ANALYSIS OF MAPPING WITHIN S-MODULE FRAMEWORK

Krzysztof Goczyla, Aleksander Waloszek, Wojciech Waloszek and Teresa Zawadzka

Dept. of Software Engineering, Gdansk University of Technology, Narutowicza 11/12, Gdansk, Poland

Keywords: Ontology, Ontology modularization, Description logics, Ontology mapping, Ontology importing.

Abstract: In this paper we present the results of our work on s-module (semantic modules) framework. The

framework, introduced recently, consists of a high-level semantic description of a modular knowledge base

accompanied by an algebra for manipulating module contents. The main contribution of the article is the

presentation of the process of expressing Distributed Description Logics knowledge base within the

s-module framework. As the two methods exhibit two different approaches to modularization, analysis of

this procedure is helpful in capturing the specifics of DDL, comparing it to other methods, and discussing

the completeness of the s-module framework.

1 INTRODUCTION

Recently significant amount of effort has been put in

the area of on ontology modularization. Ontologies

gain importance in Computer Science, and use of

modularization techniques broadens the possibilities

of their efficient development and deployment.

In this paper we continue our work from

(Goczyla et al., 2009a) on analyzing spaces of

semantic modules (s-modules). In (Goczyla et al.,

2009a) we described a procedure for constructing a

space of possibly useful modules. Construction of

the space is algebraic: we specify a set of base

modules and a set of operators. Therefore,

assimilation of knowledge by one module from

another can be depicted as a “shift” in this space and

described as a sequence of algebraic operations.

The s-module space was introduced as a

common framework for describing properties and

characteristics of a modular knowledge base or a

specific modularization approach. In this paper we

present a procedure of expressing Distributed

Description Logics (DDL; Borgida and Serafini,

2003) knowledge base in this space. The conclusions

are rather encouraging: such a translation is possible

with a minimal number of additional assumptions

and with choose of very natural base modules. The

description is a source of interesting observations

about DDL and s-module space, moreover, it

provides an alternative way of proving soundness

and completeness of the method.

2 PRELIMINARIES

Due to space limitations we cannot present the full

introduction to ontologies formulated in Description

Logic (DL) ALC. Here we only review basic terms

to establish the notation used henceforth.

In all DLs we assume that we have three sets of

names: constants (individual names), concepts

(unary predicates), and roles (binary predicates).

The full signature

contains all the valid names.

Other signatures S are subsets of

.

The names are interpreted, and each interpre-

tation I = (Δ

I

, ·

I

) consists of a non-empty domain

Δ

I

and an interpretation function ·

I

which assigns

each constant an element of Δ

I

, each concept a

subset of Δ

I

, and each role a subset of Δ

I

ä Δ

I

. We

assume that every base interpretation I of every in

fact interprets all the valid names from

.

Projection I|S of a base interpretation to some

selected signature S produces a set of interpretations

with the same domain as I and interpreting all the

names from S in the same way:

I|S = {J: Δ

J

= Δ

I

∀X ∈ S: X

J

= X

I

} (1)

An ontology O is simply a set of sentences. A

signature of O, denoted Sig(O), is the set of all

names used in any sentence of O. An interpretation

I satisfies O (is its model; denoted I O) iff it

satisfies all the sentences in O. Naturally if I O

then every J ∈ I|Sig(O) also satisfies O.

267

Goczyla K., Waloszek A., Waloszek W. and Zawadzka T..

ANALYSIS OF MAPPING WITHIN S-MODULE FRAMEWORK.

DOI: 10.5220/0003665402670272

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2011), pages 267-272

ISBN: 978-989-8425-80-5

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

3 S-MODULE SPACE

In this section we describe s-module approach

introduced in (Goczyla et al., 2009a). The semantic

modules are defined in a way which disregards the

exact form of a language (like DL) and focuses only

on interpretations. Each semantic module is in fact a

set (more precisely, a class) of base interpretations.

Each semantic module also has a finite signature S

which expresses the range of names about which we

want to reason using the module:

Definition 1. A s-module M = (S, W) is a pair of a

signature S and a class W of base interpretations,

such that W|S = W. Each interpretation from W is

called a model of M.

Henceforth we use S(M) and W(M) to describe

the two parts of a s-module M.

For any ontology O we might construct a module

M(O) such that W(M(O)) = {I: I O} and

S(M(O)) = Sig(O). However, while M(O) holds all

the possible (base) models of the ontology O, it

“forgets” the exact form of sentences; e.g.

M({A B}) = M({A

≡ A B}).

