=Paper=
{{Paper
|id=None
|storemode=property
|title=Compatible and Incompatible Ontology Mappings
|pdfUrl=https://ceur-ws.org/Vol-1001/paper1.pdf
|volume=Vol-1001
|dblpUrl=https://dblp.org/rec/conf/caise/Abbas13
}}
==Compatible and Incompatible Ontology Mappings==
https://ceur-ws.org/Vol-1001/paper1.pdf
Compatible and Incompatible Ontology Mappings
Muhammad Aun Abbas1 (Supervisor: Giuseppe Berio)
1
LabSTICC, University of South Brittany, 56000 Vannes, France
Muhammad-Aun.Abbas@univ-ubs.fr
Abstract. Upper ontologies are interesting because describing and representing
concepts that can be used across various domains (as opposed to domain ontol-
ogies). This feature may enable increased correctness of mappings between
domain ontologies, conceptual schema and languages. Unfortunately, there ex-
ist various upper ontologies and it is quite difficult to decide or to assess which
of them should be used in one application: the main reason is that upper ontolo-
gies are complex artifacts possibly specified in specific logics providing formal-
ization of highly abstract concepts. Researchers have been therefore interested
in understanding the similarities between upper ontologies by establishing map-
pings between key ontology concepts. In this paper, we review mappings pro-
posed in literature and we establish a notion of compatibility between these
mappings by introducing a method based on Galois connections. We then con-
clude with a synthesis of the results obtained by using the proposed method.
The key findings put in evidence some key differences leading to incompatibil-
ity among proposed mappings. These differences are worth to be further inves-
tigated.
Keywords: Upper ontologies, Ontology mappings, Galois connection
1 Introduction
In literature, what can be referred to as mappings between ontologies has been
studied in various contexts, especially in the case of domain ontologies. Mappings are
usually abstract or concrete functions or relations between ontology artifacts (often
concepts); mappings correspond to, precise or approximate, similarities, equality,
subsumption [1]. Mappings have been discussed and formalized in domain ontology
matching, alignment and merging; distinct approaches have been then proposed based
on logics [2], categories [3], argumentations , [4], and practice [5]. All these ap-
proaches require to use a specific formalization and not really usable for upper ontol-
ogies because mainly requiring description logics and/or, not very generic, easily
understandable, concepts belonging to ontologies). Some of those approaches are
focusing on the mapping correctness, trying to understand if a mapping is not logical-
ly contradictory when combined with the whole set of ontology axioms while some
other approaches focuses on what we name mapping acceptability, i.e. looking for
mappings that, according to formulated arguments, are more suitable than others.
We undertake an intermediate approach base on the key notion of mapping com-
patibility and incompatibility. Mappings are incompatible if they cannot be used con-
sistently together in one single application. However, it is quite important to note that
two incompatible mappings may be both correct (w.r.t. the whole set of ontology
axioms). Compatible and incompatible mappings are relevant at both run-time and
design time. At run-time, compatible and incompatible mappings are especially rele-
vant in peer to peer applications, distributed applications and agent based applications
and finally open interoperability focused applications. Indeed, all of such applications
often comprise several autonomous entities without any central point of control and
even without any common management procedure: each of these entities may use its
own vocabulary, reference schema, ontology and so on to map each message, flow,
variables and so on coming from other entities. At design-time, compatible and in-
compatible mappings can be used to understand distinct perspectives underlying in-
terpretations of ontological artifacts. For instance, to accomplish alignment, one con-
cept can be mapped on to another concept belonging to another ontology due to label
based similarities (and therefore the mapping is quite loose) while another mapping
can map one concept on to another one because their logical equivalence (within
some theoretical frameworks) can be established. Within the former mapping, the
various ontological artifacts (as concepts) are interesting because of their labels; with-
in the second mapping, the various ontological artifacts are interesting as logical arti-
facts (to perform for instance, reasoning).
Throughout the paper, we therefore provide a complete method for checking map-
ping incompatibility, especially in the case of upper ontologies. We consider this
notion very useful whenever there ontologies are formalized in distinct logics and
whenever the key point is not to evaluate if a mapping is correct but just if distinct
mappings can be used together (because they are not contradictory). Therefore, map-
ping compatibility and incompatibility will be formalized by using mechanisms which
are logic-independent (as for instance, the case of e-connections [6]).
