=Paper=
{{Paper
|id=Vol-1517/JOWO-15_ontolp_paper_12
|storemode=property
|title=Merging Incommensurable Possibilistic DL-Lite Assertional Bases
|pdfUrl=https://ceur-ws.org/Vol-1517/JOWO-15_ontolp_paper_12.pdf
|volume=Vol-1517
|dblpUrl=https://dblp.org/rec/conf/ijcai/BenferhatBLR15
}}
==Merging Incommensurable Possibilistic DL-Lite Assertional Bases==
https://ceur-ws.org/Vol-1517/JOWO-15_ontolp_paper_12.pdf
Merging Incommensurable Possibilistic DL-Lite Assertional Bases
Salem Benferhat and Zied Bouraoui and Sylvain Lagrue Julien Rossit
Univ Lille Nord de France, F-59000 Lille, France Univ Paris Descartes, LIPADE-France
UArtois, CRIL - CNRS UMR 8188, F-62300 Lens, France julien.rossit@parisdescartes.fr
{benferhat,bouraoui,lagrue}@cril.fr
Abstract merging the assertional bases using some aggregation strate-
gies.
This short paper studies the problem of merging of differ- Knowledge bases merging or belief merging (e.g. (Bloch
ent independent data sources linked to a lightweight ontol-
et al. 2001; Konieczny and Pérez 2002)), is a problem
ogy under the incommensurability assumption. In general,
data are often provided by several and potentially conflicting largely studied within the propositional logic setting. It fo-
sources of information having different levels of priority. To cuses on aggregating pieces of information issued from dis-
encode prioritized assertional bases, we use possibilistic DL- tinct, and may be conflicting or inconsistent, sources in or-
Lite logic. We investigate an egalitarist merging strategy that der to obtain a unified point of view by taking advantage
minimize dissatisfaction between the source involved in the of pieces of information provided by each source. Sev-
merging process. We provide a safe way to merge incom- eral merging approaches have been proposed which depend
mensurable possibilistic DL-Lite assertional bases using the on the nature and the representation of knowledge such as
notion of compatible scales. merging propositional knowledge bases (e.g. (Konieczny
and Pérez 2002)), prioritized knowledge bases (e.g. (Del-
grande, Dubois, and Lang 2006)) or weighted logical knowl-
Introduction edge bases (e.g. (Benferhat, Dubois, and Prade 1997)). Re-
Description Logics (DLs) are a well-known family of logic- cently, some works (e.g. (Noy and Musen 2000; Kotis,
based formalisms used to represent knowledge of a partic- Vouros, and Stergiou 2006; Moguillansky and Falappa 2007;
ular domain and make it available for reasoning. DLs are Cóbe, Resina, and Wassermann 2013)), have proposed to
recognized as powerful frameworks that support ontologies. merge ontologies.
A DL knowledge base is formed by a terminological base, In (Benferhat, Bouraoui, and Loukil 2013), the authors
called TBox, and an assertional base, called ABox. The study the counterpart of the min-based merging (Benfer-
TBox contains intentional (or generic) knowledge of the ap- hat, Dubois, and Prade 1997) when uncertain pieces of in-
plication domain whereas the ABox stores data (individuals formation are represented by a possibilistic DL-Lite knowl-
or constants) that instantiate terminological knowledge. edge base. The min-based merging operator, well-known as
In the last years, there has been an increasingly interest idempotent conjunctive operator, is suitable when sources
in Ontology-based Data Access (OBDA) that studies how to are assumed to be dependent. In (Benferhat et al. 2014),
query a set of data linked to a unified TBox (ontology). A a min-based merging operator based on conflict resolution
lot of attention was given to DL-Lite, a family of lightweight is proposed to merge uncertain DL-Lite assertional bases
DLs specifically dedicated for applications that use huge linked to the same terminological base (i.e. a TBox) seen
volumes of data, in which query answering is the most im- as merging integrity constraints.