(Goczyla et al., 2009a) define a number of ope-

rations for s-modules (M, L denote arbitrary s-mo-

dules, S any signature, γ a function

ö

):

M L = (S(M) S(L), W(M) W(L)) (2)

M L = (S(M) S(L), W(M) W(L))

(3)

M ‒ L = (S(M) S(L), W(M) ‒ W(L))

(4)

ρ

γ

(M) = (γ(S(M)), γ(W(M)))

(5)

π

S

(M) = (S, W(M)|S)

(6)

The operations , ρ, π form the backbone of the s-

module algebra. The intersection () of s-modules

representing ontologies corresponds to adding all

sentences (importing) from one ontology to another:

M(O

1

) M(O

2

) = M(O

1

O

2

).

Simple importing is possible only if there is no

name conflict between two ontologies. In the

presence of name conflicts we can use the rename

operator (ρ). Rename operation (ρ) uses the notion

of a signature mapping, which is a function γ:

ö

. We also (like in (5)) apply γ to an

interpretation in which case γ(I) = J such that

Δ

I

= Δ

J

and ∀X ∈

: γ(X)

J

= X

I

.

Sometimes we might not want to import all the

names from an ontology. To restrict a set of names,

but to preserve the relations between extensions of

the remaining names, we use projection operator (π).

Example 1 Let O

1

= {Teacher Employee, Publica-

tion Achievement}, and assume that O

2

is much

larger and contains axioms like: Book Publication,

Chapter Publication, Monograph Book etc. We

like to reuse O

2

in O

1

, however, in O

2

the notion of

Publication does not include position papers.

Moreover, we find the term Book from O

2

too gene-

ral and do not want to include it the ontology. To

meet our goals we take O

1

′ = O

1

{NPPaper ≡

Publication ŸPositionPaper} and construct M:

M = M(O

1

′) π

S ‒ Book

(ρ

Publication Ø NPPaper

(M(O

2

))).

In Ex. 1 we introduce some intuitive shortcuts to

notation that we also exploit further in the paper. For

example by ρ

A Ø B

we mean that corresponding γ

function changes only the name A to B and by

π

S ‒ X

(M) we mean π

S(M) ‒ {X}

(M). Occasionally for de-

noting names we might use wildcards: e.g. *

Ø 1:*.

While , , ‒ are taken directly from Boolean

algebra of sets,, π and ρ are equivalents of cylindrifi-

cation and substitution from the cylindric algebra

(Henkin, Monk and Tarski, 1971). Any space of

module (class of modules M closed under π, ρ, , ,

‒) can be used to construct a cylindric algebra.

A space of modules can be constructed by

choosing a base space and closing it wrt. π, ρ, , ,

‒. A very natural choice of a base space is {M(α)}

where α is a sentence valid in a selected language L.

Several auxiliary operators can be introduced for

such a space. The selection operator σ is a shortcut

for σ

α

(M) = M M(α). The further two operators

“put under” (υ) and restriction (ξ) are defined below

(I S denotes an interpretation J: Δ

J

= Δ

I

S and

∀X ∈

: X

J

= X

I

S):

L υ

C

M = (S(L) S(M), {I ∈ W(M):

I C

I

∈ W(L)})

(7)

ξ

C

(M) = (S(M), {I C

I

: I ∈ W(M)

I C

I

∈ W(M)})

(8)

“Put under” (Goczyla et al., 2009a) correlates the

domains of two modules by introducing relation-

ships between extensions of terms in L only to a

fragment of M. The restriction operator ξ is an

operator complementing υ. Namely it restricts the

domain of the module to the extension given concept

C. For ALC it may be simply treated as a shortcut

for ξ

C

(M) = σ

C ≡

(M).

Theorem 1. For every module M from M(ALC)

obtained from the basic space {M(α)} with use of

operators (π, ρ, , , σ, υ, ξ) it is decidable whether

the module is satisfiable (i.e. W(M) ≠ ∅).

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

268

4 DISTRIBUTED DL

Distributed Description Logics (DDL) is one of the

most prominent modularization methods for DL

ontologies. Originally proposed by Borgida and

Serafini in (2003), it was extended and adapted in

many works. The presentation in this Section is

mainly based on (Homola and Serafini, 2010).

DDLs focus on mapping the terms from a source

module to a target module. We assume there exists a

collection of modules {O

i

}

i ∈ I

, indexed by a set I.

