A Polynomial Algorithm for Merging Lightweight Ontologies in
Possibility Theory Under Incommensurability Assumption
Salem Benferhat
1
, Zied Bouraoui
2
, Ma Thi Chau
3
, Sylvain Lagrue
1
and Julien Rossit
4
1
Artois University, CRIL - CNRS UMR 8188, Lens, France
2
Cardiff University, Cardiff, U.K.
3
HMI Laboratory, VNU - University of Engineering and Technology, Ha Noi, Vietnam
4
Paris Descartes University, LIPADE, Paris, France
Keywords:
Lightweight Ontologies, Possibility Theory, Belief Merging, Incommensurability.
Abstract:
The context of this paper is the one of merging lightweight ontologies with prioritized or uncertain assertional
bases issued from different sources. This is especially required when the assertions are provided by multiple
and often conflicting sources having different reliability levels. We focus on the so-called egalitarian merging
problem which aims to minimize the dissatisfaction degree of each individual source. The question addressed
in this paper is how to merge prioritized assertional bases, in a possibility theory framework, when the uncer-
tainty scales are not commensurable, namely when the sources do not share the same meaning of uncertainty
scales. Using the notion of compatible scale, we provide a safe way to perform merging. The main result of
the paper is that the egalitarian merging of prioritized assertional bases can be achieved in a polynomial time
even if the uncertainty scales are not commensurable.
1 INTRODUCTION
In many applications, information pieces are provided
by several and potentially conflicting sources where
gathering them leads to inconsistent information. In-
formation merging aims to define combination opera-
tions that take as input information provided by differ-
ent sources and produce a consistent and unified point
of view that syntheses the best of the sources. Knowl-
edge bases merging or belief merging (e.g. (Bloch
et al., 2001; Konieczny and Pino P
´
erez, 2002)), is a
problem largely studied within the propositional logic
setting. Several merging approaches have been pro-
posed which depend on the nature and the represen-
tation of knowledge such as merging propositional
knowledge bases and prioritized or weighted logical
knowledge bases.
In this paper we place ourselves in the context of
Ontology-Based Data Integration (Wache et al., 2001)
and we investigate merging of uncertain assertional
bases provided by different sources. In such a set-
ting the ontology is assumed to be coherent and fully
reliable. However the data, although refer to the same
coherent ontology, are often provided by conflicting
sources generally affected with uncertainty due for in-
stance to the reliability of sources. Uncertainty here
is represented in the framework of possibility theory.
This theory is particularly appropriate when one uses
a finite ordinal scale {0, α
1
,...,α
n
,1} to asses the cer-
tainty degrees associated with each assertion. As on-
tology language, we use the well-known lightweight
description logics DL-Lite. DL-Lite is recognized
as powerful logical-based frameworks for Ontology-
Based Data Access. In such a setting, we use a knowl-
edge base formed of a terminological base, called
TBox, and an assertional base, called ABox. The
TBox contains ontological (or generic) knowledge of
the application domain whereas the ABox stores data
(or individuals or constants) that instantiate generic
knowledge.
When priorities or uncertainty degrees are at-
tached with assertional facts, there exist two main ap-
proaches (Konieczny and Pino P
´
erez, 2002) to merge
or aggregate uncertain information: utilitariant (ma-
jority) approaches and egalitarian (or egalitarest) ap-
proaches. Within possibilistic DL-Lite, an egalitarian
merging operator was proposed in (Benferhat et al.,
2013) to merge DLs knowledge bases when uncer-
tain pieces of information are represented in the pos-
sibility theory framework. However, the presented
work in (Benferhat et al., 2013) is based on the as-
sumption that the scales used to represent uncertainty
in the merged knowledge bases are commensurable,
namely the sources share the same meaning of un-
certainty scales. In some applications and especially
in the Web applications ones, the commensurability
Benferhat S., Bouraoui Z., Thi Chau M., Lagrue S. and Rossit J.
A Polynomial Algorithm for Merging Lightweight Ontologies in Possibility Theory Under Incommensurability Assumption.
DOI: 10.5220/0006120804150422
In Proceedings of the 9th International Conference on Agents and Artificial Intelligence (ICAART 2017), pages 415-422
ISBN: 978-989-758-220-2
Copyright
c
 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
415

assumption may appear to be strong and needs to be
dropped. In the propositional setting (Benferhat et al.,
2007) a merging under incommensurability assump-
tion has been proposed. However, this approach is
very computationally hard (at least in ∆
2
p
).
In this paper, we propose an efficient egalitarian-
based merging of uncertain assertional bases under
the incommensurability assumption. We assume that
generic knowledge (the TBox) is fully coherent and
fully certain and seen as a constraint to be satisfied
during the merging process. Namely, the TBox should
be present in the merging result (i.e. the resulting DL
knowledge base), while assertional facts can be either
accepted, ignored or weakened during the merging
process. The assertional facts (ABox), issued from
different sources, may be affected with uncertainty.
Sources are not assumed to share the same meaning
of uncertainty scales. To tackle the incommensurabil-
ity problem, we use the concept of compatible scales.
