=Paper=
{{Paper
|id=Vol-2668/paper21
|storemode=property
|title=Quantified Formal-Concept Operators
|pdfUrl=https://ceur-ws.org/Vol-2668/paper21.pdf
|volume=Vol-2668
|authors=M. Eugenia Cornejo,Juan Carlos Díaz-Moreno,Juan López-Rodríguez,Jesús Medina
}}
==Quantified Formal-Concept Operators==
https://ceur-ws.org/Vol-2668/paper21.pdf
Quantified Concept-Forming Operators?
M. Eugenia Cornejo, Juan Carlos Dı́az-Moreno, Juan López-Rodrı́guez, and
Jesús Medina
Department of Mathematics, University of Cádiz, Spain
{mariaeugenia.cornejo,juancarlos.diaz,juan.lopezrodriguez,jesus.medina}@uca.es
Abstract. Generalized quantifiers allow to decrease the outermost char-
acter of the universal and existential quantifiers. Concept-forming oper-
ators in Formal Concept Analysis have a universal character. As a con-
sequence, some noise in the data can dramatically alter the final result.
This paper introduces a first approximation to quantified formal-concept
operators with the main goal of decreasing this universal character and
be less sensitive to noisy data.
Keywords: Concept lattices, fuzzy sets, general quantifier, adjoint triples
1 Introduction
Formal Concept Analysis (FCA), introduced by Ganter and Wille in the eight-
ies, has become an appealing research topic both from theoretical [18, 22] and
applicative perspectives [1, 16, 17, 19, 24, 26]. Specifically, FCA is a mathemati-
cal tool in charge of extracting information from databases containing a set of
attributes A and a set of objects B related to each other by a binary relation
R ⊆ A × B. These pieces of information are called concepts and a hierarchy can
be established on them providing an algebraic structure called concept lattice.
From the concept lattice, a mathematical development for the conceptual data
analysis and processing of knowledge can be carried out. Soon after its introduc-
tion, a number of different approaches for its generalization were introduced [2,
5, 7]. One of the most general fuzzy approaches is the multi-adjoint concept
lattice [20, 21, 23], which will be considered in this paper. The multi-adjoint con-
cept lattice framework considers different adjoint triples in the definition of the
concept-forming operations, providing interesting properties [11, 13].
Generalized quantifiers [6, 8, 9, 14, 15, 25] have been proposed in order to solve
the theoretical drawback of the usual universal and existencial quantifiers. Specif-
ically, these quantifiers try to introduce intermeditate quantifiers between the
existential quantifier, where just one element is enough to result the truth, and
?
Partially supported by the the 2014-2020 ERDF Operational Programme in collab-
oration with the State Research Agency (AEI) in projects TIN2016-76653-P and
PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Busi-
ness and University of the Regional Government of Andalusia. in project FEDER-
UCA18-108612, and by the European Cooperation in Science & Technology (COST)
Action CA17124.
Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). Published in Francisco
J. Valverde-Albacete, Martin Trnecka (Eds.): Proceedings of the 15th International
Conference on Concept Lattices and Their Applications, CLA 2020, pp. 273–280,
2020.
274 M. Eugenia Cornejo et al.
the universal quantifier, where all elements have to fulfill a given formula in or-
der to result the truth. As a consequence, more proper quantifiers can be used
in applications where the outermost quantifiers are very restrictive. For exam-
ple, notions as “Most” or “Many” can be implemented and so, considered in
applications.
This paper is inspired by [9, 25] in order to develop different notions and
results on multi-adjoint concept lattices, based on the monadic quantifiers of
type h1i determined by fuzzy measures [15]. Specifically, we have introduced
the definition of generalized quantifier in this general framework as a natural
extension, which also satisfies the main properties. Moreover, we have used the
extended definition to present a generalization of the usual fuzzy concept-forming
operators [3, 4, 7, 23] in order to decrease the drastic (universal) character of these
operators, which are very sensitive to noise data. As a consequence, the new
introduced operators offer a complementary alternative to the original concept-
forming operators to obtain more robust, stable and valuable information from
datasets. Although the new operators generalize the original one, they do not
form an antitone Galois connection, in general. Therefore, different sufficient
conditions and alternative definitions will be studied in the future in order to
ensure Galois connections.
