=Paper=
{{Paper
|id=Vol-2519/paper1
|storemode=property
|title=A Category-theoretic Approach for the Detection of Conservativity Violations in Ontology Alignments
|pdfUrl=https://ceur-ws.org/Vol-2519/paper1.pdf
|volume=Vol-2519
|authors=Cauã Roca Antunes,Alexandre Rademaker,Mara Abel
|dblpUrl=https://dblp.org/rec/conf/ontobras/AntunesRA19
}}
==A Category-theoretic Approach for the Detection of Conservativity Violations in Ontology Alignments==
https://ceur-ws.org/Vol-2519/paper1.pdf
A Category-theoretic Approach for the Detection of
Conservativity Violations in Ontology Alignments
Cauã Roca Antunes1, Alexandre Rademaker2, Mara Abel1
1
Instituto de Informática – Universidade Federal do Rio Grande do Sul (UFRGS)
Caixa Postal 15.064 – 91.501-970 – Porto Alegre – RS – Brazil
2
Escola de Matemática Aplicada – Fundação Getúlio Vargas (FGV)
Rio de Janeiro – RJ – Brazil
crantunes@inf.ufrgs.br, arademaker@gmail.com, marabel@inf.ufrgs.br
Abstract. Ontologies are formal specifications that enable inferential
processes over shared knowledge. In distributed contexts, applications
frequently need to access information from multiple ontologies. For this end,
concepts of two different ontologies must be matched through an alignment. If
the alignment is not semantically sound, however, the integration of the
ontologies may lead to unintended consequences. One type of possible
consequences is the introduction of new subsumption relations between
concepts from one of the input ontologies, which violate the conservativity
principle. We propose a method based on the mathematical formalism of
Category Theory for detecting such violations.
1. Introduction
The Semantic Web is an extension of the traditional World Wide Web where
information is given well-defined meaning [Berners-Lee et al. 2001]. Such meaning is
specified in ontologies, i.e., formal and explicit specifications of a shared
conceptualization [Studer et al. 1998]. However, different people and groups may build
distinct ontologies dealing with the same subject or domain. Applications frequently
have to access multiple related ontologies in order to integrate all required information.
In order to allow this, modelers must create alignments between the ontologies, either
manually or automatically. Such alignments frequently match concepts imperfectly,
causing inconsistencies.
The conservativity principle states that the merge of two ontologies through an
alignment should not introduce new subsumption relations between concepts from the
same source ontology. We say that a concept c subsumes a concept d if every instance
of d is also an instance of c. If an alignment violates the conservativity principle, the
merged ontology does not preserve the original meaning specified by the source
ontology. For example, if an alignment matches concepts person, client and company
from two ontologies A and B, where ontology A specifies that person subsumes client
(that is, every client is a person) and ontology B states that client subsumes company
(i.e., every company is a client), a query for person in the merged ontology will return
every instance of company in the knowledge base, which is clearly not the expected
result. We distinguish two types of conservativity principle violations:
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
1. Violation of subsumption conservativity. An alignment violates subsumption
conservativity if it introduces new subsumption relations between concepts from
the same input ontology.
2. Violation of equivalence conservativity. An alignment violates equivalence
conservativity if it introduces new equivalence relations between concepts from
the same input ontology. This may happen due to the introduction of a circular
chain of subsumption relations or due to two concepts in one of the source
ontologies being mapped to a single concept in the other ontology.
Category theory is a branch of mathematics that studies the structure present in
systems of composable relations. These relations, called morphisms, are abstractions
from several distinct mappings between mathematical objects, including functions, set-
theoretic relations, graph homomorphisms, linear mappings between vector spaces, and
others. Category theory and its morphisms provide a sound formal basis for the study of
ontologies and their alignments. We use the formalisms of category theory to reduce the
problem of detecting conservativity principle violations to the computation of two
pullbacks1 followed by a verification of the existence of a particular morphism between
them.
The remainder of this paper is organized as follows. Section 2 introduces the
fundamentals of category theory and describes the concepts that are relevant to this
work. Section 3 presents works from the literature that deal with categories of
ontologies and their constructions, and discuss other approaches to the problem of
detecting conservativity violations. We present a category of ontologies in Section 3 and
describe our method in Section 5. Section 6 contains a brief discussion of the merits of
our approach and aspects for future improvement.