A compatible scale is a re-assignment of certainty de-
grees to assertional facts such that the initial plausi-
bility ordering inside each ABox is preserved. Using
the notion of compatible scale, we provide a safe way
to perform merging. The nice result of this paper is
that merging uncertain assertional bases, without the
commensurability assumption, can be achieved in a
polynomial time. The last part of the paper proposes a
way to deal with incommensurability assumptions by
normalizing uncertainty scales.
Before presenting our results, let us give a brief
refresher on possibilistic lightweight ontologies and
on merging under commensurability assumption.
2 PRIORITIZED LIGHTWEIGHT
ONTOLOGIES
In this section we briefly introduce the logical-based
formalism used to represent prioritized ontologies.
2.1 Lightweight Description Logics:
DL-Lite
One of the well-known description logics for query-
ing data is DL-Lite (Calvanese et al., 2007). This is
due to the so-called first-order rewritability property
that separates the TBox and the ABox when reason-
ing. Such property guarantees a very low computa-
tional complexity for query answering. This makes
DL-Lite well-suitable for applications that use large
volume of data. The following briefly reviews the core
fragment of all the DL-Lite family, lightweight ontolo-
gies, called DL-Lite
core
. However, results of this paper
are valid for DL-Lite
R
and DL-Lite
F
, two important
fragments of the DL-Lite family.
2.1.1 Syntax and Semantics
A standard DL-Lite knowledge base K
s
=
h
T
s
,A
s
i
is
composed of a set of atomic concepts (i.e. unary predi-
cates), a set of atomic roles (i.e. binary predicates) and
a set of individuals (i.e. constants). Complex concepts
and roles are built as follows:
B −→ A|∃R R −→ P|P
−
C −→ B|¬B
where A (resp. P) is an atomic concept (resp. role). B
(resp. C) is called basic (resp. complex) concept and
role R is called basic role. The TBox T
s
includes a
finite set of inclusion assertions of the form B v C
where B and C are concepts. The ABox A
s
contains
a finite set of assertions on atomic concepts and roles
of the form A(a) and P(a,b) where a and b are two
individuals.
The semantics of a DL-Lite knowledge base is
given in term of first order logic interpretations. An
interpretation I = (∆
I
,.
I
) consists of a non-empty do-
main ∆
I
and an interpretation function .
I
that maps
each individual a to a
I
∈ ∆
I
, each A to A
I
⊆ ∆
I
and
each role P to P
I
⊆ ∆
I
× ∆
I
. Furthermore, the inter-
pretation function .
I
is extended in a straightforward
way for complex concepts and roles: (¬B)
I
= ∆
I
\B
I
,
(P
−
)
I
= {(y, x)|(x,y) ∈ P
I
} and (∃R)
I
= {x|∃y s.t.
(x,y) ∈ R
I
}. An interpretation I is said to be a model
of a concept inclusion axiom, denoted by I |= B v C,
iff B
I
⊆ C
I
. Similarly, we say that I satisfies a
concept (resp. role) assertion, denoted by I |= A(a)
(resp. I |= P(a,b)), iff a
I
∈A
I
(resp. (a
I
,b
I
) ∈ P
I
).
An interpretation I is said to be a model of K =
h
T,A
i
,
denoted by I |= K
s
, iff I |= T
s
and I |= A
s
where
I |= T
s
(resp. I |= A
s
) means that I is a model of all
axioms in T
s
(resp. A
s
). A knowledge base K
s
is said
to be consistent if it admits at least one model, other-
wise K
s
is said to be inconsistent. A DL-Lite TBox T
is said to be incoherent if there exists at least a concept
C such that for each interpretation I which is a model
of T , we have C
I
=
/
0.
2.1.2 Query Answering
A query is a first-order logic formula, denoted
q={~x |φ(~x)}, where ~x=(x
1
,...,x
n
) are free variables, n
is the arity of q and atoms of φ(~x) are of the form A(t
i
)
or P(t
i
,t
j
) with A∈N
C
and P∈N
R
and t
i
, t
j
are terms,
i.e. constants of N
I
or variables. When φ(~x) is of the
form ∃~y.con j(~x,~y) where ~y are bound variables called
existentially quantified variables, and con j(~x,~y) is a
conjunction of atoms of the form A(t
i
) or P(t
i
,t
j
) with
A∈N
C
and P∈N
R
and t
i
, t
j
are terms, then q is said to
be a conjunctive query (CQ). When n=0, then q is said
to be a boolean query (BQ). A BQ with no bound vari-
ables is said to be a ground query (GQ). Lastly, when
q contains only one atom with no free variables, then
it is said to be an instance query (IQ) (i.e. instance
ICAART 2017 - 9th International Conference on Agents and Artificial Intelligence
416

checking). For a BQ q, we have I|=q iff (φ)
I
=true
and K|=q iff ∀I:I|=K,I|=q. For a CQ q with free vari-
ables ~x=(x
1
,...,x
n
), a tuple of constants ~a=(a
1
,..., a
n
)
is said to be the certain answer for q over K if the BQ
q(~a) obtained by replacing each variable x
i
by a
i
in
q(~x), evaluates to true for every model of K. Hence
CQ answering can be reduced to BQ answering.