The paper is organized as follows. In the second section, we recall the theory
related to multi-adjoint concept lattices. In the third section, we generalize the
notion of generalized quantifier by using adjoint triples. In addition, we use
the extended definition in order to present a generalization of the usual fuzzy
concept-forming operators. In the last section, we list some directions where our
approach could be further developed.
2 Preliminaries
To begin with, we will recall the notion of adjoint triple which play an important
role in this work. Adjoint triples arise as a generalization of triangular norms and
their residuated implications. These operators are the basic calculus operators
in different frameworks such as multi-adjoint logic programming, multi-adjoint
fuzzy relation equations, multi-adjoint fuzzy rough sets and multi-adjoint con-
cept lattices. It is worth noting that these operators increase the flexibility of
the mentioned frameworks, since the conjunctors are not required to be either
commutative or associative.
Definition 1. Let (P1 , ≤1 ), (P2 , ≤2 ), (P3 , ≤3 ) be posets and & : P1 × P2 → P3 ,
. : P3 × P2 → P1 , - : P3 × P1 → P2 be mappings. We say that (&, ., -) is an
adjoint triple with respect to (P1 , ≤1 ), (P2 , ≤2 ), (P3 , ≤3 ) if the following double
equivalence is satisfied:
x ≤1 z . y iff x & y ≤3 z iff y ≤2 z - x
for all x ∈ P1 , y ∈ P2 and z ∈ P3 . The previous double equivalence is called
adjoint property.
Quantified Formal-Concept Operators 275
The following properties are obtained straightforwardly from the adjoint
property [10, 12].
Proposition 1. Let (&, ., -) be an adjoint triple with respect to the posets
(P1 , ≤1 ), (P2 , ≤2 ) and (P3 , ≤3 ), then the following properties are satisfied:
1. & is order-preserving on both arguments.
2. . and - are order-preserving on the first argument and order-reversing on
the second argument.
3. ⊥1 & y = ⊥3 , >3 . y = >1 , for all y ∈ P2 , when (P1 , ≤1 , ⊥1 , >1 ) and
(P3 , ≤3 , ⊥3 , >3 ) are bounded posets.
4. x & ⊥2 = ⊥3 and >3 - x = >2 , for all x ∈ P1 , when (P2 , ≤2 , ⊥2 , >2 ) and
(P3 , ≤3 , ⊥3 , >3 ) are bounded posets.
5. z - ⊥1 = >2 and z . ⊥2 = >1 , for all z ∈ P3 , when (P1 , ≤1 , ⊥1 , >1 ) and
(P2 , ≤2 , ⊥2 , >2 ) are bounded posets.
Once we have introduced the notion of adjoint triple, we are in a position
to present multi-adjoint concept lattices. The philosophy of the multi-adjoint
paradigm was applied to the formal concept analysis in order to obtain a new gen-
eral approach framework that could conveniently accommodate different fuzzy
approaches given in the literature. The mentioned general approach is called
multi-adjoint concept lattices and their main notions will be recalled below [23].
Definition 2. A multi-adjoint frame is a tuple (L1 , L2 , P, &1 , . . . , &n ) where
(L1 , 1 ) and (L2 , 2 ) are complete lattices, (P, ≤) is a poset and (&i , .i , -i )
is an adjoint triple with respect to L1 , L2 , P , for all i ∈ {1, . . . , n}.
Definition 3. Let (L1 , L2 , P, &1 , . . . , &n ) be a multi-adjoint frame, a context
is a tuple (A, B, R, σ) such that A and B are non-empty sets, R is a P -fuzzy
relation R : A×B → P and σ : A×B → {1, . . . , n} is a mapping which associates
any element in A × B with some particular adjoint triple in the frame.
