Semantic Ontology Alignment: Survey and AnalysisSemantic Ontology Alignment: Survey and AnalysisThis paper was downloaded from TechRxiv (https://www.techrxiv.org).
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CC BY 4.0
SUBMISSION DATE / POSTED DATE
13-12-2020 / 17-12-2020
CITATION
Boudaoud, Lakhdar El Amine (2020): Semantic Ontology Alignment: Survey and Analysis. TechRxiv. Preprint.https://doi.org/10.36227/techrxiv.13370786.v1
DOI
10.36227/techrxiv.13370786.v1
1
Semantic Ontology Alignment: Survey and
Analysis
Boudaoud Lakhdar El Amine
Computer science Laboratory of Oran (LIO), University of Oran1 Ahmed Ben Bella, Algeria
Abstract
Ontology alignment is an important part
in the semantic web to reach its full
potential, Recently, ontologies have
become competitive common on the World
Wide Web where they generic semantics
for annotations in Web pages, This paper
aims at counting all works of the ontology
alignment field and analyzing the
approaches according to different
techniques (terminological, structural,
extensional and semantic). This can clear
the way and help researchers to choose the
appropriate solution to their issue. They
can see the insufficiency, so they can
propose new approaches for stronger
alignment and also He determines possible
inconsistencies in the state of the ontology,
which result from the userโs actions, and
suggests ways to remedy these
inconsistencies.
Key Word: String Similarity, Alignment
Method, Alignment Process, Inconsist-
ency, Similarity aggregation, Semantic
web
1. Ontology in the Web of Thing and on
the AI
We developed a plugin algorithm for
ontology merging and alignmentโAOP
(formerly Smart)โwhich code the process
as much as possible. If the decision is not
possible, the plugin guides the user to the
paragraph that information are determined,
suggests possible actions, and determines
the conflictual situation.
Ontologies are a formal way to describe
taxonomies and classification networks,
essentially defining the structure of
knowledge for various domains: the nouns
representing classes of objects and the
verbs representing relations between the
objects [1] .
2. Related Work
In the literature, we have found many
methods of the similarity aggregation task,
as follows:
Falcon-Ao [3] uses three heuristic rules to
integrate results generated by a structural
matcher called GMO, and a linguistic
matcher called LMO. The heuristic rules
constructed by measuring both linguistic
and structural comparability of two
ontologies and computing a measure of
reliability of matched entity pairs. LMO
component of Falcon-Ao combines two
linguistic similarities with homogeneous
experimental combination weights. LILY
[4] combines all separate similarities with
homogeneous experimental combination
weights. When the User use linear
Combination weights , this is Method of
Euzenat and Valtchev [5], there is also
MapPso [6] as APFEL [7] who use an
average weighted function, RIMOM [8]
who take results of multiple strategies and
use risk minimization to search for optimal
mapping, It utilizes a sigmoid function
with a set of experimental parameters,
arriving at COMA [9] , then he uses some
strategies as min, max, average. GAOM
2
[10] integrate similarities with max
strategy, LSD [11] also use max, min and
average strategy, some system put weights
of matchers using various strategies such
as Experimental. The principal definition
between ontologies is How is calculate
Homogeneous weights.
3. Classification of Ontology Alignment
Methods
The Ontology Alignment is a combination
of Technique for calculating similarity
measures, there are some parameters who
are taken in ontology as Weights,
thresholds, or External resources (Thesa-
rus, dictionary) and we take the relationsh-
ip between entities that compose
Ontologies, There are several methods for
calculating similarity between entities of
several ontologies , Methods existing in the
Ontology alignments are :
3.1 Terminological methods (A Termin-
ological matcher) [12]
These methods compared terms and strings
or texts. They are used to calculate the
value of similarity between units of text,
such as names, labels, comments,
descriptions, etc. These methods can be
further divided into two sub-categories:
methods that compare the terms based on
characters in these terms, and methods
using some linguistic knowledge. The
difficulty is, on the one hand how to select
the most appropriate similarity measures
and, on the other hand, how to effectively
combine them. We cite as an example of
matcher of this category: the edit distance.
3.2 Structural methods (String Matc-
hing) [13] :
These methods calculate the similarity
between two entities by exploiting
structural information, when the concerned
entities are connected to the others by
semantic or syntactic links, forming a
hierarchy or a graph of entities. There are
two categories :
Internal structural methods, This
Method exploit information about entity
attributes,
External structural methods, This
Method consider relations between entities,
We cite as an example of a matcher of this
category: Resnik similarity.
3.3 Extensional methods [14]
This Method infer the similarity between
two entities, especially concepts or classes,
by analyzing their extensions, i.e. their
instances.
3.4 Semantic methods
3.4.1 Techniques based on the external
ontologies [15] Techniques based on the
external ontologies: When two ontologies
have to be aligned, it is preferable that the
comparisons are done according to a
common knowledge. Thus, these
techniques use an intermediate formal
ontology to meet that need. This ontology
will define a common context for the two
ontologies to be aligned
3.4.2 Deductive techniques Semantic
methods are based on logical models, such
as propositional satisfiability (SAT), SAT
modal or description logics. They are also
based on deduction methods to deduce the
similarity between two entities. Techniques
of description logics, such as the
subsumption test, can be used to verify the
semantic relations between entities, such as
equivalence (similarity is equal to 1), the
subsumption (similarity is between 0 and
1) or the exclusion (similarity is equal to
0), and therefore used to deduce the
similarity between the entities. These
alignment techniques are integrated into
approaches for mapping ontologies. We
find approaches that combine multiple
alignment techniques. Much work has been
3
developed in the area of Ontology and
focus on the alignment techniques [17].
3.5 Instance Based Method :
These methods exploit the instances
associated to the concepts (extensions) to
calculate the similarities between them.
We cite as an example of a matcher of this
category: Jaccard similarity
4. General Alignment Approaches
The general representation and reasoning
framework that we propose includes: 1) a
declarative language to specify networks of
ontologies and alignments, with
independent control over specifying local
ontologies and complex alignment
relations, 2) the possibility to align
heterogeneous ontologies, and 3) in
principle, the possibility to combine
different alignment paradigms
(simple/integrated/contextualized) within
one network. Through category theory, we
obtain a unifying framework at various
levels [18] :
Semantic level We give a uniform
semantics for distributed networks of
aligned ontologies, using the powerful
notion of colimit, while reflecting properly
the semantic variation points indicated
above [19] .
(Meta) Language level We provide a
uniform notation (based on the distributed
ontology language DOL) for distributed
networks of aligned ontologies, spanning
the different possible semantic choices
[20].
Reasoning level Using the notion of
colimit, we can provide reasoning methods
for distributed networks of aligned
ontologies, again across all semantic
choices [21] .
Tool level The tool ontohub.org provides
an implementation of analysis and
reasoning for distributed networks of
aligned ontologies, again using the
powerful abstractions provided by category
theory[22] .
Logic level Our semantics is given for the
ontology language OWL, but due to the
abstraction power of the framework, it
easily carries over to other logics used in
ontology engineering, like RDFS, first-
order logic or F-logic. This shows that
category theory is not only a powerful
abstraction at the semantic level, but can
properly guide language design and tool
implementations and thus provide useful
abstraction barriers from a software
engineering point of view [23] .
The distributed ontology language DOL is
a metalanguage in the sense that it enables
the reuse of existing ontologies as building
blocks for new ontologies using a variety
of structuring techniques, as well as the
specification of relationships between
ontologies. One important feature of DOL
is the ability to combine ontologies that are
written in different languages without
changing their semantics. A formal
specification of the language can be found
in [24]. However note that syntax and
semantics of DOL alignments is introduced
in this paper for the first time.
The general picture is then as follows:
existing ontologies can be integrated as-is
into the DOL framework. With our new
extended DOL syntax, we can specify
different kinds of alingments. From such
an alignment, we construct a graph of
ontologies and morphisms between the min
a way depending on the chosen alignment
framework. Sometimes, this step also
involves transformations on the ontologies,
such as relativisation of the (global)
domain using predicates. A network of
alignments can then be combined to an
integrated alignment ontology via a so-
called colimit. Reasoning in a network of
aligned ontologies is then the same as
4
reasoning in the combined ontology. Thus,
in order to implement a reasoner, it is in
principle sufficient to done the
relativisation procedure for the local logics
and the alignment transformation for each
kinds of semantics.