2.2 Possibilistic DL-Lite
Let L be the DL-Lite description language described
in the previous section (Section 2.1). A prioritized
(or weighted) DLs knowledge base is made by a set
of DL axioms where each axiom is attached with a
weight that reflects its certainty/priority. In general,
the higher is the weight of an axiom the more the ax-
iom is important. Handling priorities can be conve-
niently and efficiently dealt in the possibility theory
framework (Dubois and Prade, 1988). Recently, an
extension of DL-Lite to the possibility theory frame-
work has been proposed in (Benferhat and Bouraoui,
2015). In this paper we use this framework to encode
available knowledge.
A possibilistic DL-Lite knowledge base
K={(φ,w
φ
) : 1..n}, denoted by DL-Lite
π
, is a
set of weighted axioms of the form (φ, w
φ
) where φ
is either a TBox or an ABox axiom and w
φ
∈ ]0,1] is
the degree of certainty or priority of φ. The weighted
axiom (φ,w
φ
) means that the certainty degree of φ
is at least equal to w
φ
. When ∀(φ
i
,w
φ
i
) ∈ K, we
have w
φ
i
=1 then the classical DL-Lite knowledge
base, as recalled in Section 2.1, is recovered. The
inconsistency degree of a DL-Lite
π
knowledge base
K, denoted by Inc(K), is defined as follow:
Inc(K) = max{w
φ
i
: K
≥w
φ
i
is inconsistent}
Where K
≥α
= {φ
i
: (φ
i
,w
φ
i
) ∈ K and w
φ
i
≥ α} is com-
posed of axioms having a weight greater than α. Be-
sides, Inc(K) = 0 if {φ
i
: (φ
i
,w
φ
i
) ∈ K is consistent}.
The semantics of DL-Lite
π
knowledge bases is
given by the concept of a possibility distribution, de-
noted by π. This latter is a mapping from a set of
DL-Lite interpretations Ω (namely, I = (∆,.
I
) ∈ Ω)
to the unit interval [0,1].
Definition 1. The possibility distribution induced
from a DL-Lite
π
is defined as follows: ∀I ∈ Ω :
π
K
(I ) =

1 i f ∀ (φ
i
,w
φ
i
) ∈ K,I |= φ
i
1 − max{w
φ
i
: (φ
i
,w
φ
i
) ∈ K I 6|= φ
i
} otherwise
Interpretations which have possibility degrees
equal to 1 are the most preferred ones since they are
models of {φ
i
: (φ
i
,w
φ
i
) ∈ K}. For countermodels, an
interpretation I is considered as preferred to an in-
terpretation I
0
, if the highest axiom falsified by I is
less important than the highest axiom falsified by I
0
.
It can be shown that Inc(K) = 1 − max
I ∈Ω
{π
K
(I )}. For
more details on possibilistic DL-Lite; see (Benferhat
and Bouraoui, 2015). Finally, given K a possibilis-
tic DL-Lite knowledge base, a conjunctive query q
is said to be a consequence of K iff q follows from
{φ : (φ,w
φ
) ∈ K,w
φ
> Inc(K)} using standard DL-Lite
reasoner.
Throughout this paper, we assume that the TBox
is coherent and fully certain and only assertional facts
(ABoxes) may be somewhat certain.
3 EGALITARIAN MERGING
UNDER COMMENSURABILITY
ASSUMPTION
This section briefly reviews egalitarian merging of
possibilistic DL-Lite knowledge bases in the case
where uncertainty scales used by the different sources
are commensurable, namely when all sources share
the same meaning of uncertainty scales.
Let A = {A
1
,..., A
n
} be a set of n prioritized
ABoxes issued from n distinct sources, and let T be
a common DL-Lite TBox representing the integrity
constraint (ontology) to be satisfied during the merg-
ing process. We suppose that each ABox is consistent
with T . Let π
1
,..., π
n
be the possibility distributions
provided by the n sources of information, namely a
π
i
denotes the possibility distribution associated with
each K
i
=
h
T,A
i
i
a DL-Lite knowledge base.
Given n commensurable ABoxes, the merging
process aims to compute a new DL-Lite
π
knowledge
base, denoted by ∆
T
(A), where T is the integrity con-
straint and A is an ABox representing the result of
the fusion of these ABoxes. In the literature, differ-
ent methods for merging have been proposed. In this
section, we perform merging of A
1
,...,A
n
with respect
to T using min-based merging operator. This operator
is often seen as an example of egalitarian merging and
is used when distinct sources that provide information
are assumed to be dependent. We first introduce the
notion of profile associated with an interpretation I ,
denoted by ν(I ), and defined by
ν(I ) = hπ
1
(I ),...,π
n
(I )i.
Namely, ν(I ) represents the possibility values of an
interpretation I with respect to each source. From a
semantics point of view, the result of merging is a pos-
sibility distribution π
∆
obtained in two steps:
i the possibility degrees π
i
(I )’s are first combined
with a merging operator (here we use the mini-
mum operator), and
ii the interpretations having highest degrees are con-
sidered as models of the result of merging (i.e. the
resulting DL-Lite knowledge base ∆
T
(A)).
