left continuous
t-conorm T , then the relation R?I R that results of the composition of (L, X, Y, R)
with itself, associated with the implication I, constitutes a fuzzy preorder relation
defined in the object set X.
10 C. Alcalde et al.
Proof. 1. First, we prove that it is a reflexive relation, that is, the relation
verifies:
∀x ∈ X, R ?I R(x, x) = 1.
By the definition of the composition associated with an implication operator,
we have
∀x ∈ X, R ?I R(x, x) = inf {I(R(x, y), R(x, y))},
y∈Y
and, as any residuated implication verifies that I(a, a) = 1, ∀a ∈ L, then
∀x ∈ X, R ?I R(x, x) = 1.
2. To see that R ?I R is a T -transitive relation, we have to prove that
∀x, t, z ∈ X, T (R ?I R(x, t), R ?I R(t, z)) ≤ R ?I R(x, z),
that is, the following inequality must be verified:
T inf {I(R(x, α), R(t, α))}, inf {I(R(t, β), R(z, β))} ≤
α∈Y β∈Y
inf {I(R(x, α), R(z, α))}.
α∈Y
By the monotony of the t-norm, we have:
T inf {I(R(x, α), R(t, α))}, inf {I(R(t, β), R(z, β))} ≤
α∈Y β∈Y
inf T I(R(x, α), R(t, α)), inf {I(R(t, β), R(z, β))} ≤
α∈Y β∈Y
inf {T (I(R(x, α), R(t, α)), I(R(t, α), R(z, α)))} .
α∈Y
As the used t-norm T is left-continuous, we know that [7]
∀a, b, c ∈ [0, 1], T (I(a, b), I(b, c)) ≤ I(a, c),
and it is verified that:
T inf {I(R(x, α), R(t, α))}, inf {I(R(t, β), R(z, β))} ≤
α∈Y β∈Y
inf {I(R(x, α), R(z, α))}.
α∈Y
t
u
Remark 5. The relation R ?I R is neither symmetric nor antisymmetric and
then, is neither an equivalence nor an order relation. For instance, if we take the
relation R given by the table:
Composition of L-Fuzzy contexts 11
R y1 y2 y3 y4
x1 0.1 0.3 0.5 0.1
x2 0.8 0.2 0.8 0.2
x3 0.4 0.7 0 0.1
then the relation R ?I R associated with the Lukasiewicz implication operator
is:
R ?I R x1 x2 x3
x1 1 0.9 0.5
x2 0.3 1 0.2
x3 0.5 0.5 1
and, as can be seen, is neither a symmetric nor an antisymmetric relation.
Remark 6. If we are using a non residuated implication operator, not always a
fuzzy preorder relation is obtained. For instance, if we take the previous relation
R and we do the composition R ?I R associated with the Kleene-Dienes impli-
cation (that does not verify I(x, x) = 1), then we obtain the following relation:
R ?I R x1 x2 x3
x1 0.5 0.7 0.5
x2 0.2 0.2 0.2
x3 0.3 0.3 0.3
that is neither a reflexive nor a fuzzy preorder relation.
The application of this composition can be very interesting in social or work
relations as we can see next:
Example 5. There are four different manufacture processes in a factory and we
want to organize the workers so that each of them is subordinate of another one
if its capacity to carry out each one of the processes of manufacture is smaller.
To model this problem, we are going to take the L-fuzzy context (L, X, Y, R),
where the set of objects X is formed by the workers {O1 , O2 , O3 , O4 , O5 }, the
attributes are the different manufacture processes {P1 , P2 , P3 , P4 }, and the rela-
tion R represents the capacity of each one of the workers to carry out each one
of the processes, in a scale of 0 to 1 (See Table 5).
The L-fuzzy context that results of the composition of this context with itself
allow us to define relations boss-subordinate between the workers so that the
relation R ? R(x, y) of the compound context (associated with the Lukasiewicz
implication) gives the degree in which the worker x is subordinate of the worker
y. (See Table 6).
12 C. Alcalde et al.
Table 5. Capacity of the workers to carry out each one of the manufacture processes
R P1 P2 P3 P4
O1 0.7 1 0.3 0
O2 0.3 0.8 0.9 0.4
O3 0.1 0.2 1 0.5
O4 0.5 0.3 0.2 0.4
O5 1 0.5 0.8 1
Table 6. Relation ”be subordinate of”.
R ? R O1 O2 O3 O4 O5
O1 1 0.6 0.2 0.3 0.5
O2 0.4 1 0.4 0.3 0.7
O3 0.3 0.9 1 0.2 0.8
O4 0.6 0.8 0.6 1 1
O5 0 0.3 0.1 0.4 1
This will allow us, for example, to choose bosses in the group watching the
columns of the obtained relation: In this case, we could choose as bosses of the
workers to O2 and O5 because both have as subordinate O3 and O4 and the
subordination degrees are the biggest values of the columns.
4 Conclusions and future work
This work constitutes the first approach to the problem of composition of L-
fuzzy contexts. In future works we will use these results in the resolution of
other problems that seem interesting to us:
- First, this composition will be useful to study the chained L-fuzzy contexts,
that is, to find relations between two defined contexts where the set of attributes
of the first context is the same that the set of objects of the second one.
- On the other hand, we think that it will be useful to define the compo-
sition of L-fuzzy contexts in the interval-valued case in order to study certain
situations.
Acknowledgements
This work has been partially supported by the Research Group “Intelligent Sys-
tems and Energy (SI+E)” of the Basque Government, under Grant IT519-10.
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