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

blakhdarelamine@gmail.com

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 user’s 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 alignment—AOP

(formerly Smart)—which 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]:

𝑂 = < 𝐶, 𝑅, 𝐼, 𝐻𝐶, 𝐻𝑅, 𝑋 > where:

𝐶: set of concepts.

𝑅: set of relations.

𝐼: set of instances of C and R.

𝐻𝐶: denotes a partial order relation on

C, called hierarchy or taxonomy of

concepts. It associates to each concept

its super or sub-concepts.

𝐻𝑅: denotes a partial order relation on

R, called hierarchy or taxonomy of

relations. It associates to each relation

its super or sub-relations.

𝑋: set of axioms.

𝑁𝑒𝑜 : Denotes Networks of ontologies

𝐷𝐺 ∶ 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 (𝑇𝑜𝑡𝑎𝑙) 𝐷𝑜𝑚𝑎𝑖𝑛

𝐷𝑝 ∶ 𝑃𝑎𝑟𝑡𝑖𝑎𝑙 Domain

We define as a logic syntax a tuple L = (𝐒𝐢𝐠𝐧, 𝐒𝐞𝐧, 𝐒𝐲𝐦𝐛𝐨𝐥𝐬, 𝐊𝐢𝐧𝐝𝐬, 𝐒𝐲𝐦, 𝐤𝐢𝐧𝐝, 𝐚𝐫𝐢𝐭𝐲)

𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑜𝑓:

a category Sign ,signature morphisms;

a sentence 𝑓𝑢𝑛𝑐𝑡𝑜𝑟 (Sen)

𝑆𝑒𝑛 ∶ 𝑺𝒊𝒈𝒏 → 𝑆𝑒𝑡 assigning to each

signature the set of its sentences and to

each signature morphism 𝜎: 𝛴 →

𝛴’ a sentence translation function

𝑺𝒆𝒏(𝜎) ∶ 𝑺𝒆𝒏(𝛴) → 𝑺𝒆𝒏(𝛴′);

a set Symbols of symbols and a set

Kinds of symbol kinds together with a

function kind : Symbols → Kinds

giving the kind of each symbol;

a faithful 𝑓𝑢𝑛𝑐𝑡𝑜𝑟

𝑺𝒚𝒎 ∶ 𝑺𝒊𝒈𝒏 → 𝑆 𝑒𝑡 assigning to

each signature Σ

a set of symbols 𝑺𝒚𝒎(𝜮) ⊆

𝑺𝒚𝒎𝒃𝒐𝒍𝒔 and to each signature

morphism 𝜎: 𝛴 → 𝛴′

a function 𝑺𝒚𝒎(𝝈) ∶ 𝑺𝒚𝒎(𝜮) →

𝑺𝒚𝒎(𝜮′) such that for each 𝒔 ∈

𝑺𝒚𝒎(𝜮), 𝒌𝒊𝒏𝒅(𝝈(𝒔)) = 𝒌𝒊𝒏𝒅(𝒔)

a function 𝒂𝒓𝒊𝒕𝒚 ∶ 𝑺𝒚𝒎𝒃𝒐𝒍𝒔 𝑁 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 Σ and

a set E of Σ -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 (𝐴, 𝑅, 𝐼 ) with A, R, and I Subsets of

a set of names. For two signatures 𝛴 =

(𝐴, 𝑅, 𝐼)and 𝛴′ = (𝐴′, 𝑅′, 𝐼′), a signature

morphism,ϕ: Σ → Σ′Consiste of function ∶

φA ∶ A → A′

𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜n: φR ∶ R → R′

𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜n: φI ∶ I → I′

𝑲𝒊𝒏𝒅𝒔 is the set {concept, role,

individual}.

𝑺𝒚𝒎𝒃𝒐𝒍𝒔 is the set of all pairs (𝒌, 𝒔)

where k is an element of 𝑲𝒊𝒏𝒅𝒔 and s is a

name. For each (𝑘, 𝑠) ∈ 𝑺𝒚𝒎𝒃𝒐𝒍𝒔,

𝒌𝒊𝒏𝒅(𝒌, 𝒔) = 𝒌. For each signature

𝛴 = (𝐴, 𝑅, 𝐼), 𝑆𝑦𝑚(𝛴) is the union of the

𝑠𝑒𝑡 {(𝑐𝑜𝑛𝑐𝑒𝑝𝑡, 𝑎)|𝑎 ∈ 𝐴} with

{(𝑟𝑜𝑙𝑒, 𝑟)|𝑟 ∈ 𝑅} and

{(𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙, 𝑖)| 𝑖 ∈ 𝐼}.

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 (𝑂𝑖)𝑖∈𝐼𝑛𝑑 of ontologies over a set of

indexes Ind interconnected by a set of

alignments (𝐴𝑖𝑖)𝑖∈𝐼𝑛𝑑 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

𝑡𝑟𝑖𝑝𝑙𝑒 (𝑠1, 𝑠2, 𝑅)where 𝑠1, 𝑠2 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠

and 𝑅 ∈ 𝑅such that (kind(𝑠1), kind(𝑠2)) 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 {(𝑠1𝑖 , 𝑠2

𝑖 , 𝑅𝑖)}𝑖=1,2,…𝑛

for

𝑛 ∈ 𝑁, such that for each 𝑖 = 1, . . . , 𝑛 we

have that 𝑠1𝑖 ∈ 𝑆𝑦𝑚(𝑠𝑖𝑔(𝑆)), 𝑠2

𝑖 ∈ 𝑆𝑦𝑚(𝑠𝑖𝑔(𝑇)),

𝑎𝑛𝑑 𝑅𝑖 ∈ ℜ

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)}

⊥ {(concept, concept), (role, role),

(individual, individual)}

<

{(concept, concept), (role, role)}

>

{(concept, concept), (role, role)}

∈ {(individual, concept)}

∋ {(concept, individual)}

Example 3 Similarly, in FOL we have the

following correspondences:

= {(fun, fun), (pred, pred)}

⊥ {(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

𝑆 = {(𝑂𝑖 )𝑖∈𝐼𝑛𝑑, (𝐴𝑖𝑗 )𝑖,𝑗∈𝐼𝑛𝑑}, 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)

𝐷. The relations in 𝑅 are interpreted as

relations over D, and we denote the

interpretation of R ∈ R by 𝑅𝐷. If 𝑂1, 𝑂2

are two ontologies and c = (𝑒1, 𝑒2, 𝑅) is a

correspondence between 𝑂1 and 𝑂2, we

say that c is satisfied by interpretations m1,

m2 of 𝑂1, 𝑂2 if 𝑚1 (𝑒1)) 𝑅𝐷 𝑚2(𝑒2)).