2. Category Theory Fundamentals
[Adámek et al., 1990] defines a category as a quadruple C = (O, hom, id, ◦), consisting
of:
a class O whose members are C-objects,
for each pair (A,B) of C-objects, a set hom(A,B), whose members are C-
morphisms from A to B,
for each C-object A, a morphism idA:A→A, called the C-identity on A, and
a composition law ◦ associating each pair of C-morphisms f:A→B and g:B→C to
a C-morphism g ◦ f:A→C, called the composite of f and g.
Subject to the conditions that (1) composition is associative, that is, for any three
morphisms f:A→B, g:B→C and h:C→D, h ◦ (g ◦ f) = (h ◦ g) ◦ f, (2) C-identities are
neutral with respect to composition, i.e., for any morphism f:A→B, idB ◦ f = f = f ◦ idA,
and (3) the sets hom(A,B) are pairwise disjoint.
A diagram in a category A is a selection of some of its objects and morphisms. A
source for a diagram is a pair (x,fi), consisting of an object x and a family of morphisms
fi:x→di with domain x and codomain indexed by the diagram, that is, a group of
morphisms from x to each object in the diagram. If for any morphism g:di→dj in the
diagram the triangle formed by g, fi and fj commutes, i.e., g ◦ fi = fj, then the source (x,fi)
1 We describe pullbacks, morphisms and other category-theoretic concepts in Section 2.
is called a cone. If (x,fi) is a terminal cone, that is, for every other cone (x’,fi’) there
exists a unique morphism h:x’→x such that the resulting diagram commutes, (x,fi) is a
limit. We show a cone and a limit for a diagram with the morphism g:d1→d2 in Figure
1a.
x’
f1’ h f2’
f1 x f2
d1 d2
g
Figure 1a. A limit x and a cone x’.
If we reverse the direction of the morphisms in the previous definitions, that is,
if we exchange each morphism’s domain for its codomain, we arrive at the dual
categorical constructions. Thus, the dual to a source is a sink, a pair (x,fi) consisting of
an object x and a family of morphisms fi:di→x with codomain x and domain indexed by
the diagram, that is, a group of morphisms from each object in the diagram to x. A
commutative sink is a cocone, which is dual to a cone. An initial cocone is a colimit,
i.e., the dual to a limit. We show a cocone and a colimit for a diagram with the
morphism g:d1→d2 in Figure 1b.
g
d1 d2
f1 f2
x
f1’ h f2’
x’
Figure 1b. A colimit x and a cocone x’.
In this work we are particularly interested in a specific type of limit and its dual
colimit, which are respectively pullbacks and pushouts. Pullbacks are limits of diagrams
containing two morphisms f:A→C and g:B→C with a shared codomain. Pushouts are
colimits of diagrams containing two morphisms f:A→B and g:A→C that share their
domain. Figure 2 depicts both constructions.
x’ A
B C
x x
A B
C x’
Figure 2. A pullback x with a cone x’ (left) and a pushout x with a cocone x’ (right).
3. Related Work
Traditionally, authors define categories of ontologies with total mappings as morphisms.
Since alignments are rarely complete, dealing instead with only a subset of the ontology
concepts and relations, they need to be formalized as more complex structures. [Bench-
Capon and Malcom 1999] defines relations between two ontologies O1 and O2 as
structures composed of a third ontology O and morphisms xi : O→Oi for i = 1, 2. Thus,
entities in O1 and O2 are “matched” by being mapped from the same entity in O by x1
and x2. Later, [Zimmerman et al. 2006] named such structures V-alignments due to their
shape, and defined the operation of ontology merging as a pushout over the alignment,
as well as three operations over alignments using limits and colimits, namely alignment
composition, union and intersection. In addition to the category-theoretic constructions
described in previous works, [Cafezeiro and Haeusler 2007] and [Cafezeiro et al. 2008]
demonstrated that the pullback over two ontology mappings is the intersection of two
ontologies in the context of a third, broader one. We shall build upon these
constructions in the following sections.