Each module is simply an ontology (it has its local

collection of sentences). Between each pair of

modules O

i

(as a source) and O

j

(as a target; here

and hence in this section i, j ∈ I, i ≠ j) there is

defined a (possibly empty) set of bridge rules μ

ij

.

There are three types of bridge rules (C, D are

concepts and a, b constants, resp. from O

1

and O

2

):

A distributed knowledge base (DKB) ä = ({O

i

},

{μ

ij

}), consists of modules and sets of bridge rules.

Whenever μ

ij

is non-empty, we say that O

j

uses O

i

.

A distributed interpretation à is a pair ({I

i

},

{r

ij

}), where {I

i

} are interpretations (called local in-

terpretation), and {r

ij

} are domain relations between

the domains of I

i

and I

j

. In contrast to standard DL,

each local interpretation might also be a hole, a

special interpretation I

e

with empty domain. A dis-

tributed interpretation à is a model of ä iff for each

i, j ∈ I, we have I

i

O

i

and à μ

ij

. à μ

ij

iff it satis-

fies all the rules μ

ij

according to the following:

DDL exhibits a different behavior than s-modules.

While the latter focuses on importing, DDL focuses

on mapping between terms (this distinction is based

on (Homola and Serafini, 2010)).

While in the basic DDL relation r

ij

might be of

any form, one might consider also more constrained

versions of DDL, denoted by additional symbols:

e.g. F for only functional r

ij

or I for injective r

ij

, e.g.

DDL(F) or DDL(F, I). The following well-known

“penguin” example illustrates the importance of the

relation.

Example 2 (Grau et al., 2004) Let us consider the

ontology O

1

= {Nonflying ≡ ŸFlying, Bird Flying}

and the ontology O

2

= {Penguin }. We define

the mapping μ

12

in the following way:

It might seems that Penguin is unsatisfiable “being

subsumed” by both Nonflying and Bird. But we can

still obtain a non-empty interpretation I

2

in a model

of ä, if the relation r

12

maps at least two individuals

(one Nonflying and one Bird) to a single Penguin.

Originally intended for illustrating cumbersome

behavior of DDL, in fact this example shows its

distinctive capability: to combine knowledge about

several individuals into one. In situations when such

behavior is undesirable we can turn to DDL(F).

5 DDL IN S-MODULE SPACE

In this section we present the results of our work on

expressing DDL in the s-module framework.

Starting from a bit simplified conversion for

DDL(F, I), we gradually move to less constrained

versions of DDL.

5.1 DDL(F, I) with No Cycles

At first we consider a case of DDL(F, I) in which

each individual from Δ

I

i

corresponds to at most one

individual from Δ

I

j

and vice versa.

As it turns out, such assumption significantly

reduces the difficulties of bridging DDL and s-mo-

dules. As a case-study let us consider a distributed

KB ä with two simple DDL-modules O

1

= {C ,

D } and O

2

= {E F} and a set of rules μ

21

=

{2:F 1:C, 2:E 1:D}. Such a mapping implies

that for every model à, I

1

D C.

Despite apparent simplicity of the example,

while analyzing semantics we still have to consider

several possibilities: a domain relation r

21

might map

the whole domain of I

2

to Δ

I

1

, or only a fragment of

a domain of I

2

, or r

21

might even be empty,

resulting in empty interpretation for D. The second

case is depicted in Fig. 1a with use of Venn

diagrams: we can mentally visualize that with

shrinking of r

21

(E

I

2

) the area of D

I

1

is also reduced.

To reflect this effect for s-modules, we have to

simulate the behavior of r

21

. The constructions of a

s-module M

1

representing possible models of O

1

in

ä proceeds as follows. First, we create a s-module

with two special concepts: O

1

and O

2

. Second, we

put under these concepts modules M(O

1

) and M(O

2

)

respectively (if they contain repeating names, we

have to add prefixes). Subsequently, we enforce the

bridge rules by using selection. Then, we project the

signature only to the terms from M(O

1

). Finally, we

1: Bird

2:Penguin

1: Nonflyin

g

2:Penguin

i:

C

j: D (into

b

ridge rule)

i:

C

j: D (onto

b

ridge rule)

i: a ö

j: b (individual correspondence)

à i:

C

j: D iff

r

i

j

(

C

I

i

) Œ D

I

j

à i:

C

j

: D iff

r

i

j

(

C

I

i

)

û D

I

j

à i: a ö

j: b iff

r

i

j

(a

I

i

) d

I

j

ANALYSIS OF MAPPING WITHIN S-MODULE FRAMEWORK

269

restrict the module to the concept O

1

and remove the

concept from the signature.