5. Network of Ontologies (Distributed
Ontologies Language) and there
Semantics
5.1 Preliminaries
In this section we present some
preliminary notions of ontology alignment
in order to facilitate the reading of the
paper content. We outline the notions of
ontology, similarity calculation techniques
and alignment, respectively. We refer the
reader, for more details, to the following
references [25] [26].
Definition 1: Ontology is a six tuple [27]:
๐ = < ๐ถ, ๐ , ๐ผ, ๐ป๐ถ, ๐ป๐ , ๐ > where:
๐ถ: set of concepts.
๐ : set of relations.
๐ผ: set of instances of C and R.
๐ป๐ถ: denotes a partial order relation on
C, called hierarchy or taxonomy of
concepts. It associates to each concept
its super or sub-concepts.
๐ป๐ : denotes a partial order relation on
R, called hierarchy or taxonomy of
relations. It associates to each relation
its super or sub-relations.
๐: set of axioms.
๐๐๐ : Denotes Networks of ontologies
๐ท๐บ โถ ๐บ๐๐๐๐๐๐ (๐๐๐ก๐๐) ๐ท๐๐๐๐๐
๐ท๐ โถ ๐๐๐๐ก๐๐๐ Domain
We define as a logic syntax a tuple L = (๐๐ข๐ ๐ง, ๐๐๐ง, ๐๐ฒ๐ฆ๐๐จ๐ฅ๐ฌ, ๐๐ข๐ง๐๐ฌ, ๐๐ฒ๐ฆ, ๐ค๐ข๐ง๐, ๐๐ซ๐ข๐ญ๐ฒ)
๐๐๐๐ ๐๐ ๐ก๐๐๐ ๐๐:
a category Sign ,signature morphisms;
a sentence ๐๐ข๐๐๐ก๐๐ (Sen)
๐๐๐ โถ ๐บ๐๐๐ โ ๐๐๐ก assigning to each
signature the set of its sentences and to
each signature morphism ๐: ๐ด โ
๐ดโ a sentence translation function
๐บ๐๐(๐) โถ ๐บ๐๐(๐ด) โ ๐บ๐๐(๐ดโฒ);
a set Symbols of symbols and a set
Kinds of symbol kinds together with a
function kind : Symbols โ Kinds
giving the kind of each symbol;
a faithful ๐๐ข๐๐๐ก๐๐
๐บ๐๐ โถ ๐บ๐๐๐ โ ๐ ๐๐ก assigning to
each signature ฮฃ
a set of symbols ๐บ๐๐(๐ฎ) โ
๐บ๐๐๐๐๐๐ and to each signature
morphism ๐: ๐ด โ ๐ดโฒ
a function ๐บ๐๐(๐) โถ ๐บ๐๐(๐ฎ) โ
๐บ๐๐(๐ฎโฒ) such that for each ๐ โ
๐บ๐๐(๐ฎ), ๐๐๐๐ (๐(๐)) = ๐๐๐๐ (๐)
a function ๐๐๐๐๐ โถ ๐บ๐๐๐๐๐๐ ๐ giving
the arity of each symbol.
Before giving examples of logic syntaxes,
we introduce the concept of logical theory
Definition 2 [28] Let L be a logic syntax.
A theory of L consists of signature ฮฃ and
a set E of ฮฃ -sentences.
For the purposes of this paper, it suffices to
regard an ontology as a theory. In DOL,
ontologies can be written using more
complex structuring mechanisms.
Example 1 In ALC, the signatures are
tuples (๐ด, ๐ , ๐ผ ) with A, R, and I Subsets of
a set of names. For two signatures ๐ด =
(๐ด, ๐ , ๐ผ)and ๐ดโฒ = (๐ดโฒ, ๐ โฒ, ๐ผโฒ), a signature
morphism,ฯ: ฮฃ โ ฮฃโฒConsiste of function โถ
ฯA โถ A โ Aโฒ
๐ ๐๐ข๐๐๐ก๐๐n: ฯR โถ R โ Rโฒ
๐ ๐๐ข๐๐๐ก๐๐n: ฯI โถ I โ Iโฒ
๐ฒ๐๐๐ ๐ is the set {concept, role,
individual}.
๐บ๐๐๐๐๐๐ is the set of all pairs (๐, ๐)
where k is an element of ๐ฒ๐๐๐ ๐ and s is a
name. For each (๐, ๐ ) โ ๐บ๐๐๐๐๐๐,
๐๐๐๐ (๐, ๐) = ๐. For each signature
๐ด = (๐ด, ๐ , ๐ผ), ๐๐ฆ๐(๐ด) is the union of the
๐ ๐๐ก {(๐๐๐๐๐๐๐ก, ๐)|๐ โ ๐ด} with
{(๐๐๐๐, ๐)|๐ โ ๐ } and
{(๐๐๐๐๐ฃ๐๐๐ข๐๐, ๐)| ๐ โ ๐ผ}.
5
5.2 Networks of Ontologies
In this section we recall networks of
aligned ontologies and introduce syntax for
them in DOL. Networks of aligned
ontologies (here denoted NeO) [29], called
distributed systems [43] , consist of a
family (๐๐)๐โ๐ผ๐๐ of ontologies over a set of
indexes Ind interconnected by a set of
alignments (๐ด๐๐)๐โ๐ผ๐๐ between them.
Definition 3 Let L be a logic syntax and
let R be a family of correspondence
relations for L. A correspondence is a
๐ก๐๐๐๐๐ (๐ 1, ๐ 2, ๐ )where ๐ 1, ๐ 2 โ ๐๐ฆ๐๐๐๐๐
and ๐ โ ๐ such that (kind(๐ 1), kind(๐ 2)) is
in the set of kinds of R
Definition 4 For two ontologies S, T , an
alignment between S and T is a set of
correspondences {(๐ 1๐ , ๐ 2
๐ , ๐ ๐)}๐=1,2,โฆ๐
for
๐ โ ๐, such that for each ๐ = 1, . . . , ๐ we
have that ๐ 1๐ โ ๐๐ฆ๐(๐ ๐๐(๐)), ๐ 2
๐ โ ๐๐ฆ๐(๐ ๐๐(๐)),
๐๐๐ ๐ ๐ โ โ
Example 2 Below are the types of
relations that can appear in
correspondences between ALC symbols,
together with their kinds:
=
{(concept, concept), (role, role),
(individual, individual)}
โฅ {(concept, concept), (role, role),
(individual, individual)}
<
{(concept, concept), (role, role)}
>
{(concept, concept), (role, role)}
โ {(individual, concept)}
โ {(concept, individual)}
Example 3 Similarly, in FOL we have the
following correspondences:
= {(fun, fun), (pred, pred)}
โฅ {(fun, fun), (pred, pred)}
< {(pred, pred)}
> {(pred, pred)}
6 Three Semantics of Relational
Networks of Ontologies
We will now generalise the three semantics
for networks of aligned ontologies
introduced in [43] to an arbitraries logic.
A semantics of relational NeOs is given in
terms of local interpretation of the
ontologies and alignments In consists of.
To be able to give such a semantics, one
needs to give an interpretation of the
relations between symbols that are
expressed in the correspondences. let
๐ = {(๐๐ )๐โ๐ผ๐๐, (๐ด๐๐ )๐,๐โ๐ผ๐๐}, be a NeO
(in any logic) over a set of indexes Ind.
6.1 Simple Semantics
In the simple semantics, the assumption is
that all ontologies are interpreted over the
same domain (or universe of interpretation)
๐ท. The relations in ๐ are interpreted as
relations over D, and we denote the
interpretation of R โ R by ๐ ๐ท. If ๐1, ๐2
are two ontologies and c = (๐1, ๐2, ๐ ) is a
correspondence between ๐1 and ๐2, we
say that c is satisfied by interpretations m1,
m2 of ๐1, ๐2 if ๐1 (๐1)) ๐ ๐ท ๐2(๐2)).
This is written ๐1, ๐2| =๐ c.
Definition 5 Given a model theory for a
logic L, the interpretation of
correspondence relations relative to a set is
an interpretation function .๐ผ taking as
arguments a relation ๐ โ ๐ , ๐๐๐๐ (๐1, ๐2)๐๐ ๐ ๐๐๐ a set X and giving
as result a relation ๐ ๐ผ ๐๐๐๐๐๐(๐2, ๐).