1.1 Motivating Example
We have two agents and , they are using ontologies as their knowledge base.
First agent is based on DOLCE ontology and the second agent is based on GFO on-
tology and they have ontology mappings for others agent’s ontology. These ontology
mappings are independent to each other, i.e., these mappings may have different on-
tology correspondences for any concept. Fragment of DOLCE Ontology is as fol-
lows
And Fragment of GFO Ontology is as follows
Fragment of Alignment of ontologies and is as follows
( )
( )
Fragment of Alignment of ontologies and is as follows
( )
( )
Agent is based on Ontology and has alignment , while Agent is based on
ontology and has alignment . Agent sends two messages to Agent : a)
msg(Particular) b) msg(Event). We have to check whether the ontology artifacts used
in these messages are interpreted by the receiving agent are compatible with the sent
messages. For this we have to check whether the ontology correspondences for these
ontology artifacts are compatible in mappings and . For msg(Particular) both
alignments have compatible mappings for this message because they are same, while
for msg(Event) both alignments have incompatible mappings for this message be-
cause the mapped concepts are disjoint and nothing common between them.
This paper is structured as follows. In the next section we describe about Galois
connections and Ontology mapping compatibility and incompatibility and we present
a table about available upper ontology mappings. We present our proposed method in
Section 3. A synthesis of our work is presented in Section 4. We then present Related
Work in 5. Finally we conclude in Section 6.
2 Preliminaries
In this section we will describe about Galois connections and Ontology mapping
compatibility and incompatibility.
2.1 Galois connections
In the literature, two definitions of Galois connections are reported.
Definition 1: Given ordered structures A, B with partial order relationship and
antitone mappings and , we say that the pair ( ) establishes an
order reversing Galois connection between and if ( ) and ( ),
.
Definition 2: Given ordered structures with partial order relationship and
isotone mappings and , we say that the pair ( ) establishes
an order preserving Galois connection between and if ( ) and ( ),
.
The first definition can be seen as symmetric where the two mappings γ and α can-
not be differentiated; if order relationships are information orders then if a ≽A b this
means that a is less informed than b (the same as for instance in subsumption); γ and
α can be interpreted abstraction mappings, because applying one mapping result in
some information loss. The second definition is not symmetric because γ and α can be
differentiated; under the same interpretation of order relationships, α is an abstraction
mapping while γ is a concretization mapping because resulting in information en-
richment.
Galois connections are interesting whenever dealing with mappings between on-
tologies because at least:
1. Independent of the kind of formalization (such as the kind of logics) used to
represent ontologies, ontologies comprise at least one taxonomy to organize
artifacts, which corresponds to an information order as well;
2. Can be applied to concepts but also to relationships/properties [7] and to their
taxonomies.
2.2 Ontology mapping incompatibility and compatibility
Except approximated mappings, precise mappings between ontologies are basically
represented as functions or relationships between ontology artifacts. Representing
mappings as relationships enables to map one artifact on to several artifacts, which is
the position we are undertaking in the remainder. Let now suppose that , are two
ontologies and , are the sets of their concepts, ordered according to concept tax-
onomies. Starting from two concept mappings ( ) ,
( ), being used to complete the two ontology mappings f and g when-
ever they are undefined. The same can be done for relationships/properties defined in
the ontologies. Starting from mappings, for building functions on ordered sets as re-
quired by Galois connection, lets now define orders in power sets and in the
same way as follows:
where
represents the concept taxonomies in .
Two functions can then be easily defined as:
with ( ) ( ) and ( ) ( ) .
Functions α, γ may or may not respect conditions required in definitions 1 and 2.
Figures 1(a) to 1(d) show the relevant situations depending on the conditions required
for Galois connections. Each situation depicted in the figures can be associated with a
logical meaning (according to [8] as better explained in the remainder), which is
based on the fact that the ontology source of one mapping may be re-interpreted in the
ontology target of the same mapping.