A Polynomial Algorithm for Merging Lightweight Ontologies in Possibility Theory Under Incommensurability Assumption
417

This leads to define a total pre-order relation, denoted
by ≤
min
, between interpretations as follows: an inter-
pretation I is preferred to another interpretation I
0
if
the minimum element of the profile of I is higher than
the minimum element of the profile of I
0
. Formally:
Definition 2 (Definition of ≤
Min
). Let
A = {A
1
,..., A
n
} be a set of ABoxes and T be an on-
tology. Let {π
1
,..., π
n
} be the possibility distributions
associated with {K
1
=
h
T,A
1
i
,..., K
n
=
h
T,A
n
i
}. Let
I and I
0
be two interpretations and ν(I ) and ν(I
0
)
be their associated profiles. Then:
I
0
≤
A
min
I ⇐⇒ Min(ν(I )) > Min(ν(I
0
))
where
Min(ν(I )) = Min{π
i
(I ) : i ∈ {1,...,n}}.
The result of the merging ∆
min
T
(A) is a DL-Lite
π
knowledge base whose models are interpretations
which are models of T and which are maximal with
respect to ≤
Min
. More formally:
Definition 3 (Min-Based Merging Operator). Let A =
{A
1
,..., A
n
} be a set of ABoxes and T be an ontology.
Let {π
1
,..., π
n
} be the possibility distributions associ-
ated with {K
1
=
h
T,A
1
i
,..., K
n
=
h
T,A
n
i
}. The result
of merging is a DL-Lite
π
knowledge base, denoted by
∆
min
T
(A) is such that its models are defined by:
Mod(∆
min
T
(A)) = {I ∈ Mod(T ) : @I
0
∈
Mod(T ),I ≤
A
Min
I
0
}
Example 1. Let T = {A v B, B v ¬C} be a TBox,
where the certainty degree of each axioms is set to
1 (the weights 1 in axioms pf T are omitted for sake
of simplicity). Let us consider the following set of
ABox to be linked to T : A
1
={(A(a),.6), (C(b),.5)},
A
2
={(C(a),.4), (B(b),.8), (A(b), .7)}. Table 1 con-
siders an example of four interpretations. It gives the
possibility degrees associated with
h
T,A
1
,A
2
i
using
Definition 1 and the result of combining these four in-
terpretations with the minimum operator.
Table 1: Example of merging of possibility distributions us-
ing min-based operator.
I .
I
π
A
1
π
A
2
π
∆
(A)
I
1
A={a},B={a},C={b} 1 .2 .2
I
2
A={},B={},C={a,b} .4 .2 .2
I
3
A={a,b},B={a,b},C={} .5 .6 .5
I
4
A={b},B={b},C={a} .4 1 .4
From a syntactic point of view, the min-based
merging operator, denoted by ∆
min
T
(A) is simply the
union of all ABox that are above the inconsistency de-
gree. More formally:
Definition 4. Let A = {A
1
,..., A
n
} be a set of ABoxes
and T be an ontology. Then:
∆
min
T
(A) = {φ
i j
: (φ
i j
,w
φ
i j
) ∈
h
T,A
1
∪ ... ∪ A
n
i
and
w
φ
i j
> Inc(
h
T,A
1
∪ ... ∪ A
n
i
)}
where Inc(
h
T,A
1
∪ ... ∪ A
n
i
) is defined in Section 2.2
Proposition 1. Let A = {A
1
,..., A
n
} be a set of ABoxes
and T be an ontology. Let π
∆
(A) be the possibility
distribution associated. Then ∆
min
T
(A) represents the
result of merging.
The following definition introduces query answer-
ing using min-based merging operator under com-
mensurability assumption.
Definition 5. A query q is said to be a egalitarian con-
sequence relation of hT,A
1
,..., A
n
i iff q follows from
∆
min
T
(A) using standard DL-Lite (see Section 2.1).
Example 2 (continued). At syntactic level, we have
∆
min
T
(A)=hT, {(A(a),.6), (C(b),.5),(C(a),.4),
(B(b),.8), (A(b),.7)}i. We have Inc(∆
min
T
(A))=.5
and ∆
min
T
(K) = T,{(A(a), .6),(B(b),.8),(A(b),.7)}.
4 EGALITARIAN MERGING
UNDER
INCOMMENSURABILITY
ASSUMPTION
The min-based merging operator presented in the pre-
vious section is based on the assumption that all the
sources providing the ABoxes use the same scale to
encode uncertainties between facts. In Example 1,
when dealing with assertions, we assumed that the
weights attached to a fact φ ∈ A
i
can be compared with
the weight associated with ϕ ∈ A
j
with j 6= i. In this
section, we analyse the situation where the sources are
incommensurable, namely the weights used between
ABoxes assertions are not commensurable.
A natural way to tackle the incommensurabil-
ity assumption is to use the notion of ”compatible
scales” on existing scales used by each source. An un-
certainty scale is said to be compatible with all sources
if it preserves the original total pre-orders between as-
sertions inside each ABox.