After fixing a multi-adjoint frame and a context for that frame, the concept-
forming operators are denoted as ↑ : LB A ↓ A
2 −→ L1 and : L1 −→ L2 and are
B
B A
defined, for all g ∈ L2 , f ∈ L1 and a ∈ A, b ∈ B, as
^
g ↑ (a) = R(a, b) .σ(a,b) g(b) (1)
b∈B
^
↓
f (b) = R(a, b) -σ(a,b) f (a) (2)
a∈A
A multi-adjoint concept is a pair hg, f i satisfying that g ∈ LB A
2 , f ∈ L1 and
↑ ↓
that g = f and f = g. A hierarchy can be defined on the whole set of multi-
adjoint concepts, which gives rise to the concept lattice.
Definition 4. Given a multi-adjoint frame (L1 , L2 , P, &1 , . . . , &n ) and a con-
text (A, B, R, σ), a multi-adjoint concept lattice is the set
A ↑ ↓
M = {hg, f i | g ∈ LB
2 , f ∈ L1 , g = f, f = g}
in which the ordering is defined by hg1 , f1 i hg2 , f2 i if and only if g1 2 g2
(equivalently f2 1 f1 ).
276 M. Eugenia Cornejo et al.
3 Formal-concept operators with generalized quantifiers
This section will generalize the notion of generalized quantifier given in [25] by
using adjoint triples. Specifically, we will consider adjoint triples defined on the
complete lattice ([0, 1], ≤) assuming that their conjunctors have 1 as left and
right identity element.
Before presenting the notion of quantifier, we need to present the concept of
fuzzy measure invariant with respect to the cardinality.
Definition 5. Let U be a finite universe and P(U) be the powerset of U. We will
say that the mapping µ : P(U) → [0, 1] is a fuzzy measure, if it is an increasing
mapping satisfying that µ(∅) = 0 and µ(U) = 1. We will say that the fuzzy
measure µ is invariant with respect to the cardinality, if the following condition
holds:
If |A| = |B| then µ(A) = µ(B), for all A, B ∈ P(U)
where | · | denotes the cardinality of a set.
An example of fuzzy measure invariant with respect to the cardinality is given
by the mapping µrc : P(U) → [0, 1] defined as µrc (A) = |A| |U | , for all A ∈ P(U).
Notice that, if ϕ : [0, 1] → [0, 1] is an increasing mapping such that ϕ(0) = 0 and
ϕ(1) = 1, then the mapping µϕ : P(U) → [0, 1] defined as µϕ (A) = ϕ(µrc (A)), for
all A ∈ P(U), is also a fuzzy measure invariant with respect to the cardinality.
The former fuzzy measure µrc is called relative cardinality whereas the latter
fuzzy measure µϕ is called relative cardinality modified by ϕ or simply modified
relative cardinality. Both fuzzy measures were exemplified in [25].
Definition 6. Let U be a non-empty finite universe, F(U) = [0, 1]U be the set of
fuzzy sets of U on [0, 1], P(U) be the powerset of U, µ : P(U) → [0, 1] be a fuzzy
measure invariant with respect to the cardinality and (&, ., -) be an adjoint
triple w.r.t ([0, 1], ≤) such that x & 1 = 1 & x = x, for all x ∈ [0, 1].
• A mapping Qµ : F(U) → [0, 1] defined, for all C ∈ F(U), as:
! !
_ ^
Qµ (C) = C(u) & µ(D) (3)
D∈P(U )\{∅} u∈D
is called right quantifier determined by the fuzzy measure µ.
• A mapping µ Q : F(U) → [0, 1] defined, for all C ∈ F(U), as:
!!
_ ^
µ Q(C) = µ(D) & C(u) (4)
D∈P(U )\{∅} u∈D
is called left quantifier determined by the fuzzy measure µ.
Quantified Formal-Concept Operators 277
From now on, all results will be formulated with respect to a right quantifier
determined by a fuzzy measure µ. For that reason, the right quantifier will
be called simply quantifier. Notice that, all presented results can be translated
analogously for a left quantifier.