Fig. 1.a Fig. 1.b
Fig. 1.c Fig. 1.d
Fig. 1.a provides the situation of an order preserving the Galois connection. It can
be shown that A,B,C,D (for instance, concepts) are interpreted as first order logics
symbols, (and f mapping by definition) leads to necessary conditions for being
interpreted into (a function is said to be an interpretation of into iff is
satisfied in all models of , by interpreting each symbol ‘s’ of as models of ( ).
This means that allows to infer at least the same formulas then when substitut-
ing symbols A, C with B and D (for instance, it is possible to infer because
). Because and are symmetric, also leads to necessary conditions for
being interpreted into . Fig. 1.b corresponds to the situation of an order reversing
Galois connection. By interpreting A,B,C,D as first order logic symbols, it is possible
to define an interpretation of into such that ( ) and
( ) ( is a negation connective). In practice, it is a negative inter-
pretation saying what is not instead of what is. Fig. 1.c depicts one acceptable situa-
tion in which Galois connection conditions are not satisfied. In this case, is an in-
terpretation of into , but cannot be interpreted in by therefore, f and g
mappings (on which and are built) are fundamentally distinct. Finally, Fig. 1.d
depicts another acceptable situation in which Galois connection conditions are not
satisfied. In this case, and bound concepts in completely distinct and independent
way. For instance, interprets A as B, therefore what is satisfied by B should be sat-
isfied in by A. Vice versa, interprets B as C, therefore what is satisfied by C
should be also satisfied by B in . Mappings f and g convey two distinct and inde-
pendent interpretations (without considering those concepts may be proved equiva-
lent).
According to the discussion above, the notion of “mapping incompatibility” can
now be introduced by the following definition.
Definition 3. Two mappings f and g between two ontologies are “compatible” iff
the corresponding functions α and γ are either an order preserving Galois connection
or an order reversing Galois connection. Two mappings that are not compatible are
said to be “incompatible”.
It should be noted that “mapping incompatibility” and “mapping compatibility” are
distinct notions that “mapping correctness” mentioned in the Introduction. Indeed,
compatible or incompatible mappings can be either correct or incorrect when com-
bined with additional ontology axioms.
Compatibilities and incompatibilities can also be stated at the level of ontology ar-
tifacts according to the following definition.
Definition 4. Given mappings f and g, functions α and γ built as above, an ontolo-
gy artifact A is compatible with ontology artifacts ( ) iff Galois connections
conditions are respected between A and ( ). Symmetrically, an ontology artifact
B is compatible with ontology artifacts ( ) iff Galois connections conditions are
respected between B and ( ). Otherwise involved artifacts are incompatible.
2.3 Existing upper Ontology Mappings
Comparing upper ontologies is usually performed by establishing (explicitly or im-
plicitly) some mappings between concepts belonging to distinct ontologies. We have
defined a methodology for collecting existing mappings (Table 1), then for analyzing
them according to Galois connections.
PROTON
Chisholm
WordNet
DOLCE
UEMO
SOWA
SUMO
Source
BWW
GFO
UFO
BFO
PDL
Cyc
PSI
[9] × ×
[9] × ×
[10] × ×
[11] × × × × ×
[12] × ×
[13] × × × ×
[14] × ×
[15] × ×
[16] × × ×
[17] × ×
[18] × ×
[19] × × × ×
Table 1. Existing Comparison of Upper ontologies concepts
Fig. 2. Steps for finding compatibilities and incompatibilities in ontology mappings
3 Proposed method
We propose a method to find compatibilities and incompatibilies between ontology
mappings (see Fig. 2). It has two main steps. (a) Collecting existing upper ontology
mappings, (b) Analyzing collected mappings.
3.1 Collecting existing upper ontology mappings
In most of the cases found in the literature, relationships between mapped concepts
are not qualified i.e. it remains unclear if the authors consider them as equivalence,
subsumption, similarity and so on. Our proposal based on Galois connections does not
require any information about the type of mapping such as equivalence, similarity and
so on. In some cases, the authors do not specify any mapping for some concepts. In
our proposal, we consider that the authors have tried to map all concepts, except if
otherwise stated.