๐ท1 ๐ท2 ๐ท4 ๐ท5
๐1
๐ท3
๐2 ๐4 ๐5
๐3
๐5
๐5โฒ ๐1โ
๐3โฒ ๐4โฒ
๐1
๐2โฒ
๐ท๐บ
๐3 ๐4 ๐2
6
๐ผ๐ ๐1, ๐2 ๐๐๐ two ontologies and ๐ =(๐ 1, ๐ 2, ๐ ) is a correspondence between
๐1 ๐๐๐ ๐2, we say that c is satisfied by the
models ๐1, ๐2, ๐๐๐1, ๐2, written
๐1, ๐2| =๐ ๐ถ, if and only if ๐๐1
1 ๐ ๐ท1 ๐๐2
2
model of an alignment A between
ontologies ๐1๐๐๐ ๐2 ๐๐ then a pair
๐1, ๐2 ๐๐ interpretations of ๐1, ๐2 such
that for all c โ A, ๐1, ๐2| =๐ ๐ถ. We
denote this by ๐1, ๐2| =๐ ๐ด. An
interpretation of S is a family
(๐๐)๐โ๐๐๐ of ๐๐๐๐๐ ๐๐ ๐๐ ๐๐. A simple
interpretation of S is an interpretation
(๐๐)๐โ๐๐๐
๐๐ ๐๐ over the same universe
D.
Definition 6 [80] A simple model of a
NeO S is a simple interpretation
(๐๐)๐โ๐๐๐
of S such that for each i, j
โ I , ๐๐, ๐๐| =๐ ๐ด๐๐. This is written
(๐๐)๐โ๐๐๐ | =๐ ๐
Example 4 (Interpretation of correspond-
ence relations in SROIQ)
The interpretation of correspondence
relations in SROIQ relative to a global
universe D is given in the table below,
where on the first column we have the
correspondence, on the second the relation
that interprets it and on the third its domain
of interpretation. (๐๐, ๐๐, =) = ๐ท(๐ซ) ๐ฟ ๐ท(๐ซ)
(๐๐, ๐๐, =) = ๐ท(๐ซ ๐ฟ๐ซ) ๐ฟ ๐ท(๐ซ๐ฟ๐ซ)
(๐๐, ๐๐, =) = ๐ซ ๐ฟ ๐ซ
(๐๐, ๐๐, โฅ) ๐ด๐ช๐
๐ โฉ ๐ด๐๐
๐= โ ๐ท(๐ซ) ๐ฟ ๐ท(๐ซ)
(๐๐, ๐๐, โฅ) ๐ด๐น๐
๐ โฉ ๐ด๐น๐
๐= โ ๐ท(๐ซ ๐ฟ๐ซ) ๐ฟ ๐ท(๐ซ๐ฟ๐ซ)
(๐๐, ๐๐, โฅ) โ ๐ซ ๐ฟ ๐ซ
(๐๐, ๐๐, <) โ ๐ท(๐ซ) ๐ฟ ๐ท(๐ซ)
(๐๐, ๐๐, <) โ ๐ท(๐ซ ๐ฟ๐ซ) ๐ฟ ๐ท(๐ซ๐ฟ๐ซ)
(๐๐, ๐๐, >) โ ๐ท(๐ซ) ๐ฟ ๐ท(๐ซ)
(๐๐, ๐๐, >) โ ๐ท(๐ซ ๐ฟ๐ซ) ๐ฟ ๐ท(๐ซ๐ฟ๐ซ)
(๐๐, ๐๐, โ) โ ๐ท(๐ซ)๐ฟ ๐ซ
(๐๐, ๐๐, โ) โ ๐ซ ๐ฟ ๐ท (๐ซ)
where ck, rk, ikare class, role and individual symbols from an ontology Ok and Mk โ Model(Ok) for k = 1,2 Example 6 (Interpretation of
correspondence relations in FOL) The
interpretation of correspondence relations
in FOL relative to a global universe D is
(๐๐, ๐๐, =) = ๐ญ ๐๐(๐ซ) ร ๐ญ ๐๐(๐ซ)
(๐๐, ๐๐, โฅ) โ ๐ญ ๐๐(๐ซ) ร ๐ญ ๐๐(๐ซ)
(๐๐, ๐๐, =) = ๐ท ๐๐๐ (๐ซ) ร ๐ท ๐๐๐ (๐ซ)
(๐๐, ๐๐, โฅ) โ ๐ท ๐๐๐ (๐ซ) ร ๐ท ๐๐๐ (๐ซ)
(๐๐, ๐๐, <) โ ๐ท ๐๐๐ (๐ซ) ร ๐ท ๐๐๐ (๐ซ)
(๐๐, ๐๐, >) โ ๐ท ๐๐๐ (๐ซ) ร ๐ท ๐๐๐ (๐ซ)
where ๐๐ , ๐๐ ๐๐๐ function and predicate
symbols from an ontology ๐๐, with
๐ = 1,2 6.2 Integrated Semantics:
Another possibility is to consider that the
domain of interpretation of the ontologies
of a NeO is not constrained, and a global
domain of interpretation U exists, together
with a family of equalising functions
๐พ๐ โถ ๐ท๐ โ ๐, where Di is the domain of
๐๐ , for each ๐ โ ๐ผ. A relation R in R is
interpreted as a relation RU on the global
domain. Satisfaction of a correspondence
๐ = (๐1, ๐2, ๐ ) by two models ๐1 ๐๐ ๐1
and ๐2 ๐๐ ๐2 means that
๐พ๐(๐๐(๐1))๐ ๐ ๐พ๐ (๐๐ (๐2)).
Definition 7 [80] An integrated
interpretation of
a NeO S, {(๐๐)๐โ๐๐๐
, (๐พ๐)๐ โ ๐๐๐}is an
integrated model of ๐ ๐ff ๐๐๐ ๐๐๐โ ๐, ๐ โ
๐ผ ๐๐, ๐1, ๐2|=๐พ1,๐พ2๐ผ ๐ด๐๐ |
6.3 Contextualised Semantics
The functional notion of contextualised
semantics in [38] is not very useful and has
been replaced by a more flexible relational
notion subsequently [8], closely related to
๐1 ๐2 ๐3 ๐4 ๐5
๐ท1 ๐ท2 ๐ท3 ๐ท4 ๐ท5
๐1
๐2
๐3
๐4
๐5
๐
๐พ3 ๐พ4 ๐พ2 ๐พ1 ๐พ3
7
the semantics of DDLs [9] and ๐-
connections [41]. The idea is to relate the
domains of the ontologies by a family of
relations ๐ = (๐๐๐ )๐, ๐ โ ๐ผ. The relations R
in R are interpreted in each domain of the
ontologies in the NeO. Satisfaction of a
correspondence ๐ = (๐1, ๐2, ๐ ) by two
models ๐1 of ๐1 and ๐2 of ๐2 means that
๐๐(๐1)๐ ๐ ๐๐๐(๐๐ (๐2)), where ๐ ๐ is the
interpretation of R in ๐ท๐
Contextualized semantics gives up the
notion of a global universe, and instead lets
each ontology in a network be interpreted
with its own local universe. However, in
order to give semantics to alignments,
these universes need to be related
somehow. The approach of [43] to use
mappings between local universes has a
number of limitations and has been
replaced by a more flexible approach
subsequently [29], which uses relations
between local universes. This is closely
related to the semantics of DDLs [9] and
E -connections [37].
Example 5 (Interpretation of
correspondence relations in SROIQ) The
interpretation of correspondences in
SROIQ relative to a set D in the
contextualized semantics is (๐1, ๐2, =) ๐๐1
1 = ๐21(๐๐22 )
(๐1, ๐2, =) ๐๐11 = ๐21(๐๐2
2 )
(๐1, ๐2, =) ๐๐11 = ๐21(๐๐2
2 ) ๐. ๐ ๐๐11 , ๐๐2
1 โ ๐21
(๐1, ๐2, โฅ) ๐๐11 โฉ ๐21(๐๐2
2 ) = โ
(๐1, ๐2, โฅ) ๐๐11 โฉ ๐21(๐๐2
2 ) = โ
(๐1, ๐2, โฅ) (๐๐22 , ๐๐1
1 ) โ ๐21
(๐1, ๐2, <) ๐๐11 โ ๐21(๐๐2
2 )
(๐1, ๐2, <) ๐๐11 โ ๐21(๐๐2
2 )
(๐1, ๐2, >) ๐๐11 โ ๐21(๐๐2
2 )
(๐1, ๐2, >) ๐๐11 โ ๐21(๐๐2
2 )
(๐1, ๐2, โ) ๐21(๐๐22 ) โ ๐๐1
1
(๐1, ๐2, โ) ๐๐11 = ๐21(๐๐2
2 )
where ๐1, ๐1, ๐1๐๐๐ class, role and
individual symbols from an ontology
๐1, ๐2, ๐2, ๐2 ๐๐๐ class, role and individual
symbols from an ontology ๐2, ๐1 and
๐2 are models of ๐1and ๐2 with domains
๐ท1 and ๐ท2 and ๐21 is the domain relation
between ๐ท1 ๐๐๐ ๐ท2 .