The following proposition shows that the universal and existential quantifiers
can be obtained from Definition 6 considering the minimum and maximum fuzzy
measures µ∀ and µ∃ , respectively, which are defined as follows:
( (
1 if D = U 0 if D = ∅
µ∀ (D) = µ∃ (D) =
0 otherwise 1 otherwise
Proposition 2. Given a non-empty finite universe U, the quantifiers Q∀ and
Q∃ determined by the minimum and maximum fuzzy measures µ∀ and µ∃ , re-
spectively, we have that Q∀ and Q∃ represent the universal and existencial quan-
tifiers. That is, for all C ∈ F(U), the following equalities are satisfied:
^
Q∀ (C) = C(u)
u∈U
_
Q∃ (C) = C(u)
u∈U
From a computational point of view, the notion of quantifier given in Equa-
tion (3) of Definition 6 is not suitable. This fact is due to the computation over
all sets from P(U) \ {∅} is required. In order to compute the quantifier in a more
efficient way, we propose an alternative computation procedure which makes
use of the property of being invariant with respect to the cardinality of fuzzy
measures.
Theorem 1. Let U = {u1 , . . . , un } be a universe, with |U| = n, and Qµ be a
quantifier determined by a fuzzy measure µ invariant with respect to the cardi-
nality. Then,
n
_
Qµ (C) = C(uπ(i) ) & µ({u1 , ..., ui }), C ∈ F(U)
i=1
where π is a permutation on {1, 2, . . . , n} such that C(uπ(1) ) ≥ C(uπ(2) ) ≥ · · · ≥
C(uπ(n) ).
As a direct consequence of Theorem 1, the following result proposes to com-
pute the quantifier considering a fuzzy measure built from the relative cardinal-
ity, which was illustrated at the beginning of Section 3.
Corollary 1. Let U = {u1 , . . . , un } be a universe, ϕ : [0, 1] → [0, 1] be an in-
creasing mapping such that ϕ(0) = 0, ϕ(1) = 1 and µ be a fuzzy measure built
from the relative cardinality, by using ϕ. Then,
n
_
Qµ (C) = C(uπ(i) ) & ϕ(i/n), C ∈ F(U)
i=1
278 M. Eugenia Cornejo et al.
where π is a permutation on {1, 2, . . . , n} such that C(uπ(1) ) ≥ C(uπ(2) ) ≥ · · · ≥
C(uπ(n) ).
After introducing a generalization of the definition of generalized quantifier
in the multi-adjoint concept lattices framework and showing its main properties,
we will focus on using the extended definition to present a generalization of the
usual fuzzy concept-forming operators.
Definition 7. Given a multi-adjoint frame and a context for that frame, µA ,
µB two fuzzy measures on A and B, respectively, which are invariant with respect
to the cardinality and QA , QB two quantifiers determined by the fuzzy measures
µA and µB , respectively, the quantified concept-forming operators are denoted
Q
as ↑QA : [0, 1]B −→ [0, 1]A and ↓ B : [0, 1]A −→ [0, 1]B , where [0, 1]B and [0, 1]A
denote the set of fuzzy subsets g : B → [0, 1] and f : A → [0, 1], respectively, and
are defined, for all g ∈ [0, 1]B , f ∈ [0, 1]A and a ∈ A, b ∈ B, as:
!
_ ^
↑QB σ(a,b)
g (a) = R(a, b) . g(b) & µB (X) (5)
X∈P(B)\{∅} b∈X
!
_ ^
↓QA
f (b) = R(a, b) -σ(a,b) f (a) & µA (Y ) (6)
Y ∈P(A)\{∅} a∈Y
Applying Corollary 1, we obtain the following characterization of the quanti-
fied concept-forming operators, where the fuzzy measures µA and µB are defined
from two fuzzy sets ϕA and ϕB , as follows:
µA ({a1 , ..., aj }) = ϕA (j/m)
µB ({b1 , ..., bi }) = ϕB (i/n)
for all subset {a1 , ..., aj } ⊆ A, {b1 , ..., bi } ⊆ B, with |A| = m and |B| = n.