To the best of our knowledge, no other work has dealt with the problem of
detecting conservativity violations in a category-theoretic context. However, several
approaches based on different formalisms can be found in the literature. The approach
proposed by [Jiménez-Ruiz et al. 2009] checks only for violations of equivalence
conservativity (type 2 described above) by verifying if the alignment maps directly two
different concepts from one source ontology to a single concept in the other, ignoring
the cases when such violations arise indirectly from the inclusion of circular
subsumption relations.
The method applied by [Ivanova and Lambrix 2013] and [Lambrix and Liu
2013] computes the integrated ontology over a network of alignments and use a
reasoner to infer new subsumption relations. Nevertheless, they treat the introduction of
new subsumption relations as evidence of incompleteness in the source ontologies, and
not of an incorrect alignment.
[Solimando et al. 2014] reduce the problem of detecting conservativity
violations to one of concept satisfiability. In order to do so, the authors follow the
assumption of disjointness, which states that all concepts that do not share subsumees
are disjoint. For many ontologies, however, this is not a reasonable assumption, since
expliciting common subsumees for every pair of concepts frequently leads to a
combinatorial explosion of concepts. The same authors later introduced a technique for
the detection of equivalence conservativity violations by searching for loops in graphs
where each node represent a concept and each arc a subsumption relation, using the two
approaches together in a multi-strategy method for detecting conservativity violations of
both types [Solimando et al. 2017].
4. A Category of Ontologies
We begin by defining our category Ont of ontologies. Objects in Ont are ontologies in
the form of tuples (C, R, S, A), where C is a set of concepts, R is a set of relations
between concepts, S ⊆ C×C C×C is a transitive, reflexive and antisymmetric subsumption
relation given by S(c, d), i.e., c subsumes d ↔ ∀ x, instanceOf (x, d) → instanceOf (x, x, instanceOf (x, d) → instanceOf (x,
c), and A is a set of axioms governing such concepts and relations. With this definition
we intend to abstract from representational aspects and therefore we assume that every
concept and relation is explicit in the ontology tuple. If the actual representation of the
ontology (in some ontology representation language) contains implicit knowledge that
needs to be inferred, the required reasoning tasks must be performed as a preparatory
step. Morphisms in Ont are total ontology mappings f:A→B between ontologies A and
B with components fC:CA→CB and fR:RA→RB which map respectively concepts and
relations2, such that the mappings preserve relations, that is:
(1)∀ x, instanceOf (x, d) → instanceOf (x, c, d ∈ C CA, SA(c, d) → SB (fC(c), fC(d)), and
(2)∀ x, instanceOf (x, d) → instanceOf (x, r ∈ C RA, ∀ x, instanceOf (x, d) → instanceOf (x, c, d ∈ C CA, r(c, d) → ∃ r’ ∈ R r’ ∈ C RB, fR (r) = r’ ∧ r’(fC (c), fC (d)).
Composition in Ont is usual function composition on each component of the
morphisms. Since function composition is associative and always exists for any two
functions f and g with Codomain(f) = Domain(g), in order to prove that the category
laws for composition hold in Ont, we must prove that the composition of two
morphisms is always a morphism and that identities exist and are neutral with respect to
composition. This means that given two morphisms f:A→B and g:B→C, which follow
rules (1) and (2), g ◦ f must also follow such rules. Since f preserves relations, for any
relation r in RA that holds between concepts c and d in CA, fR(r) must hold between fC(c)
and fC(d). Since g is a total mapping on both concepts and relations, it maps fR(r) to a
relation in RC and both fC(c) and fC(d) to concepts in CC. Additionally, since g also
preserves relations, we have that gR(fR(r)) must hold between gC(fC(c)) and gC(fC(d)).
Similarly, since both f and g preserve subsumption, for any pair of concepts c and d in
CA, if c subsumes d, then fC(c) subsumes fC(d) and gC(fC(c)) subsumes gC(fC(d)). Identities
simply map each concept and relation to itself, i.e., iC(c) = c and iR(r) = r. As required
for identities, such mappings are neutral on composition, since for any f, g and x,
i(f(x)) = f(x) and g(i(x)) = g(x).