7 Normalization of Alignments
In this section we describe how relational
(and therefore also general) networks can
be normalized into functional ones. Part of
this normalization process generalizes to
an arbitrary institution, while certain parts
(namely relativisation of ontologies and the
construction of bridges) are institution-
specific and have to be provided separately
for each institution.
A central motivation behind this
construction is the following: We will
prove representation theorems showing
that the semantics of a relational network
coincides with that of its normalization,
This implies that reasoning in the colimit
of the normalized network is complete and
(in case of logics admitting amalgamation)
also sound for reasoning about the network
7.1 Structure of the Normalization
Process
Relational DOL networks (i.e. networks
involving alignments) can be normalized to
purely functional networks. In this section,
we lay out the structure of this
normalization process, while in the next,
we will provide details for each of the four
possible assumptions about the semantics.
Example 6 We illustrate the four
approaches to semantics with the help of a
simple example. Let us consider the
following two ontologies:
๐๐๐ก๐๐๐๐๐ฆ ๐ ๐ถ๐๐๐ ๐ ๐ต๐๐๐_๐ธ๐ก๐๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐ ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ ๐ต๐๐๐_๐๐ก๐๐
๐1 ๐2 ๐3 ๐4 ๐5
๐ท2 ๐ท3 ๐ท4 ๐ท5
๐1,2 ๐5,4
โฆ
.. ๐1,3
โฆโฆโฆโฆโฆ
๐ท1
8
๐ถ๐๐๐ ๐ ๐ธ๐๐๐๐๐ก
๐๐๐ก๐๐๐๐๐ฆ ๐ = ๐ถ๐๐๐ ๐ ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐๐๐ ๐๐ข๐๐๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐๐๐๐๐กโ๐๐ ๐ค๐๐กโ ๐กโ๐ ๐๐๐๐๐๐ค๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐: ๐ต๐๐_๐ธ๐ก๐๐ = ๐: ๐น๐๐๐๐๐ก ๐: ๐ด๐๐๐๐ โ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐: ๐ธ๐๐๐๐๐ก โฅ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ where we prefix with S : the symbols
coming from S and with T : the symbols
coming from T . Using the DOL syntax,
we can write this alignment as
๐ด๐๐๐๐๐๐๐๐๐ก ๐ด ๐ ๐ก๐ ๐ = ๐ต๐๐๐_๐ธ๐ก๐๐ =
๐น๐๐๐๐๐ก
Amine โ Masculin,
Enfant โฅ Travailleur
Note that so far we have not specified
which kind of semantics is assumed for A.
Depending on the choice for the assumed
semantics, the normalisation of A will be
constructed in a different way. The idea is
to introduce for each correspondence a
theory that captures its semantics. This is
done differently for four possible semantics
of the alignment. Using these theories, we
then construct a diagram that gives the
semantics of the alignment. In all four
semantics, the diagram is a W-alignment in
the sense of [41]:
Definition 8 Let S and T be two ontologies
in a logic L and
A={๐๐ = (๐ 1๐ , ๐ 2
๐ , ๐ ๐)|๐ โ ๐ผ๐๐} ๐๐ ๐๐๐๐๐๐๐๐๐๐ก ๐๐๐ก๐ค๐๐๐ ๐ ๐๐๐ ๐ , the
diagram of the alignment A is :
Where the ontologies :
B, Sฬ , Sฬโ, Tฬโฒ, Tฬ and the Morphism t1, t2, ฯ1, ฯ2
depend on the choice of semantics for the
alignment A, in a way to be made precise
for each possible option.
Intuitively, ๏ฟฝฬ๏ฟฝโฒ and ๏ฟฝฬ๏ฟฝโฒ are either the
ontologies S and T being aligned, or a
transformation of them, involving their
translation along a comorphism. B is a
bridge ontology that formalises the
intended meaning of the correspondences
of A. It will be constructed as a union of
smaller theories, each internalizing the
semantics of a correspondence of A. This
means, intuitively, that the models ๐ of a
theory that internalises the semantics of
(๐ 1, ๐ 2 , ๐ ), are precisely those for which
the relation ๐๐ holds for ๐๐ 1 ๐๐๐ ๐๐ 2
, in a
way that takes into account the possible
semantics of the alignment. We will define
this formally for each choice of semantics.
It is possible that some correspondence
cannot be internalised in the logic of the
ontologies being aligned. In this case, we
will have to look for a more expressive
logic, where such a theory internalising the
semantics of that correspondence can be
constructed.. ๏ฟฝฬ๏ฟฝโ and ๐โฒฬ are interface of S
and T , respectively, with B , meaning that
they connect the symbols from the aligned
ontologies with their correspondents in the
bridge ontology along ๐ก๐ ๐๐๐ ๐๐ These
diagrams will be used in the construction
of the normalisation of a network:
Definition 9 Given a general NeO, its
normalization is defined as the union of its
functional part with the normalization of
its relational part.
For each ontology ๐๐ in a network of
aligned ontologies, let ๏ฟฝฬ๏ฟฝ๐ be its corres-
ponding ontology in the diagram of the
network. Let ฮฃ๐ = ๐๐๐(๐๐) and ฮฃ๐โฒ =
๐๐๐(๐๐โฒ). In each of the four cases that
correspond to the different choices of
semantics we can define:
(i)a sentence translation functor ฮฑโ
โ
:
๐๐๐(๐ด๐) โ ๐๐๐(ฮฃ๐โฒ)
(ii) model reduct factor ฮฒโ
โ
:
B ๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝโฒ ๏ฟฝฬ๏ฟฝโฒ
๐ก1 ๐2 ๐ก2 ๐1
9
๐๐๐(๐ด๐) โ ๐๐๐(ฮฃ๐โฒ)
such that the condition ๐ฝโ(๐โฒ)| = ๐
โ ๐โฒ| = ๐ผโ(๐) โ๐๐๐ for each ๐ โ
(๐๐๐(๏ฟฝฬ๏ฟฝ๐)) ๐๐๐ ๐๐๐โ ๐ โ ๐๐๐ (ฮฃ๐) Using
these functors allows us to formulate the
results about reasoning in a NeO in a
uniform way. In all four cases, we can
define a signature morphism in a
Grothendieck logic from ฮฃ๐ ๐ก๐ ฮฃ๐โฒ
Such that ฮฑโ and ฮฒโ translation and model
reduction functors corresponding to it.
Thus, the expected condition follows from
the satisfaction condition of the
Grothendieck logic. We now proceed with
discussing how these diagrams are
obtained for each of the four possible
semantics.
7.2 Simple Semantics Alignment
We start with defining what it means for a
theory to capture the semantics of a
correspondence. In this section, ๐๐๐ก ๐ด =
{๐๐ = (๐ 1๐ , ๐ 2
๐ , ๐ ๐)|๐ โ ๐ผ๐๐} Be an alignment
between two ontologies S and T in a
logic L, where Ind is a set of indices.
First we define the signature of the theory.
Definition 10 The bridge signature
ฮฃ๐ต ๐๐ ๐ต is defined as the union of
๐๐๐1(๐ด) ๐๐๐ ๐๐๐2(๐ด) where ฮฃ1is the
smallest subsignature of Sig(S ) such that
Symbols(ฮฃ1) includes ๐ 1๐ ๐๐๐ ๐๐๐โ ๐ โ
๐ผ๐๐ is the signature obtained by renaming
every ๐ ๐ฆ๐๐๐๐ ๐ โ ๐๐ฆ๐๐๐๐๐ (ฮฃ1) to S:s
and ฮฃ2 is the smallest subsignature of
Sig(T ) such that Symbols(ฮฃ2) includes
๐๐๐ ๐๐๐โ , ๐๐๐ ๐2๐ ๐๐๐2(๐ด) is the signat-
ure obtained by renaming every symbol
๐ โ ๐๐ฆ๐๐๐๐๐ (๐ด2) ๐ก๐ ๐: ๐ .