portant reasoning task. DL-Lite offers a very low computa- This paper goes one step further by extending the min-
tional complexity for the reasoning process. based possibilistic merging operator in the case where un-
In many applications, data are often provided by several certainty scales used by different sources are incommensu-
and potentially conflicting sources having different relia- rable. We will follow the idea of egalitarist merging operator
bility levels. Moreover, a given source may provide dif- proposed in (Benferhat, Lagrue, and Rossit 2007) based on
ferent sets of uncertain data with different confidence lev- the concept of comparable scales. In this paper, we assume
els. In such situation, there are two main attitudes that that the TBox is coherent and fully certain and only asser-
may be followed: the first attitude consists first in gath- tional facts (ABoxes) issued from distinct sources may be
ering sets of assertions provided by each sources which somewhat certain.
gives generally an inconsistent (prioritized or flat) asser- A compatible scale is a re-assignment of certainty degrees
tional base and then coping with inconsistencies when per- to assertional facts such that the initial plausibility order-
forming inference using different inconsistency-tolerant in- ing inside each ABox (source) is preserved. We show, in
ference strategies (e.g. (Lembo et al. 2010; Bienvenu 2012; particular, that merging a set of ABoxes under incommen-
Bienvenu and Rosati 2013)). The second one consists in surable assumption comes down to apply min-based possi-
bilistic merging of ABox with respect to each compatible ensure reasoning under inconsistency while keeping a com-
scale. putational complexity identical to the one used in standard
The rest of the paper is organized as follows: Section 2 DL-Lite.
gives brief preliminaries on DL-Lite. Section 3 recalls DL-
Liteπ an extension of DL-Lite within a possibility theory Possibility Distribution over DL-Lite Interpretation
setting. Section 4 investigates min-based merging of multi-
ple and uncertain DL-Lite ABoxs under the incommensura- Let Ω be a universe of discourse composed by a set of DL-
bility assumption. Section 5 concludes the paper. Lite interpretations (I=(∆, .I ) ∈ Ω). The semantic coun-
terpart of a DL-Liteπ is given by a possibility distribution,
denoted by π, which is a mapping from Ω to the unit inter-
A brief refresh on DL-Lite val [0, 1] that assigns to each interpretation I ∈ Ω a possi-
For the sake of simplicity, we only present DL-Litecore bility degree π(I) ∈ [0, 1] that represents its compatibility
the core fragment of all the DL-Lite family (Calvanese et or consistency with respect to the set of available knowl-
al. 2007) and we will simply use DL-Lite instead of DL- edge. When π(I)=0, we say that I is impossible and it
Litecore . However, results of this paper are valid for DL- is fully inconsistent with the set of available knowledge,
LiteR and DL-LiteF , two important members of the DL- whereas when π(I)=1, we say that I is totally possible
Lite family. The DL-Lite language is defined as follows: and it is fully consistent with the set of available knowl-
edge. For two interpretations I and I 0 , when π(I) > π(I 0 )
R −→ P |P − B −→ A|∃R C −→ B|¬B we say that I is more consistent or more preferred than I 0
where A is an atomic concept, P is an atomic role and w.r.t available knowledge. Lastly, π is said to be normal-
P − is the inverse of P . B (resp. C) is called basic (resp. ized if there exists at least one totally possible interpreta-
complex) concept and role R is called basic role. A DL-Lite tion, namely ∃I ∈ Ω, π(I)=1, otherwise, we say that π is
knowledge base (knowledge base) is a pair K=hT , Ai where sub-normalized. The concept of sub-normalization reflects
T is the TBox and A is the ABox. The TBox T includes a the presence of conflicts in the set of available information.
finite set of inclusion assertions of the form B v C where Given a possibility distribution π defined on a set of inter-
B and C are concepts. The ABox A contains a finite set pretations Ω, one can define two measures on a DL-Lite ax-
of assertions on atomic concepts and roles of the form A(a) iom ϕ: A possibility measure Π(ϕ)=max{π(I) : I |= ϕ}
I∈Ω
and P (a, b) where a and b are two individuals. that evaluates to what extent an axiom ϕ is compatible with
The semantics of a DL-Lite knowledge base is given in the available knowledge encoded by π and a necessity mea-
term of interpretations. An interpretation I = (∆I , .I ) con- sure N (ϕ)=1 − max{π(I) : I 6|= ϕ} that evaluates to what
sists of a non-empty domain ∆I and an interpretation func- I∈Ω
tion .I that maps each individual a to aI ∈ ∆I , each A to extent ϕ is certainty entailed from available knowledge en-
AI ⊆ ∆I and each role P to P I ⊆ ∆I × ∆I . Furthermore, coded by π.