The concept of compatible scale is very natural
and has been used in different settings. Intuitively,
when one has to deal with an imprecise or an un-
known variables, then using compatible scales con-
sists in considering all possible values of the variables.
This is the case with interval-based probability (Au-
gustin et al., 2003; Wallner, 2007) or possibility distri-
butions (Benferhat et al., 2011), where for each event
instead of specifying a single probability degree, one
specifies an interval. A compatible scale in this case
is a value of the interval. In our framework, assume
that one has two sources A
1
= {(φ,w
φ
),(ϕ, w
ϕ
)} and
A
2
= {(ψ,w
ψ
)} each of them provides some uncer-
tain facts. If the sources are incommensurable (do
ICAART 2017 - 9th International Conference on Agents and Artificial Intelligence
418

not share the same meaning of the scale), then a com-
patible scale, denoted by R , is any re-assignement of
uncertainty degrees of φ, ϕ and ψ such that R (φ) >
R (ϕ) (if w
φ
> w
ϕ
). Namely, the only requirement is
that we preserve the relative plausibility-ordering in-
side each assertional base A
i
.
Definition 6 (Compatible Scales). Let
A = {A
1
,..., A
n
} be a set of ABoxes where
A
i
= {(φ
j
,w
φ
j
)}. Then a compatible uncertainty
scale R is defined by:
R : A
1
∪ ... ∪ A
n
→ ]0,1]
(φ
j
,w
φ
j
) 7→ r
φ
j
An uncertainty scale R is said to be compatible
with A iff: ∀A
i
∈ A, ∀(φ,w
φ
) ∈ A
i
, (ϕ,w
ϕ
) ∈ A
i
, w
φ
≤
w
ϕ
iff r
φ
≤ r
ϕ
.
Example 3 (Example Continued). Let us consider
again the following set of ABox to be linked to T
given in Example 1: A
1
={(A(a),.6), (C(b),.5)}, A
2
=
{(C(a),.4), (B(b),.8), (A(b),.7)}. The following ta-
ble gives three examples of uncertainty scales.
Table 2: Examples of uncertainty scales.
φ
j
w
φ
j
R
1
φ
j
R
2
φ
j
R
3
φ
j
A
1
A(a) .6 .5 .4 .6
C(b) .5 .2 .7 .5
A
2
C(a) .4 .3 .3 .4
B(b) .8 .7 .6 .8
A(b) .7 .4 .2 .7
The scaling R
1
is a compatible one because it pre-
serves the total pre-order inside each ABox. However,
the scaling R
2
is not a compatible one since it in-
verses priorities inside A
1
and A
2
. R
3
is the com-
patible scale where the same uncertainty degrees are
used.
Given a compatible scales R , we denote by A
R
i
the
assertional base obtained from A
i
by replacing each
assertion (φ
j
,w
φ
j
) by (φ
j
,r
φ
j
). Similarly, we denote
by A
R
the set obtained from A by replacing each A
i
in A by A
R
i
. According to Example 3, it is clear that
the set of compatible uncertainty scales is not unique.
Let us denote by R (A) the set of compatible uncer-
tainty scales associated with A = {A
1
,..., A
n
}. Note
that the concept of compatible scale has been used in
the propositional logic setting. However, the compu-
tational complexity of reasoning process is very hard
(at least ∆
2
p
), even for simple knowledge bases like
Horn clauses.
Now, given the set of all compatible scales R (A),
different possibilities may exist in order to merge the
ABoxes. For instance, one can only select one scale
to perform merging (a credulous merging) or one can
consider all the compatible ranking in R (A) to define
result of merging (skeptical merging). We first con-
sider the case where all compatible uncertainty scales
are used to perform merging. When considering the
set of all compatible scales, an interpretation I is said
to be more plausible than I
0
, if for each compatible
scale R ∈R (A), I is considered more plausible than
I
0
using Definition 2 . More precisely,
Definition 7. Let A = {A
1
,..., A
n
} be a set of DL-Lite
π
ABoxes and R (A) be the set of all compatible scalings
associated with A. Let I and I
0
be two interpretations.
Then:
I
0
<
A
∀
I iff ∀R ∈ R (A), I
0
≤
A
R
min
I
where ≤
A
R
min
is the result of applying Definition 2 on
A
R
.
Definition 7 is illustrated by Figure 1 where m rep-
resents the size of R (A) (which may be infinite).
h
T, A
1
,...,A
n
i
D
T, A
R
m
1
,...,A
R
m
n
E
R
m
is a compatible
uncertainty scale
D
T, A
R
1
1
,...,A
R
1
n
E
R
1
is a compatible
uncertainty scale
I <
A
R
m
min
I
0
I <
A
R
1
min
I
0
I <
A
∀
I
0
iff ∀R
i
∈ R (A),I <
A
R
min
I
0
...
...
Figure 1: Merging process using the notion of compatible
scale.
According to Definition 7, models of the result of
merging the ABoxes (using compatible scales), ∆
∀
T
(A)
are those which are models of T and minimal for <
A
∀
:
Mod(∆
∀
T
(A))={I ∈Mod(T ): @I
0
∈Mod(T ), I
0
<
A
∀
I }.