This is written 𝑚1, 𝑚2| =𝑅 c.

Definition 5 Given a model theory for a

logic L, the interpretation of

correspondence relations relative to a set is

an interpretation function .𝐼 taking as

arguments a relation 𝑅 ∈ 𝑅, 𝑘𝑖𝑛𝑑 (𝑘1, 𝑘2)𝑜𝑓 𝑅 𝑎𝑛𝑑 a set X and giving

as result a relation 𝑅𝐼 𝑑𝑜𝑚𝑎𝑖𝑛(𝑘2, 𝑋).

𝐷1 𝐷2 𝐷4 𝐷5

𝑂1

𝐷3

𝑂2 𝑂4 𝑂5

𝑂3

𝑚5

𝑚5′ 𝑚1’

𝑚3′ 𝑚4′

𝑚1

𝑚2′

𝐷𝐺

𝑚3 𝑚4 𝑚2

6

𝐼𝑓 𝑂1, 𝑂2 𝑎𝑟𝑒 two ontologies and 𝑐 =(𝑠1, 𝑠2, 𝑅) is a correspondence between

𝑂1 𝑎𝑛𝑑 𝑂2, we say that c is satisfied by the

models 𝑀1, 𝑀2, 𝑜𝑓𝑂1, 𝑂2, written

𝑀1, 𝑀2| =𝑆 𝐶, if and only if 𝑀𝑆1

1 𝑅𝐷1 𝑀𝑆2

2

model of an alignment A between

ontologies 𝑂1𝑎𝑛𝑑 𝑂2 𝑖𝑠 then a pair

𝑀1, 𝑀2 𝑜𝑓 interpretations of 𝑂1, 𝑂2 such

that for all c ∈ A, 𝑀1, 𝑀2| =𝑆 𝐶. We

denote this by 𝑀1, 𝑀2| =𝑆 𝐴. An

interpretation of S is a family

(𝑀𝑖)𝑖∈𝑖𝑛𝑑 of 𝑚𝑜𝑑𝑒𝑙 𝑀𝑖 𝑜𝑓 𝑂𝑖. A simple

interpretation of S is an interpretation

(𝑀𝑖)𝑖∈𝑖𝑛𝑑

𝑜𝑓 𝑆𝑖 over the same universe

D.

Definition 6 [80] A simple model of a

NeO S is a simple interpretation

(𝑀𝑖)𝑖∈𝑖𝑛𝑑

of S such that for each i, j

∈ I , 𝑀𝑖, 𝑀𝑗| =𝑆 𝐴𝑖𝑗. This is written

(𝑀𝑖)𝑖∈𝑖𝑛𝑑 | =𝑆 𝑆

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. (𝒄𝟏, 𝒄𝟐, =) = 𝑷(𝑫) 𝑿 𝑷(𝑫)

(𝒓𝟏, 𝒓𝟐, =) = 𝑷(𝑫 𝑿𝑫) 𝑿 𝑷(𝑫𝑿𝑫)

(𝒊𝟏, 𝒊𝟐, =) = 𝑫 𝑿 𝑫

(𝒄𝟏, 𝒄𝟐, ⊥) 𝑴𝑪𝟏

𝟏 ∩ 𝑴𝒄𝟐

𝟐= ∅ 𝑷(𝑫) 𝑿 𝑷(𝑫)

(𝒓𝟏, 𝒓𝟐, ⊥) 𝑴𝑹𝟏

𝟏 ∩ 𝑴𝑹𝟐

𝟐= ∅ 𝑷(𝑫 𝑿𝑫) 𝑿 𝑷(𝑫𝑿𝑫)

(𝒊𝟏, 𝒊𝟐, ⊥) ≠ 𝑫 𝑿 𝑫

(𝒄𝟏, 𝒄𝟐, <) ⊆ 𝑷(𝑫) 𝑿 𝑷(𝑫)

(𝒓𝟏, 𝒓𝟐, <) ⊆ 𝑷(𝑫 𝑿𝑫) 𝑿 𝑷(𝑫𝑿𝑫)

(𝒄𝟏, 𝒄𝟐, >) ⊇ 𝑷(𝑫) 𝑿 𝑷(𝑫)

(𝒓𝟏, 𝒓𝟐, >) ⊇ 𝑷(𝑫 𝑿𝑫) 𝑿 𝑷(𝑫𝑿𝑫)

(𝒄𝟏, 𝒊𝟐, ∋) ∋ 𝑷(𝑫)𝑿 𝑫

(𝒊𝟏, 𝒄𝟐, ∈) ∈ 𝑫 𝑿 𝑷 (𝑫)

where ck, rk, ikare class, role and individual symbols from an ontology Ok and Mk ∈ 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

(𝒇𝟏, 𝒇𝟐, =) = 𝑭 𝒖𝒏(𝑫) × 𝑭 𝒖𝒏(𝑫)

(𝒇𝟏, 𝒇𝟐, ⊥) ≠ 𝑭 𝒖𝒏(𝑫) × 𝑭 𝒖𝒏(𝑫)

(𝒑𝟏, 𝒑𝟐, =) = 𝑷 𝒓𝒆𝒅(𝑫) × 𝑷 𝒓𝒆𝒅(𝑫)

(𝒑𝟏, 𝒑𝟐, ⊥) ≠ 𝑷 𝒓𝒆𝒅(𝑫) × 𝑷 𝒓𝒆𝒅(𝑫)

(𝒑𝟏, 𝒑𝟐, <) ⊆ 𝑷 𝒓𝒆𝒅(𝑫) × 𝑷 𝒓𝒆𝒅(𝑫)

(𝒑𝟏, 𝒑𝟐, >) ⊇ 𝑷 𝒓𝒆𝒅(𝑫) × 𝑷 𝒓𝒆𝒅(𝑫)

where 𝑓𝑘 , 𝑝𝑘 𝑎𝑟𝑒 function and predicate

symbols from an ontology 𝑂𝑘, with

𝑘 = 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

𝛾𝑖 ∶ 𝐷𝑖 → 𝑈, where Di is the domain of

𝑂𝑖 , for each 𝑖 ∈ 𝐼. A relation R in R is

interpreted as a relation RU on the global

domain. Satisfaction of a correspondence

𝑐 = (𝑒1, 𝑒2, 𝑅) by two models 𝑚1 𝑜𝑓 𝑂1

and 𝑚2 𝑜𝑓 𝑂2 means that

𝛾𝑖(𝑚𝑖(𝑒1))𝑅𝑈 𝛾𝑗 (𝑚𝑗 (𝑒2)).