Proposition 3. In the framework of Definition 7, the quantified concept-forming
Q
operators ↑QA : [0, 1]B −→ [0, 1]A and ↓ B : [0, 1]A −→ [0, 1]B , satisfy:
n
_
g ↑QB (a) = R(a, bπ(i) ) .σ(a,bπ(i) ) g(bπ(i) ) & ϕB (i/n) (7)
i=1
_m
QA
f↓ (b) = R(aλ(j) , b) -σ(aλ(j) ,b) f (aλ(j) ) & ϕA (j/m) (8)
j=1
for all g ∈ [0, 1]B , f ∈ [0, 1]A and a ∈ A, b ∈ B, where π and λ are permutations
such as:
R(a, bπ(i+1) ) .σ(a,bπ(i+1) ) g(bπ(i+1) ) ≤ R(a, bπ(i) ) .σ(a,bπ(i) ) g(bπ(i) )
R(aλ(j+1) , b) -σ(aλ(j+1) ,b) f (aλ(j+1) ) ≤ R(aλ(j) , b) -σ(aλ(j) ,b) f (aλ(j) )
Quantified Formal-Concept Operators 279
The next result shows that the quantified concept-forming operators ↑QA and
↓QB
form an antitone Galois connection when the quantifiers QA and QB are
determined by the minimum fuzzy measure µ∀ .
Proposition 4. Given the quantifiers Q∀A and Q∀B determined by the minimum
↑ ∀ ↑ ∀
fuzzy measure µ∀ on the universes A and B, respectively, then the pair ( QB , QA )
is an antitone Galois connection.
However, this result does not need to be satisfied for arbitrary quantifiers.
Hence, it would be interesting to study sufficient conditions to ensure that the
pair (↑QB , ↑QA ) be an antitone Galois connection.
4 Conclusions and future work
In this paper, we have introduced the monadic quantifiers of type h1i deter-
mined by fuzzy measures [15] in the general framework of multi-adjoint lattices.
Moreover, we have proven that the new definitions also have the universal and
existential quantifiers as particular cases. The second important result is the sim-
plification of the generalized quantifier definition taking into account that fuzzy
measures are invariant with respect to the cardinality. This result allows to de-
velop more simple and efficient proofs in this framework. Finally, the natural
and simple expressions of the quantified concept-forming operators have been
presented, and we have also shown that they form a Galois connection when
the minimum fuzzy measure is considered. However, this is not true in general.
Hence, in the future, sufficient conditions or new definitions will be studied in
order to provide Galois connections.
References
1. C. Alcalde, A. Burusco, H. Bustince, A. Jurio, and J. A. Sanz. Evolution in time
of L-fuzzy context sequences. Information Sciences, 326:202 – 214, 2016.
2. L. Antoni, S. Krajci, O. Kridlo, B. Macek, and L. Pisková. On heterogeneous
formal contexts. Fuzzy Sets and Systems, 234:22–33, 2014.
3. L. Antoni, S. Krajči, and O. Krı́dlo. Constraint heterogeneous concept lattices and
concept lattices with heterogeneous hedges. Fuzzy Sets and Systems, 303:21 – 37,
2016.
4. R. Bělohlávek. Lattices of fixed points of fuzzy Galois connections. Mathematical
Logic Quartely, 47(1):111–116, 2001.
5. R. Bělohlávek, T. Funiokova, and V. Vychodil. Galois connections with hedges. In
Proc. IFSA 2005 World Congress, volume II, pages 1250–1255. Springer, 2005.
6. P. Bosc, L. Lietard, and O. Pivert. Flexible database querying and the division of
fuzzy relations. Scientia Iranica, 2(4):–, 1996.
7. A. Burusco and R. Fuentes-González. Construction of the L-fuzzy concept lattice.
Fuzzy Sets and Systems, 97(1):109–114, 1998.
8. N. Cao, M. Holčapek, and M. Štěpnička. Extensions of fuzzy relational composi-
tions based on generalized quantifiers. Fuzzy Sets and Systems, 339:73 – 98, 2018.