the interpretation function .I is extended in a straightforward
way for complex concepts and roles: (¬B)I = ∆I \ B I , DL-Liteπ Knowledge Base
(P − )I = {(y, x)|(x, y) ∈ P I } and (∃R)I = {x|∃y s.t. Let L be a DL-Lite description language, a DL-Liteπ
(x, y) ∈ RI }. An interpretation I is said to be a model knowledge base is a set of possibilistic axioms of the form
of a concept inclusion axiom, denoted by I |= B v C, iff (ϕ, W (ϕ)) where ϕ is an axiom expressed in L and W (ϕ) ∈
B I ⊆ C I . Similarly, we say that I satisfies a concept (resp. ] 0, 1] is the degree of certainty/priority of ϕ. Namely, a DL-
role) assertion, denoted by I |= A(a) (resp. I |= P (a, b)), Liteπ knowledge base K is such that K={(ϕi , W (ϕi )) : i =
iff aI ∈AI (resp. (aI , bI ) ∈ P I ). 1, ..., n}. Only somewhat certain information are explicitly
An interpretation I is said to be a model of K=hT , Ai, represented in a DL-Liteπ knowledge base. Namely, ax-
denoted by I |= K, iff I |= T and I |= A where I |= T ioms with a null weight (W (ϕi ) = 0) are not explicitly
(resp. I |= A) means that I is a model of all axioms in T represented in the knowledge base. The weighted axiom
(resp. A). A knowledge base K is said to be consistent if it (ϕ, W (ϕ)) means that the certainty degree of ϕ is at least
admits at least one model, otherwise K is said to be incon- equal to W (ϕi ). A DL-Liteπ knowledge base K will also
sistent. A DL-Lite TBox T is said to be incoherent if there be represented by a couple K=hT , Ai where both elements
exists at least a concept C such that for each interpretation in T and A may be uncertain. It is important to note that, if
I which is a model of T , we have C I =∅. Note that within a we consider all W (ϕi ) = 1 then we found a classical DL-
DL-Lite setting, the inconsistency problem is always defined Lite knowledge base: K∗ ={ϕi : (ϕi , W (ϕi )) ∈ K}.
with respect to some ABox since a TBox may be incoherent Given K=hT , Ai a DL-Liteπ knowledge base, we define
but never inconsistent. the α-cut of K (resp. T and A), denoted by K≥α (resp. T≥α
and A≥α ), the subbase of K (resp. T and A) composed of
Possibilistic DL-Lite axioms having weights at least greater than α.
In this section, we recall the main notions of possibilistic We say that K is consistent if the standard knowledge base
DL-Lite framework (Benferhat and Bouraoui 2013), denoted obtained from K by ignoring the weights associated with ax-
by DL-Liteπ , as an adaptation of DL-Lite within a possibil- ioms is consistent. In case of inconsistency, we attach to K
ity theory setting (Dubois and Prade 1988). DL-Liteπ pro- an inconsistency degree. The inconsistency degree of a DL-
vides an excellent mechanism to deal with uncertainty and to Liteπ knowledge base K, denoted by Inc(K), is syntacti-
cally defined as follow: Inc(K)=max{W (ϕi ):K≥W (ϕi ) is possibility distribution πi that encodes Ki = hT , Ai i is nor-
inconsistent}. malized. For the sake of simplicity, we use πAi instead of
Given a DL-Liteπ knowledge base K, one can associate πKi to denote the possibility distribution associated to each
to K a joint possibility distribution, denoted by πK , defined Ki = hT , Ai i
over the set of all interpretations I=(∆, .I ) by associating Given n commensurable ABoxes, merging aims to com-
to each interpretation its level of consistency with the set of pute ∆T (A), an ABox representing the result of the fusion
available knowledge, that is, with K. Namely: of these ABoxes. In the literature, different methods for
merging have been proposed. In this section, we perform
Definition 1. The possibility distribution induced from a merging of A1 ,...,An a set of ABoxes with respect to a TBox
DL-Liteπis defined as follows: ∀I ∈ Ω: T using min-based merging operator proposed to aggregate
1 if ∀ (ϕi , W (ϕi )) ∈ K, I |= ϕi DL-Liteπ knowledge bases. This operator is a direct ex-
πK (I)=
1-max{W (ϕi ):(ϕi ,W (ϕi ))∈K,I6|=ϕi } otherwise tension of the well-known idempotent conjunctive operator
A DL-Liteπ knowledge base K is said to be consistent (e.g. (Benferhat, Dubois, and Prade 1997)) within possibilis-
if its joint possibility distribution πK is normalized. If not, tic DL-Lite setting. It is recommended when distinct sources
K is said to be inconsistent and its inconsistency degree is that provide information are assumed to be dependent.