Figure 1: Different but equivalent effects of DDL(F)

mapping (a) and combining s-modules (b).

The result of the first three steps is depicted in

Fig. 1b. We can see that the outcome gives a similar

effect as in the case of DDL. The area r

21

(Δ

I

2

) is

represented by the intersection O

1

O

2

. Since the

interpretation of the intersection may vary in size,

and may even be empty, all the possible forms of

r

21

(Δ

I

2

) are reflected by models of M. The fact that

r

21

is injective and functional is advantageous here:

each instance of O

1

O

2

represents one element e of

Δ

I

1

and simultaneously one element r

21

‒

(e) of Δ

I

2

.

The procedure sketched above can be general-

ized and formalized as follows:

Definition 2. For a given DKB ä = ({O

i

}, {μ

ij

}),

i, j ∈ I, i ≠ j, a converting function c is a function

that assigns each O

i

a s-module.

Definition 3. A bridge-rule operation β

b

for a bridge

rule b and a module M is:

for b = :

β

b

(M) = σ

α

(M),

α = γ

* Ø i:*

(C) γ

* Ø j:*

(D)

for b = :

β

b

(M) = σ

α

(M),

α = γ

* Ø j:*

(D) γ

* Ø i:*

(C)

for b = i: a ö j: b :

β

b

(M) = γ

i:a Ø j:b

(M)

where by γ(C) we understand a new concept with all

the names substituted with use of γ.

A bridge-rule operation β

μ

for a set of bridge rules

μ and a module M is a composition of β

b

, b ∈ μ.

Definition 4. For given two modules O

i

and O

j

from

a DKB ä, such that O

j

uses O

i

, and a converting fun-

ction c, a s-module integrating O

i

and O

j

wrt. c is:

M

ij

c

= β

μ

ij

(

ρ

* Ø i:*

(c(O

i

)) υ

Ο

i

M({O

i

})

ρ

* Ø j:*

(M(O

j

)) υ

Ο

j

M({O

j

}) )

The construction of the integrating s-module

corresponds to executing the three first steps of the

described procedure (see also Fig. 1b).

Definition 5. For a given DKB ä = ({O

i

}, {μ

ij

}), a

module O

j

, and a converting function c, an integra-

ted s-module for O

j

wrt. c is M

j

c

= …

i ∈ {i: μ

ij

≠ ∅}

M

ij

c

.

A fully integrated module for O

j

wrt. c is FM

j

c

=

ρ

j:* Ø *

(π

S ‒ O

j

(

ξ

Ο

j

(M

j

c

)))

.

The notion of integrated module generalize the

described procedure to the case when more modules

are used. Full integration corresponds to the last two

steps of the procedure. A fully integrated s-module

is indeed useful for describing DDL semantics, as

the following lemma shows.

Lemma 1. For a DKB ä = ({O

i

}, {μ

ij

}), i, j ∈ I,

i ≠ j, expressed in ALC and DDL(F, I), in which O

1

uses all the other modules, and all the other modules

use none, and a converting function c(O

i

) = M(O

i

)

for all i ≠ 1, c(O

1

) = FM

1

c

, a module O

i

is satisfiable

(i.e. for some model it has a local interpretation

which is not a hole) iff c(O

i

) ≠ M({ ^}).

The proof, omitted for brevity, consists of showing

that a model à with non-empty I

1

exists iff there

exists a model of M

1

c

.

The result from Lemma 8 can be generalized to

any acyclic DKB (i.e. DKB ä for which the relation

U

ä

= {(O

i

, O

j

): μ

ij

≠ ∅} forms a forest).

Proposition 1. For any acyclic DKB ä = ({O

i

},

{μ

ij

}), i, j ∈ I, i ≠ j, expressed in ALC and

DDL(F, I), and a converting function c(O

i

) defined

recursively as c(O

i

) = M(O

i

) for leaves, and c(O

i

) =

FM

i

c

for other modules, a module O

i

is satisfiable iff

c(O

i

) ≠ M({ ^}).

With use of Lemma 1 the proof is straightforward,

by induction on each tree of using relation (Lemma

8 forms the induction base, and gives means for

proving the induction hypothesis).

5.2 DDL(F, N

n

) with No Cycles

Here we extend the results from the previous

subsection towards slightly more expressive DDL,

by adapting the introduced notion to the case when

the domain relations are not necessarily injective.

Once again we start with a motivation example.