Definition 7 [80] An integrated

interpretation of

a NeO S, {(𝑀𝑖)𝑖∈𝑖𝑛𝑑

, (𝛾𝑖)𝑖 ∈ 𝑖𝑛𝑑}is an

integrated model of 𝑆 𝑖ff 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖, 𝑗 ∈

𝐼 𝑛𝑑, 𝑀1, 𝑀2|=𝛾1,𝛾2𝐼 𝐴𝑖𝑗 |

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

𝑂1 𝑂2 𝑂3 𝑂4 𝑂5

𝐷1 𝐷2 𝐷3 𝐷4 𝐷5

𝑚1

𝑚2

𝑚3

𝑚4

𝑚5

𝑈

𝛾3 𝛾4 𝛾2 𝛾1 𝛾3

7

the semantics of DDLs [9] and 𝜀-

connections [41]. The idea is to relate the

domains of the ontologies by a family of

relations 𝑟 = (𝑟𝑖𝑗 )𝑖, 𝑗 ∈ 𝐼. The relations R

in R are interpreted in each domain of the

ontologies in the NeO. Satisfaction of a

correspondence 𝑐 = (𝑒1, 𝑒2, 𝑅) by two

models 𝑚1 of 𝑂1 and 𝑚2 of 𝑂2 means that

𝑚𝑖(𝑒1)𝑅𝑖 𝑟𝑗𝑖(𝑚𝑗 (𝑒2)), where 𝑅𝑖 is the

interpretation of R in 𝐷𝑖

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 (𝑐1, 𝑐2, =) 𝑀𝑐1

1 = 𝑟21(𝑀𝑐22 )

(𝑟1, 𝑟2, =) 𝑀𝑟11 = 𝑟21(𝑀𝑟2

2 )

(𝑖1, 𝑖2, =) 𝑀𝑖11 = 𝑟21(𝑀𝑖2

2 ) 𝑖. 𝑒 𝑀𝑖11 , 𝑀𝑖2

1 ∈ 𝑟21

(𝑐1, 𝑐2, ⊥) 𝑀𝑐11 ∩ 𝑟21(𝑀𝑐2

2 ) = ∅

(𝑟1, 𝑟2, ⊥) 𝑀𝑟11 ∩ 𝑟21(𝑀𝑟2

2 ) = ∅

(𝑖1, 𝑖2, ⊥) (𝑀𝑖22 , 𝑀𝑖1

1 ) ∉ 𝑟21

(𝑐1, 𝑐2, <) 𝑀𝑐11 ⊆ 𝑟21(𝑀𝑐2

2 )

(𝑟1, 𝑟2, <) 𝑀𝑟11 ⊆ 𝑟21(𝑀𝑟2

2 )

(𝑐1, 𝑐2, >) 𝑀𝑐11 ⊇ 𝑟21(𝑀𝑐2

2 )

(𝑟1, 𝑟2, >) 𝑀𝑟11 ⊇ 𝑟21(𝑀𝑟2

2 )

(𝑐1, 𝑖2, ∋) 𝑟21(𝑀𝑖22 ) ⊆ 𝑀𝑐1

1

(𝑖1, 𝑐2, ∈) 𝑀𝑖11 = 𝑟21(𝑀𝑐2

2 )

where 𝑐1, 𝑟1, 𝑖1𝑎𝑟𝑒 class, role and

individual symbols from an ontology

𝑂1, 𝑐2, 𝑟2, 𝑖2 𝑎𝑟𝑒 class, role and individual

symbols from an ontology 𝑂2, 𝑀1 and

𝑀2 are models of 𝑂1and 𝑂2 with domains

𝐷1 and 𝐷2 and 𝑟21 is the domain relation

between 𝐷1 𝑎𝑛𝑑 𝐷2 .

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:

𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑆 𝐶𝑙𝑎𝑠𝑠 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠 𝐵𝑖𝑒𝑛_𝑒𝑡𝑟𝑒

𝑂1 𝑂2 𝑂3 𝑂4 𝑂5

𝐷2 𝐷3 𝐷4 𝐷5

𝑟1,2 𝑟5,4

…

.. 𝑟1,3

……………

𝐷1

8

𝐶𝑙𝑎𝑠𝑠 𝐸𝑛𝑓𝑎𝑛𝑡

𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑇 = 𝐶𝑙𝑎𝑠𝑠 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝑇𝑜𝑔𝑒𝑡ℎ𝑒𝑟 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒𝑠 𝑆: 𝐵𝑒𝑛_𝐸𝑡𝑟𝑒 = 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝑆: 𝐴𝑚𝑖𝑛𝑒 ∈ 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 ⊥ 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 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

𝐴𝑙𝑖𝑔𝑛𝑒𝑚𝑒𝑛𝑡 𝐴 𝑆 𝑡𝑜 𝑇 = 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 =

𝐹𝑜𝑟𝑚𝑎𝑡

Amine ∈ Masculin,

Enfant ⊥ 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={𝑐𝑖 = (𝑠1𝑖 , 𝑠2

𝑖 , 𝑅𝑖)|𝑖 ∈ 𝐼𝑛𝑑} 𝑎𝑛 𝑎𝑙𝑖𝑔𝑛𝑚𝑒𝑛𝑒𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑆 𝑎𝑛𝑑 𝑇 , the

diagram of the alignment A is :

Where the ontologies :

B, S̃ , S̃’, T̃′, T̃ and the Morphism t1, t2, σ1, σ2

depend on the choice of semantics for the

alignment A, in a way to be made precise

for each possible option.