In our proposal, we have also taken care of the “format”' that the authors use for
representing mappings. For mappings implicitly explained in the text, we have explic-
itly built a two column table for each couple of ontologies. For mappings directly
provided as two column tables, we have just considered the same tables.
When dealing with a multicolumn table involving more than two ontologies, we
have applied transitivity by assuming that, without specific assumptions provided by
authors, the authors have used the same types of mapping for all ontologies.
To arrange the collected mappings, we have used several four column tables
providing one mapping from some authors and an inverse mapping from some other
authors. Extract of those tables is shown in Fig. 3. Each of these four column tables
possibly provides distinct mappings between distinct ontologies, and can be used for
the further analysis step.
DOLCE map- BWW mapping by DOLCE mapping BWW map-
ping by [15] [15] by [13] ping by [13]
Entity Thing Entity Thing
System Endurant
Fig. 3. Arrangement of ontology mappings in four column table
3.2 Analyzing collected upper ontology mappings
The objective of this step is to verify if the couples of collected mappings respect
Galois connection conditions. Compatible mappings have been further distinguished
in weak compatible mappings and compatible mappings. The former rises whenever
Galois connection conditions are trivially respected because some concepts are
mapped to ⊥.
Hereinafter, the reader can find in several situations, how compatibilities and in-
compatibilities have been established. According to definition 4, compatibilities and
incompatibilities are stated per ontology artifacts. .
1. Trivial compatibility case
α ({Category})GFO = ({Abstract})Sowa
γ({Abstract})Sowa=({Category})GFO
α ᵒ γ ({Abstract})Sowa =α ({Category})GFO = ({Abstract})Sowa
γ ᵒ α({Category})GFO =γ({Abstract})Sowa=({Category})GFO
The situation above corresponds to (one type of) Galois connection for specific con-
cepts.
2. Weak compatibility case
α({MaterialStructure})GFO = ( )DOLCE γ({PhysicalEndurant})DOLCE = ({Materi-
alStructure})GFO
α ᵒ γ({PhysicalEndurant})DOLCE =α({MaterialStructure})GFO = ( )DOLCE ⊑ (Physi-
calEndurant)DOLCE
γ ᵒ α({MaterialStructure})GFO = γ( )DOLCE = ( )GFO ⊑ ({MaterialStructure})GFO
This means that the two mappings are compatible. It should be noted that the situation
does not much change if instead of the o tology root ⊤ woul h ve bee use .
Indeed:
α({MaterialStructure})GFO = (⊤)DOLCE
γ({MaterialStructure})GFO = ({PhysicalEndurant})DOLCE
α ᵒ γ ({PhysicalEndurant})DOLCE =α(MaterialStructure})GFO = (⊤)DOLCE
γ ᵒ α({MaterialStructure})GFO = γ (⊤)DOLCE = ⊤ ⊒({PhysicalEndurant})DOLCE
3. Compatibility case
A concept X in one ontology is mapped to some concept Y in other ontology, but in
another mapping performed by some other author(s), X is mapped to Z which is sub-
sumed by Y. i.e. Z Y. For instance
α({Process})GFO = ({Stative})DOLCE
γ({Process})DOLCE=({Process})GFO
α ᵒ γ ({Process})DOLCE= α({Process})GFO = ({Stative})DOLCE ({Process})DOLCE
Stative is more general than Process, and Process is an immediate descendant of
Stative.
γ ᵒ α({Process})GFO = γ({Stative})DOLCE = ({Process})GFO
which corresponds to a reverse ordering Galois connection.
4. Incompatibility case
A concept X in one ontology is mapped to some concept Y in another ontology, but X
is mapped to Z, while Y and Z are not ordered.
α({Region})GFO = ({Space Region})DOLCE
γ({Spatial Location})DOLCE=({Region})GFO
α ᵒ γ ({Spatial Location})DOLCE= α(Region)GFO = ({Space Region})DOLCE
γ ᵒ α({Region})GFO = γ({Space Region})DOLCE = ( )GFO =
Spatial Location and Space Region are subsumed by Physical Quality and Abstract
Region respectively that are not ordered. This situation therefore does correspond to
neither order reversing not order preserving Galois connection, rising in incompatibil-
ity.