The following example illustrates the fusion pro-
cess based on all compatible uncertainty scales.
Example 4 (continued). Let us consider again
the following set of ABoxes given in Example
1: A
1
={(A(a),.6), (C(b),.5)}, A
2
={(C(a),.4),
(B(b),.8), (A(b),.7)}. Let us consider the scaling
R
1
defined: A
R
1
1
= {(A(a),.8),(C(b),.4))} and A
R
1
2
=
{(C(a),.2), (B(b),.9), (A(b),.6)}. And the scaling R
2
defined: A
R
2
1
= {(A(a),.4),(C(b),.2))} and A
R
2
2
=
{(C(a),.3), (B(b),.6), (A(b),.5)}. Both of them are
compatible scale since they preserve for each asser-
tional base certainty degrees of assertions. Table 3
gives an example of four interpretations I
1
-I
4
and
presents their profile for each uncertainty scale.
A Polynomial Algorithm for Merging Lightweight Ontologies in Possibility Theory Under Incommensurability Assumption
419

Table 3: Merging under two compatible scales.
I ν
A
R
1
(I ) Min ν
A
R
2
(I ) Min
I
1
< 1,.1 > .1 < 1,.4 > .4
I
2
< .2,.1 > .1 < .6,.4 > .4
I
3
< .6,.8 > .6 < .8,.7 > .7
I
4
< .2,1 > .2 < .6,1 > .6
Note that in both uncertainty scales R
1
and R
2
,
I
3
is the preferred one. In fact, whatever is the con-
sidered compatible scale, it will be the preferred one.
Hence, it can be shown that if we consider all the com-
patible scales, I
3
will represent the result of merging
under the incommensurability assumption.
Once preferred models are computed, query an-
swering from a set of uncertain ABoxes under incom-
mensurability assumption, is given as follows:
Definition 8. Let A = {A
1
,..., A
n
} be a set of ABoxes
and T be an ontology. A query q(~x) is said to be con-
sequence of A under incommensurability assumption
if ∀I ,I ∈ Mod(∆
min
T
(A
R
)),I |= q(~x).
Said differently, a query follows from
h
T,A
1
,..., A
n
i
under incommensurability assumption
if and only if for each compatible uncertainty scale
R , q(~x) follows from
D
T,A
R
1
,..., A
R
n
E
under a
commensurability assumption.
Example 5 (Example Continued). From Example 4,
we have Mod(∆
min
T
(A
R
))={I
3
} where A
I
3
= {a,b},
B
I
3
= {a, b} and C
I
3
= {}. Let q
1
(x) ← A(x)∧B(x) be
a conjunctive query. One can easily check that < b >
is an answer of q
1
(x) using ∆
min
T
(A
R
). Similarly, let
B(a) be an instance query, one can check that B(a)
follows from ∆
min
T
(A
R
).
5 A POLYNOMIAL ALGORITHM
FOR QUERY ANSWER UNDER
INCOMMENSURABILITY
ASSUMPTION
In the previous section, we have seen that a query
follows from
h
T,A
1
,..., A
n
i
under incommensurability
assumption if and only if for each compatible uncer-
tainty scale R , q follows from
D
T,A
R
1
,..., A
R
n
E
under
a commensurability assumption.
The problem is that the number of compatible un-
certainty scales may be infinite. As we will show in
this section, there is no need to explicitly state all these
compatible uncertainty scales. In fact, query answer-
ing under the incommensurability assumption can be
achieved in a polynomial time. For the sake of sim-
plicity, we will illustrate our approach for the instance
checking problem. However, result of this section can
be generalized to general conjunctive query. More
precisely, we provide an algorithm that implements
Definition 8 when q is of the form A(a) or P(a,b)
where A is a concept, P is a role and a,b are individu-
als. In the following, we simply write X(z) instead of
A(a) or P(a,b) to denote an instance fact.
We first recall in standard DL-Lite that in order to
check whether an instance query of the form X(z) is
inferred from a standard (one consistent source) DL-
Lite knowledge base (i.e. K |= X(z)), we first add to K
the assumption that X(z) is false. This is encoded by
the following statements: {Y v ¬X,Y (z)} where Y is
a new concept not appearing in K. Then we check if
the augmented knowledge base is consistent or not. If
it is inconsistent then X(z) holds from K. Otherwise
X(z) does not follow from K.
The aim of this section is to adapt this reason-
ing process when the assertional bases come form
several incommensurable sources. Recall that we
are interested in checking if X (z) holds from K =
{T,A
1
,..., A
n
}. A preliminary step of the algorithm
consists in computing:
• the set of conflict C(K) and
• the set of conflicts of the augmented KB K
0
=
h
T ∪ {Y v ¬X }, A
1
∪ ... ∪ A
n
∪ {Y (z)}
i
.
Let us denote by
F
X(z)
= {φ ∈ A
1
∪ ... ∪ A
n
: (φ,Y (z)) ∈ C(K
0
)}
the set of all assertional facts from A
1
∪ ... ∪ A
n
that
directly contradict the assumption that X(z) is false.