We adapt the “penguin” example (see Ex. 3). DKB

consists of two modules O

1

= {P }, and O

2

=

{NF ŸF, B NF}, and one non-empty bridge rule

set is: μ

21

= {2:NF 1:P, 2:B 1:P}.

As already mentioned above, the concept P in O

1

may be satisfiable, though for this to happen r

21

has

to map two individuals of Δ

I

2

to a single individual

C

C

O

1

O

1

M

O

2

O

2

D

D

E

E

F

F

a)

b)

i:

C

j: D

i:

C

j: D

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

270

of Δ

I

1

. This situation is depicted in Fig. 2a.

Figure 2: Different but (almost) equivalent effects of

DDL(F, N

n

) mapping (a) and combining s-modules (b).

The strategy from the previous section is not

enough to model this situation in the realm of

s-modules. Although we can overlap the domains of

the two modules, simple overlapping (like in Fig.

1b) would render the concept P unsatisfiable. We

have to somehow model the possibility of mapping

two individuals into one.

A solution to this issue is illustrated in Fig. 2b.

The main idea is to apply the conceptual decom-

position twice to the same domain. It can be done

with prefixes (omitted in Fig. 2 for readability),

appropriate s-module operation might look like

M

2.2

= ρ

* Ø 2.2.1:*

(M(O

2

)) ρ

* Ø 2.2.2:*

(M(O

2

). After the

transformation, every element of the domain of (any

model of) M

2.2

represents in fact a pair of elements

of the domain of (some model of) M(O

2

).

There are, however, two issues connected with

this approach. First of all, the constructed module

represent pairs of the original domains. The same

approach can be used to triples, quadruples etc., but

there have to be some known and finite limit to the

cardinality of the tuples. This is the motivation

behind introducing a new constraint for DDL,

namely N

n

, n ∈ N, which implies that every domain

relation r

ij

is at most n-to-one. The discussion in this

section is thus constrained to DDL(F, N

n

).

Second issue is that elements of the domain of

M

2.2

represent in fact some pairs of elements of the

original domain, like (e, e), that we do not want to

include in our considerations. This problem can be

technically overcome (by exploiting disjoint union

satisfiability property of ALC introduced by

Serafini et al. in (2005)), but the discrepancy

between “double overlapping” and pair of domains

still exists, and should be dealt with in future

development of s-module framework (see Sec. 6).

In the following we adapt the notions from the

previous section to the case of DDL(F, N

n

).

Definition 6. A n-bridge-rule operation β

b

n

for a

bridge rule b, a module M and given n is:

for b =:

β

b

(M) = σ

α

(M),

where α is defined below:

α = +

k ∈ [1..n]

+

l ∈ [1..k]

γ

* Ø i.k.l:*

(C) γ

* Ø j:*

(D)

for b = :

β

b

(M) = σ

α

(M),

where α is defined below:

α = γ

* Ø j:*

(D) +

k ∈ [1..n]

+

l ∈ [1..k]

γ

* Ø i.k.l:*

(C)

for b = i: a ö j: b:

β

b

(M) = »

k ∈ 1..n

γ

i.k.1:* Ø j:b

(M)

A n-bridge-rule operation β

μ

n

for a set of bridge

rules μ, a module M and given n is a composition of

β

b

n

for every b ∈ μ.

Definition 7. For given two modules O

i

and O

j

from

a DKB ä, such that O

j

uses O

i

, and a converting

function c, let M be defined as follows:

M = M({O

i.k

O

i.l

^: k, l ∈ [1..n], k ≠ l})

a s-module n-integrating O

i

and O

j

wrt. c is:

n

M

ij

c

= β

μ

ij

n

( ρ

* Ø j:*

(M(O

j

)) υ

Ο

j

M({O

j

})

…

k, l ∈ [1..n], k ≠ l

(

ρ

* Ø i.k.l:*

(c(O

i

)) υ

Ο

i.k.l

M ) )

The pairwisely disjoint concepts O

j.k

represent

k-tuples of elements of the original domain.

Definition 8. For a given DKB ä = ({O

i

}, {μ

ij

}), a

module O

j

, a converting function c, and a number n

an n-integrated s-module for O

j

wrt. c is

n

M

j

c

=

…

i ∈ {i: μ

ij

≠ ∅}

n

M

ij

c

. A fully n-integrated module for O

j

wrt. c is F

n

M

j

c

= ρ

j:* Ø *

(π

S ‒ O

j

(

ξ

Ο

j

(

n

M

j

c

)))

.

Once again we show that fully integrated modules

are equisatisfiable with corresponding modules from

DKB.