4 Synthesis
The methodology presented in section 3 has been applied to available mappings
(shown in table 1) whenever the two required mappings α and γ are established by
distinct authors (however, it is possible to apply the methodology to mappings sup-
plied by same authors). Compatibilities due to mapping to are provided in italic
(because they are “weak compatibilities” due to our interpretation of partial map-
pings). Table 2 is interpreted as a compatible ontology mapping couples and incom-
patible ontology mapping couples. A compatible mapping couple observes the proper-
ties of Galois connection and Incompatible mapping couples does not respect the
properties of Galois connection. We have also applied our approach on GFO and
Sowa's ontology; DOLCE and SUMO; DOLCE and WordNET; DOLCE and BWW.
Table 2. Compatibilities and Incompatibilities of DOLCE and GFO
Mappings couple α [9], γ [11]
Compatibilities InCompatibilities
DOLCE GFO DOLCE GFO
Particular Individual Event {Change},
{Discrete Pro-
cess}
Entity Entity {Spatial Location}, Region
{Space Region}
Perdurant Occurrent
{Quality}, {Phys- Property
ical Quality}
Stative Process
Material Struc-Physical En-
ture durant
State State
Time interval Chronoid
Endurant {Presential,
Persistant}
Mappings couple α [13], γ [11]
Compatibilities InCompatibilities
DOLCE GFO DOLCE GFO
Particular Individual
Quality Quality
Entity Entity
5 Related Work
As said in the Introduction, several approaches to mappings between ontologies have
been proposed in the literature. These approaches are not intended for any kind of
ontology even if some rely on the specific formalization in which the ontology is
represented. Hereinafter, we are going to review approaches that in our opinion are
representative.
In [20] it is suggested to use Category theory approach for Ontology merging. In
[3], an algebraic approach i.e. categorical approach is used to formally describe ontol-
ogy merging, ontology alignment composition, union and intersection. They focus on
defining suitable categorical representation of ontology alignments. However they
define composition, union and intersection operation operations for ontology align-
ments without considering whether ontology alignments involve in these operations
may generate inconsistencies.
In [4] it is suggested to use Argumentation to argue about acceptability of map-
pings issued by distinct agents. The disadvantage is that in addition to mappings there
is the need to provide justification of such mappings, which in most cases are not
available. Our approach does not consider justifications of mappings, so it can handle
this kind of mappings. So However, our approach can be seen within the context of
arguments. Indeed, if it is possible to say that if two functions are incompatible,
couples ( ) or ( ) can be redefined as attacks. We do not define successful
attack, but conflict free set can contain either or , not both.
Researchers work on debugging and repairing of ontology alignment. [21] and [22]
consider that ontologies are correct but if there is some inconsistency it is caused only
by ontology mappings. [21] find the minimal conflict set that causes incoherent
alignment, and then removes the correspondences causing inconsistencies by mini-
mizing its impact. [22] uses the notion of minimal conflict sets and provides a map-
ping revision operator that modify alignment so that the result be consistent. Our in-
compatible mappings can be considered as minimal conflict set and can be first evalu-
ated by these approaches for possible incoherence.
6 Conclusion
This paper has presented an approach of comparing mappings proposed by differ-
ent authors. These mappings are usually stated for showing underlying similarities
between upper ontologies, even when these ontologies have been designed by using
fundamentally distinct design options. The methodology is based on Galois connec-
tions and on the definition of “mapping compatibility and incompatibility". We ap-
plied the methodology to available mappings (shown in Table 1) and in the paper we
have provided a synthesis of the results concerning relevant couples of upper ontolo-
gies. These results show that all the mappings established between upper ontologies
are weakly compatible or incompatible. We believe our work is useful in the context
of upper ontologies. As upper ontologies are based on different design options, so
they are very difficult to establish mappings between upper ontologies. So our work
of classifying upper ontologies correspondence into compatible and incompatible
mappings helps the user of upper ontologies when they used these ontologies in their
applications.
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