To check if X(z) is a consequence of K, we first
need to see if there exists a new conflict (φ,Y (z)) in
C(K
0
). Namely, we need to see whether there exists a
fact that contradicts the assumption that X(z) is false.
If such a conflict does not exist then X(z) is not a con-
sequence from K. More formally,
Proposition 2. Let K =
h
T,A
1
,..., A
n
i
be a DL-Lite
knowledge base issued from different sources. Let
K
0
=
h
T ∪ {Y v ¬X }, A
1
∪ ... ∪ A
n
∪ {Y (z)}
i
be the
augmented knowledge base by the assumption that
X(z) is false. If C(K)=C(K
0
) (namely F
X(z)
=
/
0) then
X(z) is not a consequence using Definition 8 of K.
Proof. The proof is immediate. Assume that F
X(z)
=
/
0 or similarly C(K) = C(K
0
). Indeed if X(z) is a
consequence of K using Definition 8, this means that
there exists a compatible uncertainty scale R such
that X(z) is a consequence of K
R
=
D
T,A
R
1
,..., A
R
n
E
using the incommensurability assumption. Hence,
there exists a consistent subset K
1
=
h
T,B
i
, of K
R
with B ⊆ {φ : (φ, w
φ
) ∈ A
R
1
∪ ... ∪ A
R
n
)} s.t X(z)
follows from K in a standard way. This means
that K
0
=
h
T ∪ {Y v ¬X }, B ∪ {Y (z)}
i
is inconsistent.
This means that there exists a conflict that involves
ICAART 2017 - 9th International Conference on Agents and Artificial Intelligence
420

Y (z) (a new conflict), but this contradicts the fact that
C(K
0
) = C(K).
The following lemma simply states that if two el-
ements are conflicting then they necessarily belong to
two different assertional bases.
Lemma 1. Let (α,β) ∈ C(K). Then α and β belongs
to two different assertional bases, namely ∃i, j such
that α ∈ A
i
,β ∈ A
j
and i 6= j.
The proof of this lemma follows from the fact
that it is assumed that for each assertional base A
i
,
h
T,A
i
i
is consistent. We now analyze the general case
where C(K) 6= C(K
0
). Namely, there exists a new
conflict arising by adding the assumption that X(z)
is false. For sake of simplicity, if R is a compati-
ble scale and if φ and ϕ are two assertions then we
simply write φ
R
> ϕ
R
(resp. φ
R
= ϕ
R
, φ
R
< ϕ
R
) to denote that the uncertainty degree of φ is more
(resp. equal, less) plausible than the certainty of
ϕ using compatible scale R . Recall that when X(z)
is a consequence of K then, by Definition 8, for all
compatible scales and for all conflicts (α,β) ∈ C(K),
one should have: φ
R
> min(α
R
,β
R
), said differently
φ
R
> α
R
or φ
R
> β
R
.
The following proposition gives, in case where
C(K
0
) 6= C(K), the two cases needed to check whether
X(z) is a consequence of K.
Proposition 3. Let (α, β) ∈ C(K) and (φ,Y (z)) be a
new conflict. Let i, j be two integers such that α ∈ A
i
and β ∈ A
j
(with i 6= j). Then
1. if φ /∈ A
i
and φ /∈ A
j
. Then X(z) is not a conse-
quence of
h
T,A
1
,..., A
n
i
.
2. if φ ∈ A
i
or φ ∈ A
j
then:
• if φ ∈ A
i
and φ ≤ α (resp. φ ∈ A
j
and φ ≤ β),
then X(z) is not a consequence of
h
T,A
1
,..., A
n
i
.
• if φ ∈ A
i
and φ > α (resp. φ ∈ A
j
and φ > β),
then X(z) is a consequence of
h
T,A
1
,..., A
n
i
.
On the basis of Propositions 2 and 3, we are now
ready to provide a polynomial algorithm (Algorithm
1) that implements Definition 8.
Algorithm 1 can be improved where rather to con-
sider C(K) one can only use Reduce(C(K)) defined
by Reduce(C(K)) = {(α, β) : (α,β) ∈ C(K),@(α
0
,β
0
)
such that α
0
> α and β
0
> β with α,α
0
∈ A
i
,β, β
0
∈ A
j
}.
Clearly the computational complexity of Algorithm 1
is polynomial. Indeed, Steps 1 and 4 are polynomial
since computing the set of conflicts in DL-Lite (core,
F and R) is polynomial (Calvanese et al., 2007). Steps
2 and 3 are trivially polynomial, as well as steps 5-7,
9-20. Step 4 is a loop with a maximal |A
1
∪ ... ∪ A
n
|
iterations. Step 8 is another loop with a maximal of n
3
where n is the maximal size of an ABox. Hence, the
whole algorithm is polynomial.
Algorithm 1: Query answering for egalitarian incommensu-
rable merging.
Input: K =
h
T,A
1
,. .. ,A
n
i
, X(z).
Output: Yes (1) if X(z) is a consequence, no (0) oth-
erwise.