Intuitively, �̃�′ and �̃�′ 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 𝑀 of a

theory that internalises the semantics of

(𝑠1, 𝑠2 , 𝑅), are precisely those for which

the relation 𝑀𝑅 holds for 𝑀𝑠1 𝑎𝑛𝑑 𝑀𝑠2

, 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.. �̃�’ and 𝑇′̃ 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 𝑡𝑖 𝑎𝑛𝑑 𝜎𝑖 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 𝑂𝑖 in a network of

aligned ontologies, let �̃�𝑖 be its corres-

ponding ontology in the diagram of the

network. Let Σ𝑖 = 𝑆𝑖𝑔(𝑂𝑖) and Σ𝑖′ =

𝑆𝑖𝑔(𝑂𝑖′). In each of the four cases that

correspond to the different choices of

semantics we can define:

(i)a sentence translation functor α∗

∗

:

𝑆𝑒𝑛(𝛴𝑖) → 𝑆𝑒𝑛(Σ𝑖′)

(ii) model reduct factor β∗

∗

:

B �̃�

�̃� �̃�′ �̃�′

𝑡1 𝜎2 𝑡2 𝜎1

9

𝑀𝑜𝑑(𝛴𝑖) → 𝑀𝑜𝑑(Σ𝑖′)

such that the condition 𝛽∗(𝑀′)| = 𝑒

⇔ 𝑀′| = 𝛼∗(𝑒) ℎ𝑜𝑙𝑑 for each 𝑀 ∈

(𝑀𝑜𝑑(�̃�𝑖)) 𝑎𝑛𝑑 𝑒𝑎𝑐ℎ 𝑒 ∈ 𝑆𝑒𝑛 (Σ𝑖) 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 Σ𝑖 𝑡𝑜 Σ𝑖′

Such that α∗ and β∗ 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, 𝑙𝑒𝑡 𝐴 =

{𝑐𝑖 = (𝑠1𝑖 , 𝑠2

𝑖 , 𝑅𝑖)|𝑖 ∈ 𝐼𝑛𝑑} 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

Σ𝐵 𝑜𝑓 𝐵 is defined as the union of

𝑆𝑖𝑔1(𝐴) 𝑎𝑛𝑑 𝑆𝑖𝑔2(𝐴) where Σ1is the

smallest subsignature of Sig(S ) such that

Symbols(Σ1) includes 𝑠1𝑖 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖 ∈

𝐼𝑛𝑑 is the signature obtained by renaming

every 𝑠𝑦𝑚𝑏𝑜𝑙 𝑠 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠(Σ1) to S:s

and Σ2 is the smallest subsignature of

Sig(T ) such that Symbols(Σ2) includes

𝑓𝑜𝑟 𝑒𝑎𝑐ℎ , 𝑎𝑛𝑑 𝑆2𝑖 𝑆𝑖𝑔2(𝐴) is the signat-

ure obtained by renaming every symbol

𝑠 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠(𝛴2) 𝑡𝑜 𝑇: 𝑠.

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 Σ𝐵 be the bridge

signature of A and ∆ a set of

Σ𝐵 sentences. We say that (Σ𝐵, ∆)

internalises the semantics of

{𝑐𝑖 = (𝑠1𝑖 , 𝑠2

𝑖 , 𝑅𝑖)|𝑖 ∈ 𝐼𝑛𝑑} denoted

(Σ𝐵, ∆) ≊𝑠𝑖𝑚 𝑐𝑖, 𝑖𝑓

𝑀 |= Σ𝐵 Δ 𝑖𝑓𝑓 (𝑀𝑆:𝑆𝑖2

, 𝑀𝑇:𝑆𝑖2

) ∈

(𝑅𝐼) 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒(𝑀) for each ΣB

Definition 12 Let Σ𝐵 be the bridge signatu-

re of A.

Assume that (Σ𝐵, ∆) ≊𝑠𝑖𝑚 𝑐𝑖 each 𝑐𝑖 ∈ 𝐴.

The diagram of A is obtained by setting

the parameters as follows:

�̃� = 𝑆 𝑎𝑛𝑑 �̃� = 𝑆

�̃� = (𝑆𝑖𝑔1(𝐴), ∅)

𝑡1 𝑚𝑎𝑝𝑠 𝑒𝑎𝑐ℎ 𝑆: 𝑠 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠(𝑆𝑖𝑔1(𝐴))𝑡𝑜 𝑠

�̃� = (𝑆𝑖𝑔1(𝐴), ∅)

𝑡2 𝑚𝑎𝑝𝑠 𝑒𝑎𝑐ℎ 𝑇: 𝑠 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠(𝑆𝑖𝑔1(𝐴))𝑡𝑜 𝑠

𝐵 = (Σ𝐵, ⋃ 𝑖 ∈ 𝐼𝑛𝑑 ∆𝑖 )

𝜎1 𝑎𝑛𝑑 𝜎2 𝑎𝑟𝑒 𝑖𝑛𝑐𝑙𝑢𝑠𝑖𝑜𝑛𝑠 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 .

(𝑐1, 𝑐2, =) 𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝑐1 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ∶ 𝑇: 𝑐2 (𝑟1, 𝑟2, =) 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑆: 𝑟1 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ∶ 𝑇: 𝑟2 (𝑖1, 𝑖2, =) 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝑖1 𝑆𝑎𝑚𝑒 𝑎𝑠 𝑇: 𝑖2 (𝑐1, 𝑐2, ⊥)𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝑐1 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑠 𝑤𝑖𝑡ℎ 𝑇: 𝑐2 (𝑟1, 𝑟2, ⊥)𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑆: 𝑟1 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑠 𝑤𝑖𝑡ℎ 𝑇: 𝑟2 (𝑖1, 𝑖2, ⊥)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝑖1 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑇: 𝑖2 (𝑐1, 𝑐2, <)𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝑐1 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑇: 𝐶2 (𝑟1, 𝑟2, <)𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑆: 𝑟1 𝑆𝑢𝑏𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑤𝑖𝑡ℎ 𝑟2 (𝑐1, 𝑐2, >)𝐶𝑙𝑎𝑠𝑠: 𝑇: 𝑐2 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑜𝑓 𝑆: 𝑐1 (𝑟1, 𝑟2, >)𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑇: 𝑟2

10

𝑆𝑢𝑏𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑂𝑓 𝑆: 𝑟1 (𝑐1, 𝑖2, ∋)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑇: 𝑖2 𝑇𝑦𝑝𝑒 𝑆: 𝐶1 (𝑖1, 𝑐2, ∈)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝑖1 𝑇𝑦𝑝𝑒 𝑆: 𝐶1 Example 8 (Simple semantics in FOL)

Similarly, in semantics of corresp-

ondences:

(𝑓1, 𝑓2 =) ∀𝑥1, . . . , 𝑥𝑛. 𝑆: 𝑓1(𝑥1, . . . , 𝑥𝑛) = 𝑇: 𝑓2(𝑥1, . . . , 𝑥𝑛) (𝑓1, 𝑓2, ⊥)∀𝑥1, . . . , 𝑥𝑛. ¬𝑆: 𝑓1(𝑥1, . . . , 𝑥𝑛) = 𝑇: 𝑓2(𝑥1, . . . , 𝑥𝑛) (𝑝1, 𝑝2, =) ∀𝑥1, . . . , 𝑥𝑛. 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ⇐⇒ 𝑇: 𝑝2(𝑥1, . . . , 𝑥𝑛) (𝑝1, 𝑝2, ⊥)∀𝑥1, . . . , 𝑥𝑛. ¬(𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑇: 𝑝2(𝑥1, . . . , 𝑥𝑛)) (𝑝1, 𝑝2, <)∀𝑥1, . . . , 𝑥𝑛. 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) =⇒ 𝑇: 𝑝2(𝑥1, . . . , 𝑥𝑛) (𝑝1, 𝑝2, >)∀𝑥1, . . . , 𝑥𝑛. 𝑇: 𝑝2(𝑥1, . . . , 𝑥𝑛) =⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) 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=

𝑊ℎ𝑒𝑟𝑒 𝑆′𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑠 𝑆: 𝐵𝑖𝑒𝑛𝐸𝑡𝑟𝑒𝑎𝑛𝑑 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑆: 𝑎𝑚𝑖𝑛𝑒 𝑎𝑛𝑑 𝑇′𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑠 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡, 𝑇: 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑒 𝑎𝑛𝑑 𝑇: 𝑀𝑎𝑙𝑒. 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒

𝑏𝑟𝑖𝑑𝑔𝑒 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐵 𝑖𝑠: 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐵 = 𝐶𝑙𝑎𝑠𝑠 𝑆 ∶ 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑇

∶ 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 𝑇 ∶ 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐶𝑙𝑎𝑠𝑠 𝑇 ∶ 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝐶𝑙𝑎𝑠𝑠 𝑆 ∶ 𝐸𝑛𝑓𝑎𝑛𝑡 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 𝑤𝑖𝑡ℎ 𝑇

∶ 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑆 ∶ 𝐴𝑚𝑖𝑛𝑒, 𝑇𝑦𝑝𝑒 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 We can combine the resulting functional

network into a single ontology. In DOL,

this is written as:

𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐶 = 𝑐𝑜𝑚𝑏𝑖𝑛𝑒 𝑁 𝑇ℎ𝑒 𝑐𝑜𝑙𝑖𝑚𝑖𝑡 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑡𝑤𝑜𝑟𝑘 𝑜𝑓 𝐴 𝑖𝑠:

𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐶 =

𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝑇𝑜:

𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡

𝐶𝑙𝑎𝑠𝑠: 𝑇: 𝑇r𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟

𝐶𝑙𝑎𝑠𝑠: 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡

𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑊𝑖𝑡ℎ: 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟

𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝑎𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠: 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛, 𝑆:

𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒

𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐶 =

𝐶𝑙𝑎𝑠𝑠 𝑆: 𝐵𝑒𝑛_𝐸𝑡𝑟𝑒

∶ 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡

𝐶𝑙𝑎𝑠𝑠 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟

𝐶𝑙𝑎𝑠𝑠 𝑇

∶ 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆𝑢𝑏 𝐶𝑙𝑎𝑠𝑠 𝑂𝑓 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡

𝐶𝑙𝑎𝑠𝑠 𝑆

∶ 𝐸𝑛𝑓𝑎𝑛𝑡 𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡 𝑤𝑖𝑡ℎ: 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟

𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑆

∶ 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆: 𝐵𝑖𝑒𝑛_𝑒𝑡𝑟𝑒

Since the original ontologies are not

modified in the diagram of the alignments,

the signature morphism from 𝑆𝑖𝑔(𝑂𝑖) to

𝑆𝑖𝑔(�̃�𝑖) is the identity, so the functors

𝛼∗ 𝑎𝑛𝑑 𝛽∗ are the identities on 𝑆𝑖𝑔(𝑂𝑖) −

𝑠𝑒𝑛𝑡𝑒𝑛𝑐𝑒𝑠, respectively on 𝑆𝑖𝑔(𝑂𝑖) −

𝑚𝑜𝑑𝑒𝑙𝑠.

Example 10 (Generalised integrated

semantics in FOL) In FOL we have the

following basic bridge ontology:

∀ 𝑥1, 𝑥2, 𝑧 . 𝑧𝑟𝑠𝑥1 ⋀ 𝑧𝑟𝑠𝑥2 ⟹ 𝑥1 = 𝑥2 ∀ 𝑥 . 𝑆:⊺ (𝑥) ⟹ ∃𝑧 . 𝑧𝑟𝑠 𝑥 ∀𝑥, 𝑧 . 𝑧𝑟𝑠𝑥 ⟹ 𝑆:⊺ (𝑥) ∧ 𝐺(𝑧)

∀ 𝑥1, 𝑥2, 𝑧 . 𝑧𝑟𝑇𝑥1 ⋀ 𝑧𝑟𝑇𝑥2 ⟹ 𝑥1 = 𝑥2 ∀ 𝑥 . 𝑇:⊺ (𝑥) ⟹ ∃𝑧 . 𝑧𝑟𝑇 𝑥 ∀𝑥, 𝑧 . 𝑧𝑟𝑠𝑥 ⟹ 𝑇:⊺ (𝑥) ∧ 𝐺(𝑧) where for each of 𝑟𝑆 and 𝑟𝑇, 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:

(𝑓1, 𝑓2, ) ∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛

S B

�̃�′ �̃�′ T

Tapez une équation ici.

𝑡1 𝑡2

𝜎1 𝜎2

11

=⇒

∃𝑧 . 𝑧 𝑟𝑠𝑆: 𝑓1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑧 𝑟𝑇𝑇: 𝑓2(𝑦1, . . . , 𝑦𝑛) (𝑓1, 𝑓2, ) ∃𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛 =⇒

∃𝑧 . 𝑧 𝑟𝑠𝑆: 𝑓1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑧 𝑟𝑇𝑇: 𝑓2(𝑦1, . . . , 𝑦𝑛)𝑟𝑇

(𝑝1, 𝑝2, ) ∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛 =⇒

𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛)

(𝑝1, 𝑝2, )¬∃𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛 ∧ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛)

(𝑝1, 𝑝2, )∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛 =⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) =⇒ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) (𝑝1, 𝑝2, )∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛, 𝑧1, . . . , 𝑧𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑧1𝑟𝑆𝑥1 ∧. . .∧ 𝑧𝑛𝑟𝑆𝑥𝑛 ∧ 𝑧1𝑟𝑇𝑦1 ∧. . .∧ 𝑧𝑛𝑟𝑇𝑦𝑛 =⇒ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) =⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) Example 11 (General integrated semantics

in SROIQ)