Lemma 2. For a DKB ä = ({O

i

}, {μ

ij

}), i, j ∈ I,

i ≠ j, expressed in ALC and DDL(F, N

n

), in which

O

1

uses all the other modules, and all the other

modules use none, and a converting function

c(O

i

) = M(O

i

) for all i ≠ 1, c(O

1

) = F

n

M

1

c

, a module

O

i

is satisfiable iff c(O

i

) ≠ M({ ^}).

The proof, which we omit for brevity, shows that

a model à with non-empty I

1

exists iff there exists a

model of

n

M

1

c

. Again, we can generalize the results

of the lemma to a case of any acyclic DKB.

Proposition 2. For any acyclic DKB ä = ({O

i

},

{μ

ij

}), i, j ∈ I, i ≠ j, expressed in ALC and

DDL(F, N

n

), and a converting function c(O

i

) defined

recursively as c(O

i

) = M(O

i

) for leaves, and c(O

i

) =

F

n

M

i

c

for other modules, a module O

i

is satisfiable

iff c(O

i

) ≠ M({ ^}).

Proof (sketch): Analogously like in proof for Prop.

1, but with use of Lemma 13.

P

O

1

M

O

2

O

2

O

1

B

B

P

NF

NF

NF

F

B

F

F

a)

b)

f

e

p

i:

C

j

: D

i:

C

j: D

ANALYSIS OF MAPPING WITHIN S-MODULE FRAMEWORK

271

5.3 Decidability

The discussion from the previous points gives us

also means for creating a procedure for deciding

satisfiability of modules in a DKB.

The decidability result from Th. 1 combined with

Prop. 2 allows for immediate stating that DDL(F,

N

n

) is decidable for acyclic DKBs. However, we can

extend this result a bit by including the DKBs which

can contain cycles.

A basic idea behind such extention is simple: we

proceed iteratively with determining c(O

i

) for each

module, assuming that in first iteration c

1

(O

i

) =

M(O

i

) and then, in the next k-th iteration taking

c

k

(O

i

) = F

n

M

i

c

k ‒ 1

(O

i

). As Serafini and Tamilin show

in (2007), the fixpoint will finally be reached, which

can be detected by adapted procedure for checking

whether an ontology is a conservative extension of

another (Lutz, Walther and Wolter, show in (2007)

that this problem for ALC is decidable).

This leads us to the following conclusion:

Proposition 3. For a given DKB ä = ({O

i

}, {μ

ij

}),

i, j ∈ I, i ≠ j, an recursive procedure for converting

modules in the following way: c

1

(O

i

) = M(O

i

), c

k

(O

i

)

= F

n

M

i

c

k ‒ 1

(O

i

), repeated until c

k

(O

i

) is a conservative

extension of c

k ‒ 1

(O

i

) for all i ∈ I, is a terminating,

sound and complete procedure for deciding satisfia-

bility of modules for ALC and DDL(F, N

n

).

6 CONCLUSIONS

In this section we summarize the main observations

and contributions of the paper and relate them to

other studies.

From the point of view of DDL, the results

allows us to show some insight in the relation

between mapping and importing (Homola and

Serafini, 2010). Here we show how different kinds

of mappings relate to specific kinds of importing

(especially “putting under”). Further work will allow

us to include also E-Connection (Kutz, Lutz,

Wolter, and Zakharyaschev, 2004) and P-DL (Bao,

Voutsadakis, Slutzki, and Honavar, 2009), two other

major methods of modularization.

The other result is an alternative way of proving

decidability of DDL(F, N

n

) for ALC. Though at the

current stage of research it does not extend the

results already available in literature, it shows the

practical application of the results from Th. 1. The

further development might result in a set of

techniques for proving decidability for a wide range

of modularization methods.

From the perspective of s-module framework the

presented discussion provides interesting hints about

its further development. The s-module framework

cannot easily handle situations in which we want to

refer to a tuple of elements of a domain. Sec. 5.3

suggests it may be useful to extend the framework

by some kind of treatment for limits (i.e. the ability

to determine bounds for an arbitrary set of modules).

Finally, the paper presents some extensions to

the framework of s-modules: definition of s-module

space, restriction operator, and a slightly extended

result for decidability (cf. Sec. 3).

ACKNOWLEDGEMENTS

This work is partially supported by the Polish

National Centre for Research and Development

under Grant No. SP/I/1/77065/10 by the strategic

scientific research and experimental development

program: „Interdisciplinary System for Interactive

Scientific and Scientific-Technical Information”.

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