5. Our Approach
Given ontologies A and B and a V-alignment (V, fi:V→i) for i = A, B, we wish to verify
if the alignment leads to a violation of the conservativity principle. Since every concept
and relation in V must be mapped by each fi to a concept or relation in the corresponding
ontology, the new subsumption relations introduced by the alignment cannot possibly be
in V. Instead, they must hold in the other aligned ontology, between concepts which are
mapped by fj, where j ≠ i. Thus, in order to answer our original question, we must find
the sub-ontologies A’ and B’, where A’ contains only the concepts and relations in V
plus every subsumption relation between the mapped concepts in A and B’ contains only
the concepts and relations in V plus every subsumption relation between the mapped
concepts in B, and check if A’ and B’ share all their subsumption relations, i.e., there is
no subsumption relation in A’ that is not in B’ and vice versa.
Therefore, first we must find A’ and B’. We achieve this by computing the
pullbacks (i’, gi :i’→i, hi :i’→V*) of the diagrams containing the morphisms ji :i→i* and
k:V*→i*, where V* is an ontology with the same concepts and relations from V but with
additional subsumption relations such that each concept subsumes every other, and i* is
the merge of V* and i computed through a pushout using V as alignment, along with fi
and the trivial inclusion v:V→V*. That is, i* is the ontology i extended with additional
subsumption relations so that each concept in the image of fi subsumes every other. We
propose that the ontology i’ in the pullback just described is the smallest sub-ontology
from i with every concept in V and every subsumption relation that holds between them
in i.
2 Axioms are not yet in the scope of this definition
Proof. Suppose there exists a concept c in i to which a concept in V is mapped
by fi and there is no concept in i’ that is mapped by gi to c. Then, we may build an
ontology i’’ that contains every concept and relation in i’ plus c, along with morphisms
q:i’’→i and r:i’’→V* which map every concept and relation to itself. Thus, (i’’, q, r) is
a cone for the diagram with morphisms ji and ki and there is no morphism p:i’’→i’ such
that gi ◦ p = q and hi ◦ p = r, since c is in the image of q but not of gi. Therefore, i’ is not
the pullback, contradicting our initial assumption. Similarly, if we suppose that there
exists a subsumption relation s in i that is not in i’, we may construct a different i’’ with
every concept and relation in i’ plus s, mapping concepts and non-subsumption relations
exactly as for i’. Thus, we again have a cone for the diagram, and again no morphism
p:i’’→i’ can be found, since such mapping would break the requirement that morphisms
should preserve subsumption relations. Therefore, i’ again is not the pullback. On the
other hand, if there exists a concept c in i’ that is not in V, then it either is mapped to a
concept in i that is also not in V, and thus there exists no hi such that the diagram
commutes, or it is mapped to a concept in i to which another concept d is also mapped,
and then there would exist an ontology i’’ where c and d are collapsed in a single
concept e, along with morphisms q:i’’→i and r:i’’→V* such that
gi (c) = gi (d) = q(e) and hi (c) = hi (d) = r(e). However, there are two morphisms p1 and
p2:i’’→i, with p1 (e) = c and p2 (e) = d, such that the diagram commutes, that is,
gi ◦ p1 = q = gi ◦ p2 and hi ◦ p1 = r = hi ◦ p1, contradicting the uniqueness restriction from
the definition of limit.
(1) (2)
i* i*
ji ki ji ki
i V* i V*
fi v gi hi
V i’
(3) A* B*
jA kA jB kB
A V* B
hA hB
gA A’ gB
B’
f’A f’B
V
(4) (5)
mA mB
A’ B’ A’ B’
fA fB fA fB
V V
Figure 3. Our approach step by step.
Since (i’, gi, hi) is the pullback and (V, fi, v) is a cone for the diagram, depicted
respectively in (2) and (1) in Figure 3, then there exists a single morphism fi’ :V→i’ such
that gi ◦ fi’ = fi and hi ◦ fi’ = v. Considering that A’ and B’ have exactly every concept and
relation in V plus any subsumption relation between those concepts in A or B
respectively, if there is a morphism mA :A’→B’ such that mA ◦ kA = k B, then every
subsumption relation between concepts in V that holds in A also holds in B. Otherwise,
i.e., if there is no such morphism, the alignment introduces at least one new
subsumption relation between concepts from B and thus violates the conservativity
principle. Symmetrically, if there is no mB :B’→A’ such that mB ◦ k B = k A, a new
subsumption relation is introduced between concepts from A.