The basic bridge ontology for general

integrated semantics in SROIQ is

𝑂𝑏𝑗𝑒𝑐𝑡𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑟𝑠 𝐶ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠 ∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙

𝐷𝑜𝑚𝑎𝑖𝑛 ∶ 𝐺 𝑅𝑎𝑛𝑔𝑒 ∶ 𝑆:⊺ 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑆:⊺ 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 ∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑠

𝑂𝑏𝑗𝑒𝑐𝑡𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑟𝑇 𝐶ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠 ∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐷𝑜𝑚𝑎𝑖𝑛 ∶ 𝐺 𝑅𝑎𝑛𝑔𝑒 ∶ 𝑇:⊺ 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇:⊺ 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 ∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇 𝐶𝑙𝑎𝑠𝑠 ∶ 𝐺 𝑇ℎ𝑒 𝑡ℎ𝑒𝑜𝑟𝑦 𝑡ℎ𝑎𝑡 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝑖𝑧𝑒 𝑡ℎ𝑒 𝑠𝑒𝑚𝑎𝑛𝑡𝑖𝑐𝑠 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 𝑎𝑟𝑒 ∶ (𝑐1, 𝑐2, =) 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠 𝑟𝑠 𝑆𝑜𝑚𝑒 𝑆: 𝑐1 𝑟𝑇 𝑆𝑜𝑚𝑒 𝑇: 𝑐2 (𝑖1, 𝑖2, =) 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠 𝑟𝑠 𝑆𝑜𝑚𝑒 {𝑆: 𝑖1} 𝑟𝑇 𝑆𝑜𝑚𝑒 {𝑇: 𝑖2} (𝑐1, 𝑐2, ⊥) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑠 𝑠𝑜𝑚𝑒 {𝑆: 𝑐1} 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑠 𝑤𝑖𝑡ℎ 𝑟𝑇 𝑆𝑜𝑚𝑒 𝑇: 𝑐2 (𝑖1, 𝑖2, ⊥) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑠 𝑠𝑜𝑚𝑒 {𝑆: 𝑖1} 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑠 𝑤𝑖𝑡ℎ 𝑟𝑇 𝑆𝑜𝑚𝑒 𝑇: 𝑖2 (𝑐1, 𝑐2, <) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑠 𝑠𝑜𝑚𝑒 {𝑆: 𝑐1} 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑟𝑇 𝑠𝑜𝑚𝑒 𝑇 ∶ 𝑐2 (𝑐1, 𝑐2, >) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑇 𝑠𝑜𝑚𝑒 {𝑇: 𝑐1} 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑟𝑆 𝑠𝑜𝑚𝑒 𝑆 ∶ 𝑐2 (𝑐1, 𝑖2, ∋) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑇 𝑠𝑜𝑚𝑒 {𝑇: 𝑖2} 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑟𝑆 𝑠𝑜𝑚𝑒 𝑆 ∶ 𝑐1 (𝑖1, 𝑐2, ∈) 𝑐𝑙𝑎𝑠𝑠 ∶ 𝑟𝑇 𝑠𝑜𝑚𝑒 {𝑆: 𝑖1} 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑟𝑆 𝑠𝑜𝑚𝑒 𝑇 ∶ 𝑐2

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

(𝑟1, 𝑟2, =), (𝑟1, 𝑟2, ⊥) , (𝑟1, 𝑟2, <) and (𝑟1, 𝑟2, >) 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

�̃�′consists of the concepts G,

S:Thing, S :Bien_etre and S :Enfant, the

object property 𝑟𝑠and the individual

S :amine and �̃�′consists of the concepts G,

T :Thing, T :Format 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 and

T :Masculin and the object property 𝑟𝑇.

The ontologies

𝑆 ̃ 𝑎𝑛𝑑 𝑇 ̃ 𝑎𝑟𝑒 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 �̃� = 𝑐𝑙𝑎𝑠𝑠 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 ∶ 𝐺 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑠 𝐶𝑙𝑎𝑠𝑠 ∶ 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝑠𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝑆: 𝑡ℎ𝑖𝑛𝑔 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 ∶ 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒, 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠: 𝐸𝑛𝑓𝑎𝑛𝑡 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 ∶ 𝑆: 𝑇ℎ𝑖𝑛𝑔

𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑇 ̃ = 𝑐𝑙𝑎𝑠𝑠 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 ∶ 𝐺 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑇 𝐶𝑙𝑎𝑠𝑠 𝐹𝑜𝑟𝑚𝑎𝑡 𝑠𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝑇: 𝑡ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝐹𝑜𝑟𝑚𝑎𝑡 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 ∶ 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝑇ℎ𝑒 𝑏𝑟𝑖𝑑𝑔𝑒 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐵 𝑜𝑓 𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝑖𝑠 ∶ 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐵 = 𝐶𝑙𝑎𝑠𝑠 ∶ 𝐺 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑠 𝑠𝑜𝑚𝑒 𝐺 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑇 𝐶ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠: 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐹𝑢𝑛𝑐𝑖𝑜𝑛𝑎𝑙 𝐷𝑜𝑚𝑎𝑖𝑛 ∶ 𝐺 𝑅𝑎𝑛𝑔𝑒 ∶ 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝐶𝑙𝑎𝑠𝑠𝑒𝑠 ∶ 𝑟𝑠 𝑠𝑜𝑚𝑒 𝑆: 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝑟𝑇 𝑠𝑜𝑚𝑒 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 𝑟𝑠 𝑠𝑜𝑚𝑒 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑤𝑖𝑡ℎ 𝑟𝑇 𝑠𝑜𝑚𝑒 ∶

𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐶𝑙𝑎𝑠𝑠 𝑟𝑠 𝑠𝑜𝑚𝑒 {𝑆: 𝐴𝑚𝑖𝑛𝑒} 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 ∶ 𝑟_𝑡 𝑠𝑜𝑚𝑒 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛

𝑇ℎ𝑒 𝑐𝑜𝑙𝑜𝑚𝑖𝑡 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑖𝑧𝑒𝑑 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑙𝑖𝑔𝑛𝑚𝑒𝑛𝑡 𝑖𝑛 𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝑖𝑠 ∶ 𝑂 𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐶 ∶ 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑮 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝑆𝑢𝐵𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑠 𝑠𝑜𝑚𝑒 𝐺 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑠 𝐶ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠: 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐹𝑢𝑛𝑐𝑖𝑜𝑛𝑎𝑙 𝐷𝑜𝑚𝑎𝑖𝑛 ∶ 𝐺 𝑅𝑎𝑛𝑔𝑒 ∶ 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 ∶

𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝑆𝑢𝐵𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 ∶