We must prefix the symbols occurring in
correspondences with the names of the
ontology where they come from to avoid
unintended identifications when making
the union of the involved signatures.
Definition 11 Let ฮฃ๐ต be the bridge
signature of A and โ a set of
ฮฃ๐ต sentences. We say that (ฮฃ๐ต, โ)
internalises the semantics of
{๐๐ = (๐ 1๐ , ๐ 2
๐ , ๐ ๐)|๐ โ ๐ผ๐๐} denoted
(ฮฃ๐ต, โ) โ๐ ๐๐ ๐๐, ๐๐
๐ |= ฮฃ๐ต ฮ ๐๐๐ (๐๐:๐๐2
, ๐๐:๐๐2
) โ
(๐ ๐ผ) ๐ข๐๐๐ฃ๐๐๐ ๐(๐) for each ฮฃB
Definition 12 Let ฮฃ๐ต be the bridge signatu-
re of A.
Assume that (ฮฃ๐ต, โ) โ๐ ๐๐ ๐๐ each ๐๐ โ ๐ด.
The diagram of A is obtained by setting
the parameters as follows:
๏ฟฝฬ๏ฟฝ = ๐ ๐๐๐ ๏ฟฝฬ๏ฟฝ = ๐
๏ฟฝฬ๏ฟฝ = (๐๐๐1(๐ด), โ )
๐ก1 ๐๐๐๐ ๐๐๐โ ๐: ๐ โ ๐๐ฆ๐๐๐๐๐ (๐๐๐1(๐ด))๐ก๐ ๐
๏ฟฝฬ๏ฟฝ = (๐๐๐1(๐ด), โ )
๐ก2 ๐๐๐๐ ๐๐๐โ ๐: ๐ โ ๐๐ฆ๐๐๐๐๐ (๐๐๐1(๐ด))๐ก๐ ๐
๐ต = (ฮฃ๐ต, โ ๐ โ ๐ผ๐๐ โ๐ )
๐1 ๐๐๐ ๐2 ๐๐๐ ๐๐๐๐๐ข๐ ๐๐๐๐ Example 7 (Simple semantics in SROIQ)
For each type of correspondence, we give
below the theory that internalises its
semantics. We have chosen to use
Manchester syntax for SROIQ [38], as it
makes more obvious the kinds of symbols
involved. We also assume that the
correspondences are between symbols
from the ontologies S and T .
(๐1, ๐2, =) ๐ถ๐๐๐ ๐ : ๐: ๐1 ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐ โถ ๐: ๐2 (๐1, ๐2, =) ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐1 ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐ โถ ๐: ๐2 (๐1, ๐2, =) ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐1 ๐๐๐๐ ๐๐ ๐: ๐2 (๐1, ๐2, โฅ)๐ถ๐๐๐ ๐ : ๐: ๐1 ๐ท๐๐ ๐๐๐๐๐ก๐ ๐ค๐๐กโ ๐: ๐2 (๐1, ๐2, โฅ)๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐1 ๐ท๐๐ ๐๐๐๐๐ก๐ ๐ค๐๐กโ ๐: ๐2 (๐1, ๐2, โฅ)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐1 ๐ท๐๐๐๐๐๐๐๐ก ๐๐๐๐ ๐: ๐2 (๐1, ๐2, <)๐ถ๐๐๐ ๐ : ๐: ๐1 ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐ถ2 (๐1, ๐2, <)๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐1 ๐๐ข๐๐๐๐๐๐๐๐ก๐ฆ ๐ค๐๐กโ ๐2 (๐1, ๐2, >)๐ถ๐๐๐ ๐ : ๐: ๐2 ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐1 (๐1, ๐2, >)๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐2
10
๐๐ข๐๐๐๐๐๐๐๐ก๐ฆ ๐๐ ๐: ๐1 (๐1, ๐2, โ)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐2 ๐๐ฆ๐๐ ๐: ๐ถ1 (๐1, ๐2, โ)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐1 ๐๐ฆ๐๐ ๐: ๐ถ1 Example 8 (Simple semantics in FOL)
Similarly, in semantics of corresp-
ondences:
(๐1, ๐2 =) โ๐ฅ1, . . . , ๐ฅ๐. ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) = ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐) (๐1, ๐2, โฅ)โ๐ฅ1, . . . , ๐ฅ๐. ยฌ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) = ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐) (๐1, ๐2, =) โ๐ฅ1, . . . , ๐ฅ๐. ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โโ ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐) (๐1, ๐2, โฅ)โ๐ฅ1, . . . , ๐ฅ๐. ยฌ(๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐)) (๐1, ๐2, <)โ๐ฅ1, . . . , ๐ฅ๐. ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) =โ ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐) (๐1, ๐2, >)โ๐ฅ1, . . . , ๐ฅ๐. ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐) =โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) Example 9 For the alignment of Ex. 9, we
start by adding the assumption that we
have a shared universe for the ontologies:
The network of A is then
Alignment S to T=
๐โ๐๐๐ ๐โฒ๐๐๐๐ ๐๐ ๐ก๐ ๐๐ ๐กโ๐ ๐๐๐๐๐๐๐ก๐ ๐: ๐ต๐๐๐๐ธ๐ก๐๐๐๐๐ ๐: ๐ธ๐๐๐๐๐ก ๐๐๐ ๐กโ๐ ๐๐๐๐๐ฃ๐๐๐ข๐๐ ๐: ๐๐๐๐๐ ๐๐๐ ๐โฒ๐๐๐๐ ๐๐ ๐ก๐ ๐๐ ๐กโ๐ ๐๐๐๐๐๐๐ก๐ ๐: ๐น๐๐๐๐๐ก, ๐: ๐ธ๐๐๐๐๐ฆ๐๐ ๐๐๐ ๐: ๐๐๐๐. ๐โ๐๐ ๐กโ๐
๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐ฆ ๐ต ๐๐ : ๐๐๐ก๐๐๐๐๐ฆ ๐ต = ๐ถ๐๐๐ ๐ ๐ โถ ๐ต๐๐๐_๐ธ๐ก๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐ ๐
โถ ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐ โถ ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐ถ๐๐๐ ๐ ๐ โถ ๐๐๐ ๐๐ข๐๐๐ ๐ถ๐๐๐ ๐ ๐ โถ ๐ธ๐๐๐๐๐ก ๐ท๐๐ ๐๐๐๐๐ก ๐ค๐๐กโ ๐
โถ ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐ ๐ โถ ๐ด๐๐๐๐, ๐๐ฆ๐๐ ๐๐๐ ๐๐ข๐๐๐ We can combine the resulting functional
network into a single ontology. In DOL,
this is written as:
๐๐๐ก๐๐๐๐๐ฆ ๐ถ = ๐๐๐๐๐๐๐ ๐ ๐โ๐ ๐๐๐๐๐๐๐ก ๐๐๐ก๐๐๐๐๐ฆ ๐๐ ๐กโ๐ ๐๐๐ก๐ค๐๐๐ ๐๐ ๐ด ๐๐ :
๐๐๐ก๐๐๐๐๐ฆ ๐ถ =
๐ถ๐๐๐ ๐ : ๐: ๐ต๐๐๐_๐ธ๐ก๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐๐:
๐: ๐น๐๐๐๐๐ก
๐ถ๐๐๐ ๐ : ๐: ๐r๐๐ฃ๐๐๐๐๐๐ข๐
๐ถ๐๐๐ ๐ : ๐: ๐๐๐ ๐๐ข๐๐๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐: ๐: ๐น๐๐๐๐๐ก
๐ถ๐๐๐ ๐ : ๐: ๐ธ๐๐๐๐๐ก ๐ท๐๐ ๐๐๐๐๐ก๐๐๐กโ: ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐
๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐๐๐๐๐ ๐๐ฆ๐๐๐ : ๐: ๐๐๐ ๐๐ข๐๐๐, ๐:
๐ต๐๐๐_๐ธ๐ก๐๐
๐๐๐ก๐๐๐๐๐ฆ ๐ถ =
๐ถ๐๐๐ ๐ ๐: ๐ต๐๐_๐ธ๐ก๐๐
โถ ๐๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐ ๐: ๐น๐๐๐๐๐ก
๐ถ๐๐๐ ๐ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐
๐ถ๐๐๐ ๐ ๐
โถ ๐๐๐ ๐๐ข๐๐๐ ๐๐ข๐ ๐ถ๐๐๐ ๐ ๐๐ ๐: ๐น๐๐๐๐๐ก
๐ถ๐๐๐ ๐ ๐
โถ ๐ธ๐๐๐๐๐ก ๐๐๐ ๐๐๐๐๐ก ๐ค๐๐กโ: ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐
๐ผ๐๐๐๐ฃ๐๐๐ข๐๐ ๐
โถ ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐: ๐ต๐๐๐_๐๐ก๐๐
Since the original ontologies are not
modified in the diagram of the alignments,
the signature morphism from ๐๐๐(๐๐) to
๐๐๐(๏ฟฝฬ๏ฟฝ๐) is the identity, so the functors
๐ผโ ๐๐๐ ๐ฝโ are the identities on ๐๐๐(๐๐) โ
๐ ๐๐๐ก๐๐๐๐๐ , respectively on ๐๐๐(๐๐) โ
๐๐๐๐๐๐ .