Figure 3 depicts our approach step by step. Step (1) is the construction of i*
trough a pushout. Step (2) is the computation of i’ as a pullback. Then, (3) shows the
complete diagram after both sides of the alignment have been analyzed. (4) and (5)
show the commutative triangles formed with mA and mB respectively. We note that this
approach is also enough to detect equivalence conservativity violations, since if two
concepts c and c’ in A are mapped to a single concept d in B, then it is impossible to
build a mapping mB:B’→A’ such that mB(d) = c ∧ mB(d) = c’ unless c = c’.
From the category-theoretic constructions previously described, we build the
Algorithm 1 to find the sub-ontologies A’ and B’. Taking advantage of the knowledge
that i* is the pushout over fi and v, and that therefore the only concepts and non-
subsumption relations in i’ are those in the image of fi, the algorithm constructs i’
directly from the mapping and then includes the subsumption relations found in i.
Algorithm 1. findSubOntology algorithm for finding minimum sub-ontology
Input: V, i: ontologies; fi:V→i: mapping.
Output: i’: minimum sub-ontology containing all subsumption relations in i between concepts in V.
fi’:V→i’: ontology mapping.
1: Ci’ ← ∅ // initialize the set of concepts of the sub-ontology as an empty set
2: Si’ ← ∅ // initialize the subsumption relation in the sub-ontology as an empty relation
3: fCi’ ← ∅
4: for each c1 ∈ C CV do // for every concept in V
5: if fCi(c1) ∉ Ci’ do // if it is not yet in i’
6: Ci’ ← Ci’ ∪ { {fCi (c1)} // add it to i’
7: for each fCi (c2) ∈ C Ci’ do // then, for every concept already in i’
8: if (fCi (c1), fCi (c2)) ∈ C Si do // check if they should subsume
9: Si’ ← Si’ ∪ { {(fCi (c1), fCi (c2))} // each other and add the relation
10: end if
11: if (fCi (c2), fCi (c1)) ∈ C Si do
12: Si’ ← Si’ ∪ { {(fCi (c2), fCi (c1))}
13: end if
14: end for
15: end if
16: fCi’ ← fCi’ ∪ { {(c1, fCi (c1))} //update fi’ with the new concept mapping
17: end for
18: Ri’ ← ∅
19: fRi’ ← ∅
20: for each r ∈ C RV do // for every relation in V
21: if fRi(r) ∉ Ri’ do // if it is not yet in i’
22: Ri’ ← Ri’ ∪ { {fRi (r)} // add it
23: end if
24: fRi’ ← fRi’ ∪ { {(r, fRi (r))} //update fi’ with the new relation mapping
25: end for
26: i’ ← (Ci’, Ri’, Si’, ∅)
27: fi’ ← (fCi’, fRi’)
28: return (i’, f i’)
Algorithm 2 takes two ontologies and a V-alignment as input and checks if the
alignment is conservative. We use the Algorithm 1 to find the corresponding sub-
ontologies and then build mappings between them by matching each entity in each
ontology to the concept or relation in the other to which it is aligned. Then, the
algorithm checks if the mappings are functional (i.e., no entity may be mapped to more
than one entity in the target ontology) and if they preserve the subsumption relations, as
required for morphisms in our category Ont of ontologies. If they do, we have a
morphism between the sub-ontologies and, as previously discussed, the V-alignment
does not violate the principle of conservativity. We note that the operator ⊕ here here
denotes exclusive logical disjunction, i.e., p ⊕ here q is true if and only if p is true or q is
true, but not both.
Algorithm 2. isConservative algorithm for detecting conservativity violations
Input: A, B: ontologies; (V, fA:V→A, fB:V→B): V-alignment.
Output: true if the alignment is conservative, false otherwise.