𝑆: 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝑆𝑢𝐵𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝐶𝑙𝑎𝑠𝑠 𝑟𝑠 𝑠𝑜𝑚𝑒 𝑆: 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝑟𝑇 𝑠𝑜𝑚𝑒 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑆: 𝐸𝑛𝑓𝑎𝑡𝑛 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑟𝑠 𝑠𝑜𝑚𝑒 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 𝑤𝑖𝑡ℎ ∶

𝑟𝑇 𝑠𝑜𝑚𝑒 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 ∶

𝑆: 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠 𝑆: 𝑃𝑒𝑟𝑠𝑜𝑛 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑟𝑠 𝑠𝑜𝑚𝑒 {𝑆: 𝐴𝑚𝑖𝑛𝑒} 𝑠𝑢𝑏𝐶𝑙𝑎𝑠𝑠 𝑂𝑓 𝑟𝑇 𝑠𝑜𝑚𝑒 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝜎1

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={𝑐𝑖 = (𝑠1𝑖 , 𝑠2

𝑖 , 𝑅𝑖)|𝑖 ∈ 𝐼𝑛𝑑} be an

alignment between two ontologies S and

T in a logic L, where Ind is a set of

indices. The basic bridge ontology (Σ𝐵, Δ𝐵)

of A in the contextualised semantics

consists of

−𝑎 𝑠𝑖𝑔𝑛𝑎𝑡𝑢𝑟𝑒 Σ𝐵 𝑡ℎ𝑎𝑡 𝑡𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑢𝑛𝑖𝑜𝑛

𝑜𝑓 𝑆𝑖𝑔1(𝐴) 𝑎𝑛𝑑 𝑆𝑖𝑔2(𝐴), 𝑤ℎ𝑒𝑟𝑒

– Σ1 is the smallest subsignature of

Sig(S ) such that Symbols(Σ1)

includes 𝑠1𝑖 and 𝑆𝑖𝑔1(A) takes the

signature obtained by renaming every

�̃� �̃�

�̃�′ B �̃�′ 𝑡1

𝜎1

𝜎2

𝑡2

13

s ∈ Symbols(Σ1) to S :s and extends it

with S :⊺ ∈ 𝑢𝑛𝑎𝑟𝑦𝐿,

– Σ2 is the smallest subsignature of

Sig(T ) such that Symbols(Σ2 ) includes

𝑆2𝑖 and 𝑆𝑖𝑔2(A) takes the signature obtain-

ed by renaming every 𝑠 ∈ 𝑆𝑦𝑚𝑏𝑜𝑙𝑠(Σ2 ) to

T :s and extends with T : ⊺ ∈ 𝑢𝑛𝑎𝑟𝑦𝐿 and

extends this union with 𝑟𝑇𝑆 in 𝑏𝑖𝑛𝑎𝑟𝑦𝐿 .

– a set Δ𝐵 𝑜𝑓 Σ𝐵-sentences that

axiomatise in a logic-dependent way that

the domain of 𝑟𝑇𝑆 is T :⊺and the range

of 𝑟𝑇𝑆 is S :⊺.

Definition 14 Let c = (𝑠1, 𝑠2, R) be a

correspondence of a contextualised

alignment A. Let (Σ𝐵, Δ𝐵) be the basic

bridge ontology of 𝐴 and let ∆ be a set of

Σ𝐵 − 𝑠𝑒𝑛𝑡𝑒𝑛𝑐𝑒𝑠 that includes Δ𝐵. We say

that (Σ𝐵, ∆) internalises the semantics of c ,

denoted (Σ𝐵, ∆)| =𝑐𝑜𝑛 𝑐 If 𝑀| =Σ𝐵∆ 𝑖𝑓𝑓 𝑀𝑆:𝑠1

𝑅𝐼 𝑀𝑆:𝑇 𝑀𝑟𝑇𝑆(𝑀𝑇:𝑆2).

Definition 15 Assume that

(Σ𝐵, ∆𝑖) ≊𝑐𝑜𝑛

𝑐𝑖for each 𝑐𝑖 ∈ 𝐴. The

parameters of Def. 14 are set as follows

𝑆 = 𝑟𝑒𝑙𝐿(𝑆)𝑎𝑛𝑑 𝑇 = 𝑟𝑒𝑙𝐿(𝑇), 𝑤ℎ𝑒𝑟𝑒 𝑟𝑒𝑙𝐿 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑔𝑖𝑐 𝐿 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑖𝑒𝑠 𝑏𝑒𝑖𝑛𝑔 𝑎𝑙𝑖𝑔𝑛𝑒𝑑, -�̃�′ = (𝑆𝑖𝑔(𝐴), ∅)

−𝑡1 𝑚𝑎𝑝𝑠 𝑒𝑎𝑐ℎ 𝑆: 𝑠 ∈𝑆𝑦𝑚𝑏𝑜𝑙𝑠(𝑆𝑖𝑔1(𝐴)) 𝑡𝑜 𝑠 𝑎𝑛𝑑 𝑆 ∶ ⊺

𝑡𝑜 𝑖𝑡𝑠𝑒𝑙𝑓

-�̃�′ = (𝑆𝑖𝑔(𝐴), ∅)

−𝑡2 𝑚𝑎𝑝𝑠 𝑒𝑎𝑐ℎ 𝑇: 𝑠 ∈𝑆𝑦𝑚𝑏𝑜𝑙𝑠(𝑆𝑖𝑔2 (𝐴)) 𝑡𝑜 𝑠𝑎𝑛𝑑 𝑇:⊺ 𝑡𝑜 𝑖𝑡𝑠𝑒𝑙𝑓, – 𝐵 = (Σ𝐵, ⋃ 𝑖 ∈ 𝐼𝑛𝑑

∆𝑖),

– 𝜎1 𝑎𝑛𝑑 𝜎2 𝑎𝑟𝑒 𝑖𝑛𝑐𝑙𝑢𝑠𝑖𝑜𝑛𝑠. Example 13 (Contextualised semantics in

FOL) In FOL we have the following

theories that internalise the semantics of

correspondences:

(𝑓1, 𝑓2, =) ∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆

𝑥𝑛

=⇒ 𝑆: 𝑓1(𝑦1, . . . , 𝑦𝑛)𝑟𝑇𝑆 𝑇: 𝑓2(𝑥1, . . . , 𝑥𝑛)

(𝑓1, 𝑓2, ⊥)¬∃𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛)

∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆𝑥𝑛

∧ 𝑆: 𝑓1(𝑦1, . . . , 𝑦𝑛)𝑟𝑇 𝑆 𝑇: 𝑓2(𝑥1, . . . , 𝑥𝑛)