Example 10 (Generalised integrated
semantics in FOL) In FOL we have the
following basic bridge ontology:
โ ๐ฅ1, ๐ฅ2, ๐ง . ๐ง๐๐ ๐ฅ1 โ ๐ง๐๐ ๐ฅ2 โน ๐ฅ1 = ๐ฅ2 โ ๐ฅ . ๐:โบ (๐ฅ) โน โ๐ง . ๐ง๐๐ ๐ฅ โ๐ฅ, ๐ง . ๐ง๐๐ ๐ฅ โน ๐:โบ (๐ฅ) โง ๐บ(๐ง)
โ ๐ฅ1, ๐ฅ2, ๐ง . ๐ง๐๐๐ฅ1 โ ๐ง๐๐๐ฅ2 โน ๐ฅ1 = ๐ฅ2 โ ๐ฅ . ๐:โบ (๐ฅ) โน โ๐ง . ๐ง๐๐ ๐ฅ โ๐ฅ, ๐ง . ๐ง๐๐ ๐ฅ โน ๐:โบ (๐ฅ) โง ๐บ(๐ง) where for each of ๐๐ and ๐๐, the first axiom
is inverse functionality, the second one is
right-totality and the third one gives the
domain and the range, and the following
theories that internalise the semantics of
correspondences:
(๐1, ๐2, ) โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐
S B
๏ฟฝฬ๏ฟฝโฒ ๏ฟฝฬ๏ฟฝโฒ T
Tapez une รฉquation ici.
๐ก1 ๐ก2
๐1 ๐2
11
=โ
โ๐ง . ๐ง ๐๐ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐ง ๐๐๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) (๐1, ๐2, ) โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐ =โ
โ๐ง . ๐ง ๐๐ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐ง ๐๐๐: ๐2(๐ฆ1, . . . , ๐ฆ๐)๐๐
(๐1, ๐2, ) โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐ =โ
๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐)
(๐1, ๐2, )ยฌโ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐ โง ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐)
(๐1, ๐2, )โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐ =โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) =โ ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) (๐1, ๐2, )โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐, ๐ง1, . . . , ๐ง๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ง1๐๐๐ฅ1 โง. . .โง ๐ง๐๐๐๐ฅ๐ โง ๐ง1๐๐๐ฆ1 โง. . .โง ๐ง๐๐๐๐ฆ๐ =โ ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) =โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) Example 11 (General integrated semantics
in SROIQ)
The basic bridge ontology for general
integrated semantics in SROIQ is
๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐๐ ๐ถโ๐๐๐๐๐ก๐๐๐๐ ๐ก๐๐๐ โถ ๐๐๐ฃ๐๐๐ ๐๐น๐ข๐๐๐ก๐๐๐๐๐
๐ท๐๐๐๐๐ โถ ๐บ ๐ ๐๐๐๐ โถ ๐:โบ ๐ถ๐๐๐ ๐ โถ ๐:โบ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐
๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐๐ ๐ถโ๐๐๐๐๐ก๐๐๐๐ ๐ก๐๐๐ โถ ๐๐๐ฃ๐๐๐ ๐๐น๐ข๐๐๐ก๐๐๐๐๐ ๐ท๐๐๐๐๐ โถ ๐บ ๐ ๐๐๐๐ โถ ๐:โบ ๐ถ๐๐๐ ๐ โถ ๐:โบ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐ ๐ถ๐๐๐ ๐ โถ ๐บ ๐โ๐ ๐กโ๐๐๐๐ฆ ๐กโ๐๐ก ๐๐๐ก๐๐๐๐๐๐๐ง๐ ๐กโ๐ ๐ ๐๐๐๐๐ก๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ ๐๐๐ โถ (๐1, ๐2, =) ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐๐๐๐ ๐: ๐1 ๐๐ ๐๐๐๐ ๐: ๐2 (๐1, ๐2, =) ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐๐๐๐ {๐: ๐1} ๐๐ ๐๐๐๐ {๐: ๐2} (๐1, ๐2, โฅ) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐1} ๐ท๐๐ ๐๐๐๐๐ก๐ ๐ค๐๐กโ ๐๐ ๐๐๐๐ ๐: ๐2 (๐1, ๐2, โฅ) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐1} ๐ท๐๐ ๐๐๐๐๐ก๐ ๐ค๐๐กโ ๐๐ ๐๐๐๐ ๐: ๐2 (๐1, ๐2, <) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐1} ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ โถ ๐2 (๐1, ๐2, >) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐1} ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ โถ ๐2 (๐1, ๐2, โ) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐2} ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ โถ ๐1 (๐1, ๐2, โ) ๐๐๐๐ ๐ โถ ๐๐ ๐ ๐๐๐ {๐: ๐1} ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ โถ ๐2
For correspondences involving roles, we
would need to be able to express
equivalences or disjointness axioms
involving complex roles, which are beyond
the expressivity of SROIQ. Therefore, the
correspondences
(๐1, ๐2, =), (๐1, ๐2, โฅ) , (๐1, ๐2, <) and (๐1, ๐2, >) cannot be internalised in
SROIQ. We will give their internalisations
in FOL
Example 12 Continuing , we add the
assumption of a global universe with
general integrated semantics:
The diagram of A is then
12
๏ฟฝฬ๏ฟฝโฒconsists of the concepts G,
S:Thing, S :Bien_etre and S :Enfant, the
object property ๐๐ and the individual
S :amine and ๏ฟฝฬ๏ฟฝโฒconsists of the concepts G,
T :Thing, T :Format ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ and
T :Masculin and the object property ๐๐.