Theme: Algebra and Set Theory.
280 M. Eugenia Cornejo et al.
9. N. Cao, M. Stepnicka, and M. Burda. Fuzzy quantifiers and compositions of partial
fuzzy relations employing dragonfly algebras. In 2019 IEEE International Confer-
ence on Fuzzy Systems, FUZZ-IEEE 2019, New Orleans, LA, USA, June 23-26,
2019, pages 1–6. IEEE, 2019.
10. M. E. Cornejo, J. Medina, and E. Ramı́rez-Poussa. A comparative study of adjoint
triples. Fuzzy Sets and Systems, 211:1–14, 2013.
11. M. E. Cornejo, J. Medina, and E. Ramı́rez-Poussa. Attribute reduction in multi-
adjoint concept lattices. Information Sciences, 294:41 – 56, 2015.
12. M. E. Cornejo, J. Medina, and E. Ramı́rez-Poussa. Multi-adjoint algebras ver-
sus non-commutative residuated structures. International Journal of Approximate
Reasoning, 66:119–138, 2015.
13. M. E. Cornejo, J. Medina, and E. Ramı́rez-Poussa. Characterizing reducts in
multi-adjoint concept lattices. Information Sciences, 422:364 – 376, 2018.
14. M. Delgado, D. Sánchez, and M. A. Vila. Fuzzy cardinality based evaluation of
quantified sentences. International Journal of Approximate Reasoning, 23(1):23 –
66, 2000.
15. A. Dvořák and M. Holčapek. L-fuzzy quantifiers of type h1i determined by fuzzy
measures. Fuzzy Sets and Systems, 160(23):3425 – 3452, 2009. Theme: Computer
Science.
16. A. Formica. Concept similarity in fuzzy formal concept analysis for semantic
web. International Journal of Uncertainty, Fuzziness and Knowlege-Based Sys-
tems, 18(2):153–167, 2010.
17. B. Galitsky and J. D. L. Rosa. Concept-based learning of human behavior for cus-
tomer relationship management. Information Sciences, 181(10):2016–2035, 2011.
18. O. Krı́dlo, S. Krajči, and L. Antoni. Formal concept analysis of higher order.
International Journal of General Systems, 45:116–134, 2016.
19. B. Li, X. Sun, and H. Leung. Combining concept lattice with call graph for impact
analysis. Advances in Engineering Software, 53(0):1–13, 2012.
20. J. Medina and M. Ojeda-Aciego. Multi-adjoint t-concept lattices. Information
Sciences, 180(5):712–725, 2010.
21. J. Medina and M. Ojeda-Aciego. On multi-adjoint concept lattices based on het-
erogeneous conjunctors. Fuzzy Sets and Systems, 208:95–110, 2012.
22. J. Medina, M. Ojeda-Aciego, J. Pócs, and E. Ramı́rez-Poussa. On the Dedekind-
MacNeille completion and formal concept analysis based on multilattices. Fuzzy
Sets and Systems, 303:1 – 20, 2016.
23. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño. Formal concept analysis via
multi-adjoint concept lattices. Fuzzy Sets and Systems, 160(2):130–144, 2009.
24. J. M. Rodrı́guez-Jiménez, P. Cordero, M. Enciso, and A. Mora. Data mining
algorithms to compute mixed concepts with negative attributes: an application
to breast cancer data analysis. Mathematical Methods in the Applied Sciences,
39(16):4829–4845, 2016.
25. M. Štěpnička and M. Holčapek. Fuzzy relational compositions based on generalized
quantifiers. In A. Laurent, O. Strauss, B. Bouchon-Meunier, and R. R. Yager,
editors, Information Processing and Management of Uncertainty in Knowledge-
Based Systems, pages 224–233, Cham, 2014. Springer International Publishing.
26. F. J. Valverde-Albacete, J. M. González-Calabozo, A. Peñas, and C. Peláez-
Moreno. Supporting scientific knowledge discovery with extended, generalized
formal concept analysis. Expert Systems with Applications, 44:198 – 216, 2016.