(𝑝1, 𝑝2, =) ∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆

𝑥𝑛

=⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ⇐⇒ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) (𝑝1, 𝑝2, ⊥)¬∃𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆

𝑥𝑛

∧ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) ∧ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) (𝑝1, 𝑝2, <)∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆

𝑥𝑛

=⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) =⇒ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) (𝑝1, 𝑝2, >)∀𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛. 𝑆:⊺ (𝑥1) ∧. . .∧ 𝑆:⊺ (𝑥𝑛) ∧ 𝑇:⊺ (𝑦1) ∧. . .∧ 𝑇:⊺ (𝑦𝑛) ∧ 𝑦1𝑟𝑇𝑆

𝑥1 ∧. . .∧ 𝑦𝑛𝑟𝑇𝑆

𝑥𝑛

=⇒ 𝑇: 𝑝2(𝑦1, . . . , 𝑦𝑛) =⇒ 𝑆: 𝑝1(𝑥1, . . . , 𝑥𝑛) 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 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑖𝑒𝑠 𝑆 𝑎𝑛𝑑 𝑇 is obtained

by applying the relativisation of the

aligned ontologies and to the

correspondences of the alignment. The

basic bridge ontology is

𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑇𝑆 𝐷𝑜𝑚𝑎𝑖𝑛 ∶ 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝑅𝑎𝑛𝑔𝑒 ∶ 𝑆: 𝑇ℎ𝑖𝑛𝑔 The theories internalising the semantics of

the correspondences extend it as follows:

(𝑐1, 𝑐2 =) 𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝑐1 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ∶ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆) 𝑠𝑜𝑚𝑒 𝑇: 𝑐2 (𝑖1, 𝑖2, =)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑇: 𝑖2 𝐹𝑎𝑐𝑡𝑠: 𝑟𝑇𝑆 𝑆: 𝑖1 (𝑐1, 𝑐2, ⊥)𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠:

14

(𝑆: 𝑐1 ) 𝑎𝑛𝑑 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝑐2 𝑁𝑜𝑡ℎ𝑖𝑛𝑔 (𝑖1, 𝑖2, ⊥)𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠: (𝑇: 𝑖2) 𝑎𝑛𝑑 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑆: 𝑇1 𝑁𝑜𝑡ℎ𝑖𝑛𝑔 (𝑐1, 𝑐2, >)𝐶𝑙𝑎𝑠𝑠: 𝑖𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆

)𝑠𝑜𝑚𝑒 𝑇: 𝑐2

𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑆: 𝑐1 (𝑟1, 𝑟2, >)𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑆: 𝑟1 𝑆𝑢𝑏𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝐶ℎ𝑎𝑖𝑛 𝐼𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆) ∘ 𝑇: 𝑟2 ∘ 𝑟𝑇𝑆 (𝑐1, 𝑐2, <)𝐶𝑙𝑎𝑠𝑠: 𝑆: 𝑐1 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓: 𝑖𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆) 𝑠𝑜𝑚𝑒 𝑇: 𝑐2

(𝑟1, 𝑟2, <)𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑆: 𝑟1 𝑆𝑢𝑏𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝐶ℎ𝑎𝑖𝑛 𝑟𝑇𝑆 ∘ 𝑇: 𝑟1 ∘ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆) (𝑐1, 𝑖2, ∋)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑇: 𝑖2 𝐹𝑎𝑐𝑡𝑠 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑆: 𝑐1 (𝑖1, 𝑐2, ∈)𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝑖1 𝐹𝑎𝑐𝑡𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒(𝑟𝑇𝑆) 𝑠𝑜𝑚𝑒 𝑇: 𝑐2 For the correspondence (𝑟1, 𝑟2, =) where

𝑟1𝑎𝑛𝑑 𝑟2 are roles, it is not possible to

express in SROIQ that

𝑟1 𝑎𝑛𝑑 𝑟𝑇𝑆−1, 𝑟2, 𝑟𝑇𝑆 𝑎𝑟𝑒 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑅𝑜𝑙𝑒,

𝑊ℎ𝑒𝑟𝑒 𝑟𝑇𝑆 𝑖𝑠 the domain relation. A

similar problem appears for the

correspondence (𝑟1, 𝑟2, ⊥).

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 :

𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐵: 𝐶𝑙𝑎𝑠𝑠 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑇 ∶ 𝑇ℎ𝑖𝑛𝑔 𝑂𝑏𝑗𝑒𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∶ 𝑟𝑇𝑆 𝐷𝑜𝑚𝑎𝑖𝑛 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝑅𝑎𝑛𝑔𝑒 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑆 ∶ 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒

𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑇𝑜∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇: 𝑇r𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠: 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝑁𝑜𝑡ℎ𝑖𝑛𝑔 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑇ℎ𝑒 𝑐𝑜𝑙𝑜𝑚𝑖𝑡 𝑜𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝑡ℎ𝑖𝑠 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑖𝑠: 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑦 𝐶 ∶ 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝑂𝑏𝑗𝑒𝑐𝑡𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦∶ 𝑟𝑇𝑆 𝐷𝑜𝑚𝑎𝑖𝑛 𝑇: 𝑇ℎ𝑖𝑛𝑔 𝑅𝑎𝑛𝑔𝑒∶ 𝑆: 𝑇ℎ𝑖𝑛𝑔 𝐶𝑙𝑎𝑠𝑠 𝑆 ∶ 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑇𝑜∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝐹𝑜𝑟𝑚𝑎𝑡 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝐶𝑙𝑎𝑠𝑠 ∶ 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛 𝑠𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂𝑓 𝑇

∶ 𝐹𝑜𝑟𝑚𝑎𝑡 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝐶𝑙𝑎𝑠𝑠𝑒𝑠: 𝑆: 𝐸𝑛𝑓𝑎𝑛𝑡 𝑎𝑛𝑑 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝑇𝑟𝑎𝑣𝑎𝑖𝑙𝑙𝑒𝑢𝑟 𝑁𝑜𝑡ℎ𝑖𝑛𝑔 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙: 𝑆: 𝐴𝑚𝑖𝑛𝑒 𝑇𝑦𝑝𝑒𝑠∶ 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑟𝑇𝑆 𝑠𝑜𝑚𝑒 𝑇: 𝑀𝑎𝑠𝑐𝑢𝑙𝑖𝑛, 𝑆: 𝐵𝑖𝑒𝑛_𝐸𝑡𝑟𝑒 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 𝛼∗ and 𝛽∗ 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

�̃� �̃�

�̃�′ B �̃�′

𝑡1 𝜎1 𝜎2 𝑡2

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