The ontologies
๐ ฬ ๐๐๐ ๐ ฬ ๐๐๐ ๐๐๐ก๐๐๐๐๐ฆ ๏ฟฝฬ๏ฟฝ = ๐๐๐๐ ๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ โถ ๐บ ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐ ๐ถ๐๐๐ ๐ โถ ๐ต๐๐๐_๐ธ๐ก๐๐ ๐ ๐ข๐๐ถ๐๐๐ ๐ ๐๐: ๐: ๐กโ๐๐๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐ โถ ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ ๐ต๐๐๐_๐ธ๐ก๐๐, ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ : ๐ธ๐๐๐๐๐ก ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ โถ ๐: ๐โ๐๐๐
๐๐๐ก๐๐๐๐๐ฆ ๐ ฬ = ๐๐๐๐ ๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ โถ ๐บ ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐ ๐ถ๐๐๐ ๐ ๐น๐๐๐๐๐ก ๐ ๐ข๐๐ถ๐๐๐ ๐ ๐๐: ๐: ๐กโ๐๐๐ ๐ถ๐๐๐ ๐ โถ ๐๐๐ ๐๐ข๐๐๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐น๐๐๐๐๐ก ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ : ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ โถ ๐: ๐โ๐๐๐ ๐โ๐ ๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐ฆ ๐ต ๐๐ ๐ธ๐ฅ๐๐๐๐๐ ๐๐ โถ ๐๐๐ก๐๐๐๐๐ฆ ๐ต = ๐ถ๐๐๐ ๐ โถ ๐บ ๐ถ๐๐๐ ๐ โถ ๐: ๐โ๐๐๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐: ๐ผ๐๐ฃ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐บ ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐ ๐ถโ๐๐๐๐๐ก๐๐๐๐ ๐ก๐๐๐ : ๐ผ๐๐ฃ๐๐๐ ๐๐น๐ข๐๐๐๐๐๐๐ ๐ท๐๐๐๐๐ โถ ๐บ ๐ ๐๐๐๐ โถ ๐: ๐โ๐๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐๐ โถ ๐๐ ๐ ๐๐๐ ๐: ๐ต๐๐๐_๐ธ๐ก๐๐ ๐๐ ๐ ๐๐๐ ๐: ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐: ๐ธ๐๐๐๐๐ก ๐ท๐๐ ๐๐๐๐๐ก๐ค๐๐กโ ๐๐ ๐ ๐๐๐ โถ
๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐ถ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ {๐: ๐ด๐๐๐๐} ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ โถ ๐_๐ก ๐ ๐๐๐ ๐: ๐๐๐ ๐๐ข๐๐๐
๐โ๐ ๐๐๐๐๐๐๐ก ๐๐๐ก๐๐๐๐๐ฆ ๐๐ ๐กโ๐ ๐๐๐๐๐ก๐๐ฃ๐๐ง๐๐ ๐๐๐๐๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐๐๐๐๐ก ๐๐ ๐ธ๐ฅ๐๐๐๐๐ ๐๐ โถ ๐ ๐๐ก๐๐๐๐๐ฆ ๐ถ โถ ๐ถ๐๐๐ ๐ โถ ๐ฎ ๐ถ๐๐๐ ๐ โถ ๐: ๐โ๐๐๐ ๐๐ข๐ต๐ถ๐๐๐ ๐ ๐๐: ๐๐๐ฃ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐บ ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐ ๐ถโ๐๐๐๐๐ก๐๐๐๐ ๐ก๐๐๐ : ๐ผ๐๐ฃ๐๐๐ ๐๐น๐ข๐๐๐๐๐๐๐ ๐ท๐๐๐๐๐ โถ ๐บ ๐ ๐๐๐๐ โถ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ โถ
๐: ๐น๐๐๐๐๐ก ๐๐ข๐ต๐ถ๐๐๐ ๐ ๐๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ โถ
๐: ๐ต๐๐๐_๐ธ๐ก๐๐ ๐๐ข๐ต๐ถ๐๐๐ ๐ ๐๐ ๐: ๐โ๐๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐: ๐ต๐๐๐_๐ธ๐ก๐๐ ๐๐ ๐ ๐๐๐ ๐: ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐น๐๐๐๐๐ก ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐: ๐ธ๐๐๐๐ก๐ ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐: ๐ธ๐๐๐๐๐ก ๐ท๐๐ ๐๐๐๐๐ก ๐ค๐๐กโ โถ
๐๐ ๐ ๐๐๐ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐ โถ
๐: ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ ๐: ๐๐๐๐ ๐๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐๐ ๐ ๐๐๐ {๐: ๐ด๐๐๐๐} ๐ ๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐1
7.5 Contextualised Semantics
Normalization
The ontologies of the network can be
interpreted using different universes, which
however are related using binary relations.
Definition 13
Let A={๐๐ = (๐ 1๐ , ๐ 2
๐ , ๐ ๐)|๐ โ ๐ผ๐๐} be an
alignment between two ontologies S and
T in a logic L, where Ind is a set of
indices. The basic bridge ontology (ฮฃ๐ต, ฮ๐ต)
of A in the contextualised semantics
consists of
โ๐ ๐ ๐๐๐๐๐ก๐ข๐๐ ฮฃ๐ต ๐กโ๐๐ก ๐ก๐๐๐๐ ๐กโ๐ ๐ข๐๐๐๐
๐๐ ๐๐๐1(๐ด) ๐๐๐ ๐๐๐2(๐ด), ๐คโ๐๐๐
โ ฮฃ1 is the smallest subsignature of
Sig(S ) such that Symbols(ฮฃ1)
includes ๐ 1๐ and ๐๐๐1(A) takes the
signature obtained by renaming every
๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝโฒ B ๏ฟฝฬ๏ฟฝโฒ ๐ก1
๐1
๐2
๐ก2
13
s โ Symbols(ฮฃ1) to S :s and extends it
with S :โบ โ ๐ข๐๐๐๐ฆ๐ฟ,
โ ฮฃ2 is the smallest subsignature of
Sig(T ) such that Symbols(ฮฃ2 ) includes
๐2๐ and ๐๐๐2(A) takes the signature obtain-
ed by renaming every ๐ โ ๐๐ฆ๐๐๐๐๐ (ฮฃ2 ) to
T :s and extends with T : โบ โ ๐ข๐๐๐๐ฆ๐ฟ and
extends this union with ๐๐๐ in ๐๐๐๐๐๐ฆ๐ฟ .
โ a set ฮ๐ต ๐๐ ฮฃ๐ต-sentences that
axiomatise in a logic-dependent way that
the domain of ๐๐๐ is T :โบand the range
of ๐๐๐ is S :โบ.
Definition 14 Let c = (๐ 1, ๐ 2, R) be a
correspondence of a contextualised
alignment A. Let (ฮฃ๐ต, ฮ๐ต) be the basic
bridge ontology of ๐ด and let โ be a set of
ฮฃ๐ต โ ๐ ๐๐๐ก๐๐๐๐๐ that includes ฮ๐ต. We say
that (ฮฃ๐ต, โ) internalises the semantics of c ,
denoted (ฮฃ๐ต, โ)| =๐๐๐ ๐ If ๐| =ฮฃ๐ตโ ๐๐๐ ๐๐:๐ 1
๐ ๐ผ ๐๐:๐ ๐๐๐๐(๐๐:๐2).
Definition 15 Assume that
(ฮฃ๐ต, โ๐) โ๐๐๐
๐๐for each ๐๐ โ ๐ด. The
parameters of Def. 14 are set as follows
๐ = ๐๐๐๐ฟ(๐)๐๐๐ ๐ = ๐๐๐๐ฟ(๐), ๐คโ๐๐๐ ๐๐๐๐ฟ ๐๐ ๐กโ๐ ๐๐๐๐๐ก๐๐ฃ๐๐ ๐๐ก๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐ ๐ฟ ๐๐ ๐กโ๐ ๐๐๐ก๐๐๐๐๐๐๐ ๐๐๐๐๐ ๐๐๐๐๐๐๐, -๏ฟฝฬ๏ฟฝโฒ = (๐๐๐(๐ด), โ )
โ๐ก1 ๐๐๐๐ ๐๐๐โ ๐: ๐ โ๐๐ฆ๐๐๐๐๐ (๐๐๐1(๐ด)) ๐ก๐ ๐ ๐๐๐ ๐ โถ โบ
๐ก๐ ๐๐ก๐ ๐๐๐
-๏ฟฝฬ๏ฟฝโฒ = (๐๐๐(๐ด), โ )
โ๐ก2 ๐๐๐๐ ๐๐๐โ ๐: ๐ โ๐๐ฆ๐๐๐๐๐ (๐๐๐2 (๐ด)) ๐ก๐ ๐ ๐๐๐ ๐:โบ ๐ก๐ ๐๐ก๐ ๐๐๐, โ ๐ต = (ฮฃ๐ต, โ ๐ โ ๐ผ๐๐
โ๐),
โ ๐1 ๐๐๐ ๐2 ๐๐๐ ๐๐๐๐๐ข๐ ๐๐๐๐ . Example 13 (Contextualised semantics in
FOL) In FOL we have the following
theories that internalise the semantics of
correspondences:
(๐1, ๐2, =) โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐
๐ฅ๐
=โ ๐: ๐1(๐ฆ1, . . . , ๐ฆ๐)๐๐๐ ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐)
(๐1, ๐2, โฅ)ยฌโ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐)
โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐๐ฅ๐
โง ๐: ๐1(๐ฆ1, . . . , ๐ฆ๐)๐๐ ๐ ๐: ๐2(๐ฅ1, . . . , ๐ฅ๐)
(๐1, ๐2, =) โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐
๐ฅ๐
=โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โโ ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) (๐1, ๐2, โฅ)ยฌโ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐
๐ฅ๐
โง ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) โง ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) (๐1, ๐2, <)โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐
๐ฅ๐
=โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) =โ ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) (๐1, ๐2, >)โ๐ฅ1, . . . , ๐ฅ๐, ๐ฆ1, . . . , ๐ฆ๐. ๐:โบ (๐ฅ1) โง. . .โง ๐:โบ (๐ฅ๐) โง ๐:โบ (๐ฆ1) โง. . .โง ๐:โบ (๐ฆ๐) โง ๐ฆ1๐๐๐
๐ฅ1 โง. . .โง ๐ฆ๐๐๐๐
๐ฅ๐
=โ ๐: ๐2(๐ฆ1, . . . , ๐ฆ๐) =โ ๐: ๐1(๐ฅ1, . . . , ๐ฅ๐) However, the following example shows
that it is not always possible to express the
semantics of a correspondence in the
contextualised semantics in the same logic
as the one used in the aligned ontologies.