1: (A’, fA’) ← findSubOntology(V, A, fA) // find sub-ontology A’
2: (B’, fB’) ← findSubOntology(V, B, fB) // find sub-ontology B’
2: mA ← ∅
3: mB ← ∅
4: flag ← true
5: for each e ∈ C CV ∪ { RV do // for each entity in the alignment
6: if fA’ (e) ∈ C Dom (mA) do // if its match in A’ is already mapped to something by mA
7: if mA (fA’ (e)) ≠ fB’ (e) do // and it does not match the mapping by fB’
8: flag ← false // then the alignment is not conservative
9: end if
10: else do: // if it is not mapped to anything by mA
11: mA ← mA ∪ { {(fA’ (e), fB’ (e))} // map it to the same entity to which e is mapped
12: end if // by fB’
13: if fB’ (e) ∈ C Dom (mB) do // if its match in B’ is already mapped to something by mB
14: if mB (fB’ (e)) ≠ fA’ (e) do // and it does not match the mapping by fA’
15: flag ← false // then the alignment is not conservative
16: end if
17: else do: // if it is not mapped to anything by mB
18: mB ← mB ∪ { {(fB’ (e), fA’ (e))} // map it to the same entity to which e is mapped
19: end if // by fA’
20: if e ∈ C CV do // if the entity is a concept
21: for each c ∈ C Dom (mA) do // for each concept already mapped
22: if (fA’ (e), c) ∈ C SA’ ⊕ here (fB’ (e), mA (c)) ∈ C SB’ do
23: flag ← false // if subsumption is different in A’ and B’
24: end if // then the alignment is not conservative
25: if (c, fA’ (e)) ∈ C SA’ ⊕ here (mA (c), fB’ (e)) ∈ C SB’ do
26: flag ← false
27: end if
28: end for
29: end if
30: end for
31: return flag
Complexity Analysis. First, we note that variable assignments, equality checks and
logical operations such as exclusive disjunction all have constant time complexity, i.e.
O(1). With the right choice of data structure, checking if an element is in a set and
inserting a new element also present the same complexity. This is the case if we use a
presence-absence array for the concepts and an adjacency matrix for the subsumption
relation, for example. For Algorithm 1, we have two nested conditional loops, Loop 1.1
in lines 4-17 and Loop 1.2 in lines 7-14, followed by another loop, Loop 1.3, in lines
20-25. Loop 1.1 runs for nV iterations, where nV is the number of concepts in the
alignment V. Loop 1.2 runs a single iteration in the first iteration of Loop 1.1, two in the
second, and so forth, up to nV iterations in the last. This sum is the result of the formula
nV*(nV+1)/2. Loop 1.3 runs mV iterations, where mV is the number of relations in V.
Since many different relations may hold between any pair of concepts in V, mV may be
greater than nV2. Every other operation has constant complexity, as previously noted.
Therefore, the time complexity of Algorithm 1.1 is O(nV2 + mV). With the data structures
we have described, auxiliary space complexity is nV for the concepts, mV for the relations
and nV2 for the adjacency matrix for subsumption relations.
In Algorithm 2, there are two nested loops, Loop 2.1 in lines 4-30 and Loop 2.2
in lines 21-28. Loop 2.1 runs for nV + mV iterations, and Loop 2.2 runs an increasing
number of iterations up to nV, but only when the element selected in Loop 2.1 is a
concept. Thus, the time complexity is nV*(nV+1)/2 + mV, which is bound by O(nV2 + mV),
the same time complexity from both calls of Algorithm 1 in lines 1 and 2. We highlight
here that the complexity of both algorithms depends only on the size of the alignment,
and not of the aligned ontologies, which are usually much larger.
6. Conclusion
We have described a method based on category theory for the detection of
conservativity violations in ontology alignments. The foundation in category theory
allows the integration of our approach with other techniques and procedures defined in
the same mathematical formalism. Further, our proposal allows the detection of both
types of conservativity violations, i.e., subsumption conservativity violations as well as
equivalence conservativity violations, without requiring the merging of the aligned
ontologies, the execution of reasoners, or the assumption of disjointness, all of which
lead to great complexity when the input ontologies are large, as discussed in Section 3.
The technique described in this work still needs to be extended to repair the
discovered violations, as well as implemented and evaluated against reference datasets,
such as the ones provided by the Ontology Alignment Evaluation Initiative [Thiéblin et
al. 2018]. The evaluation would allow us to check (1) the number of violations detected,
(2) repaired and (3) the execution time of the algorithm. Another aspect that needs to be
further investigated is the preservation of conservativity over the alignment operations
defined by [Zimmerman et al. 2006]. Intuitively, we expect that the intersection of two
conservative alignments should also be conservative – however, this proposition still
needs to be proved.
Acknowledgemments
This work was partially funded by brazilian agencies CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior) and CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico).
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