1: C ← conflict of K
2: K
0
←
h
T ∪ {Y v ¬X }, A
1
∪ ... ∪ A
n
∪ {Y (z)}
i
.
3: F
X(z)
← {α : (α,Y (z) ∈ Conflict of K
0
}.
4: while F
X(z)
6=
/
0 do
5: bool ← (1)
6: φ ← φ ∈ F
X(z)
7: F
X(z)
← F
X(z)
\ {φ}
8: for all (α,β) ∈ C do
9: i ← the assertional base that contains α
10: j ← the assertional base that contains β
11: k ← the assertional base that contains φ
12: if k 6= i and k 6= j then
13: bool ← 0
14: else
15: if k = i and φ is less certain than α in
A
i
then
16: bool ← 0
17: else
18: if k= j and φ is less certain than β
is A
j
then
19: bool ← 0
20: if bool == 1 then return 1
21: return 0
Example 6. Continue with Example 4 and
consider again B(a) as instance query.
We have C(K) = {{(A(a),.6),(C(a),.4)},
{(C(b),.5), (B(b),.8)}, {(C(b),.5), (A(b),.7)}}
and reduce(C) = {{(A(a), .6),(C(a), .4)},
{(C(b),.5), (B(b),.8)}}. After adding the
assumption that B(a) is false, we have
F
B(a)
= {(A(a),.6)}. By taking φ ← (A(a), .6)
and {(A(a), .6),(C(a), .4)} ∈ C
0
, it is easily to
check that bool ← 1. Similarly, by considering
{(C(b),.5), (B(b),.8)} ∈ C
0
. One can verify that
bool ← 1. Hence K |= B(a) under egalitarian
incommensurable merging.
6 SELECTING ONE
NORMALIZED COMPATIBLE
SCALE
Using the set of all compatible scales may lead to a
very cautious merging operation. One way to get rid
of incommensurability assumption is to use some nor-
malization function in the spirit of the ones used in
clustering methods for gathering attributes having in-
commensurable domaines. Let A
i
be an ABox and
A Polynomial Algorithm for Merging Lightweight Ontologies in Possibility Theory Under Incommensurability Assumption
421

W (A
i
) be the set of different certainty degrees used
in A
i
. Let Min(W (A
i
)) and Max(W (A
i
)) be respec-
tively the minimum and maximum certainty degrees
associated with assertional facts in W (A
i
). Then an
example of normalization function is ∀φ
j
∈ A
i
:
N(w
φ
j
) =
w
φ
j
− (Min(W (A
i
) − ε)
Max(W (A
i
)) − (Min(W (A
i
)) − ε)
(1)
Where w
φ
j
is a certainty degree belonging to W (A
i
)
and ε is a very small number (lower than Min(W (A
i
)).
ε is added to avoid to have null degrees in possibilis-
tic DL-Lite knowledge base. The main advantage of
only having one normalization function is that one can
have an immediate syntactic counterpart. More pre-
cisely, it is enough to replace for each fact (φ
j
,w
φ
j
) by
(φ
j
,N(w
φ
j
)) where N(w
φ
j
) is the normalization func-
tion given by Equation 1.
Example 7 (Example Continued). From Example 1,
we have A
1
={(A(a),.6), (C(b),.5)}, A
2
={(C(a),.4),
(B(b),.8), (A(b),.7)}. We have Min(A
1
) = .5,
Min(A
2
) = .4, MaX(A
1
) = .6 and Max(A
2
) = .8. Let
ε = .01, then applying Equation 1 on A
1
and A
2
, gives:
A
1
={(A(a),1), (C(b), .09)}, and A
2
={(C(a),0, 02),
(B(b),1), (A(b),.75)}.
Once the syntactic computation of normalized as-
sertional bases is done, it is enough the reuse merging
of commensurable possibilistic knowledge bases for
query answering recalled in Section 2.2.
Example 8 (Example Continued).
From Example 7, we have ∆
min
T
(A) =
hT,{(A(a), 1),(C(b), .09),(C(a), .02),(B(b), 1),
(A(b),.75)}i. We have Inc(∆
min
T
(A))=.09 and
∆
min
T
(K)=T,{(A(a), 1),(B(b), 1),(A(b),.75)}. Con-
sider now q
1
(x) ← A(x) ∧ B(x) and q
2
← B(a),
queries given in Example 5. One can check that
< b > is an answer of q
1
(x) from the and B(a) holds
from the resulting knowledge bases.
7 CONCLUSIONS
This paper dealt with the problem of merging pos-
sibilistic DL-Lite assertional bases under the incom-
mensurability assumption. The main result of the pa-
per is that query answering is achieved in a polyno-
mial time. This is a nice feature comparing for in-
stance with merging merging within propositional set-
ting where the problem is intractable even for simple
knowledge bases such as horn clauses.
ACKNOWLEDGEMENTS
This work has received support from the euro-
pean project H2020 Marie Sklodowska-Curie Ac-
tions (MSCA) research and Innovation Staff Ex-
change (RISE): AniAge (High Dimensional Heteroge-
neous Data based Animation Techniques for South-
east Asian Intangible Cultural Heritage Digital Con-
tent), project number 691215.
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