Example 14 (Contextualised semantics in
SROIQ)
The diagram of an alignment between two
SROIQ ๐๐๐ก๐๐๐๐๐๐๐ ๐ ๐๐๐ ๐ is obtained
by applying the relativisation of the
aligned ontologies and to the
correspondences of the alignment. The
basic bridge ontology is
๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐๐ ๐ท๐๐๐๐๐ โถ ๐: ๐โ๐๐๐ ๐ ๐๐๐๐ โถ ๐: ๐โ๐๐๐ The theories internalising the semantics of
the correspondences extend it as follows:
(๐1, ๐2 =) ๐ถ๐๐๐ ๐ : ๐: ๐1 ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐ โถ ๐ผ๐๐ฃ๐๐๐ ๐(๐๐๐) ๐ ๐๐๐ ๐: ๐2 (๐1, ๐2, =)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐2 ๐น๐๐๐ก๐ : ๐๐๐ ๐: ๐1 (๐1, ๐2, โฅ)๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ :
14
(๐: ๐1 ) ๐๐๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐2 ๐๐๐กโ๐๐๐ (๐1, ๐2, โฅ)๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ : (๐: ๐2) ๐๐๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐1 ๐๐๐กโ๐๐๐ (๐1, ๐2, >)๐ถ๐๐๐ ๐ : ๐๐๐ฃ๐๐๐ ๐(๐๐๐
)๐ ๐๐๐ ๐: ๐2
๐๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐: ๐1 (๐1, ๐2, >)๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐1 ๐๐ข๐๐๐๐๐๐๐๐ก๐ฆ๐ถโ๐๐๐ ๐ผ๐๐ฃ๐๐๐ ๐(๐๐๐) โ ๐: ๐2 โ ๐๐๐ (๐1, ๐2, <)๐ถ๐๐๐ ๐ : ๐: ๐1 ๐๐ข๐๐ถ๐๐๐ ๐ ๐๐: ๐๐๐ฃ๐๐๐ ๐(๐๐๐) ๐ ๐๐๐ ๐: ๐2
(๐1, ๐2, <)๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ: ๐: ๐1 ๐๐ข๐๐๐๐๐๐๐๐ก๐ฆ๐ถโ๐๐๐ ๐๐๐ โ ๐: ๐1 โ ๐ผ๐๐ฃ๐๐๐ ๐(๐๐๐) (๐1, ๐2, โ)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐2 ๐น๐๐๐ก๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐1 (๐1, ๐2, โ)๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐1 ๐น๐๐๐ก๐ ๐๐๐ฃ๐๐๐ ๐(๐๐๐) ๐ ๐๐๐ ๐: ๐2 For the correspondence (๐1, ๐2, =) where
๐1๐๐๐ ๐2 are roles, it is not possible to
express in SROIQ that
๐1 ๐๐๐ ๐๐๐โ1, ๐2, ๐๐๐ ๐๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ ๐๐๐,
๐โ๐๐๐ ๐๐๐ ๐๐ the domain relation. A
similar problem appears for the
correspondence (๐1, ๐2, โฅ).
To obtain a theory that internalises the
semantics of this correspondence, we must
use a more expressive logic, like first order
logic. This will be done in the next section.
Example 15 For the alignment in Ex. 4,
we add the assumption that we have
different universes for the ontologies,
which are related by relations:
Alignment A:S To T:
Assuming Contextualised Domain:
The Network of A is then
where the constituents of the diagram,
except B . The bridge ontology of
A now becomes :
๐๐๐ก๐๐๐๐๐ฆ ๐ต: ๐ถ๐๐๐ ๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐ โถ ๐โ๐๐๐ ๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆ โถ ๐๐๐ ๐ท๐๐๐๐๐ ๐: ๐โ๐๐๐ ๐ ๐๐๐๐ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐ โถ ๐ต๐๐๐_๐ธ๐ก๐๐
๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐๐โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ โถ ๐: ๐r๐๐ฃ๐๐๐๐๐๐ข๐ ๐ถ๐๐๐ ๐ โถ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ : ๐: ๐ธ๐๐๐๐๐ก ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐๐๐กโ๐๐๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐โ๐ ๐๐๐๐๐๐๐ก ๐๐๐ก๐๐๐๐๐ฆ ๐๐ ๐กโ๐๐ ๐๐๐ก๐ค๐๐๐ ๐๐ : ๐๐๐ก๐๐๐๐๐ฆ ๐ถ โถ ๐ถ๐๐๐ ๐ โถ ๐: ๐โ๐๐๐ ๐๐๐๐๐๐ก๐๐๐๐๐๐๐ก๐ฆโถ ๐๐๐ ๐ท๐๐๐๐๐ ๐: ๐โ๐๐๐ ๐ ๐๐๐๐โถ ๐: ๐โ๐๐๐ ๐ถ๐๐๐ ๐ ๐ โถ ๐ต๐๐๐_๐ธ๐ก๐๐ ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก ๐๐โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐น๐๐๐๐๐ก ๐ถ๐๐๐ ๐ โถ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐ถ๐๐๐ ๐ โถ ๐: ๐๐๐ ๐๐ข๐๐๐ ๐ ๐ข๐๐ถ๐๐๐ ๐ ๐๐ ๐
โถ ๐น๐๐๐๐๐ก ๐ธ๐๐ข๐๐ฃ๐๐๐๐๐ก๐ถ๐๐๐ ๐ ๐๐ : ๐: ๐ธ๐๐๐๐๐ก ๐๐๐ ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐๐๐๐ฃ๐๐๐๐๐๐ข๐ ๐๐๐กโ๐๐๐ ๐ผ๐๐๐๐ฃ๐๐๐ข๐๐: ๐: ๐ด๐๐๐๐ ๐๐ฆ๐๐๐ โถ ๐๐๐ฃ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐: ๐๐๐ ๐๐ข๐๐๐, ๐: ๐ต๐๐๐_๐ธ๐ก๐๐ Note That the Correspondance Of A do not
include an equivalence between roles, and
thus we can build a bridge ontology in
SROIQ
The functors ๐ผโ and ๐ฝโ are defined as in
the case of inclusive integrated semantics
8. Conclusion
In this paper, we have study various
ontologies strategies, and evaluate
partially Ontologies alignments, with
different semantics, we have showed the
alignment of two ontologies on the simple
semantics, integrated and contextualized
semantics, and I have choice one example
for all the manuscript, example who have
introduced are benefic to the syntax of
DOL Language, The goal of this analysis
paper is to give difference between
different syntax paradigm, difficult key
word uses in codification by DOL, FOL,
SROIQ , SPRQL, Therefore, these theory
are applicable to a wide range of
๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝโฒ B ๏ฟฝฬ๏ฟฝโฒ
๐ก1 ๐1 ๐2 ๐ก2
15
knowledge representation and ontology-
development systems, ontology alignment
and combination have a potentially large
impact on future alignment practices and
reasoning, Regardless of the semantic
paradigm employed, `reasoning' with
alignments involves at least three levels:
(1) the finding/discovery of alignments
(often based heavily on statistical
methods), (2) the construction of the
aligned ontology (the `colimit'), and (3)
reasoning over the aligned result,
respectively debugging and repair, closing
the loop to (1). Our contributions in this
paper address levels (2) .
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.
He is a computer science Student at the Faculty of
Exact Science of Oran 1 University (Algeria). He
earned his Master of Science degree in 2016, From
Oran 1 Ahmed Ben Bella University. His research
interests focus on subject of Artificial Intelligence