defined semantically as follow: Inc(K)=1 − max{πK (I)}. We first introduce the notion of profile associated with an
I∈Ω interpretation I, denoted by νA (I), and defined by
It was shown in (Benferhat and Bouraoui 2013) that com-
puting the inconsistency degree of a DL-Liteπ knowledge νA (I) =< πA1 (I), ..., πAn (I) > .
base comes from the extension of the algorithm presented in Namely, νA (I) represents the possibility values of an in-
(Calvanese et al. 2007) by modifying it to query for individ- terpretation I with respect to each source.
uals with a given certainty degree. From a semantics point of view, the result of merging is
Example 1. Let K=hT , Ai be a DL-Liteπ knowledge base a possibility distribution ∆T (A) obtained using two steps:
where T ={(AvB, 1), (Bv¬C, .9)} and A={(A(a), .6), i) the possibility degrees πAi (I)’s are first combined with a
(C(b), .5)}. The possibility distribution πK associated to K merging operator (here we use the minimum operator), and
is computed using Definition 1 as follows where ∆={a, b}: the interpretations with height degrees are kept. This leads
to define an order relation, denoted by /M in , between in-
I .I πK terpretations as follows: an interpretation I is preferred to
I1 A={a},B={},C={b} 0 another interpretation I 0 if the minimum element of the pro-
I2 A={a},B={a},C={b} 1 file of I is higher than the minimum element of the profile
I3 A={},B={},C={a,b} .4 of I 0 . More formally:
I4 A={a,b},B={a,b},C={} .5 Definition 2 (Definition of /M in ). Let A = {A1 , ..., An }
be a set of ABoxes linked to a TBox T . Let I and I 0 be
Table 1: Example of a possibility distribution induced from two interpretations and νA (I), νA (I 0 ) be their associated
a DL-Liteπ knowledge base profiles. Then:
I /A 0 0
min I ⇐⇒ M in(νA (I)) > M in(νA (I ))
One can observe that πK (I2 )=1 meaning that πK is normal- where
ized, and thus, K is consistent.
M in(νA (I)) = M in{πAi (I) : i ∈ {1, ..., n}}.
Fusion-based on compatible scalings The result of the merging ∆min T (A) is a DL-Liteπ
knowledge base whose models are interpretations which are
This section studies min-based possibilistic merging opera-
models of a constraint T and which are maximal with re-
tor in the case where uncertainty scales used by the different
spect to /M in . More formally:
sources are incommensurable. Throughout this section, we
assume that the TBox is coherent and fully certain and only Definition 3 (Min-based merging operator). Let A =
assertional facts (ABoxes) may be somewhat certain. We {A1 , ..., An } be a set of ABoxes and T be an integrity con-
first present merging using min-based operator of DL-Lite straint. Let {πA1 , ..., πAn } possibility distributions associ-
assertional bases under commensurability assumption. ated with (hT , A1 i , ..., hT , An i). The result of merging is a
DL-Liteπ knowledge base, denoted by ∆min T (A) where its
Merging using the min-based operator model are defined by:
Let A = {A1 , ..., An } be a set of n uncertain ABoxes issued M od(∆min T (A))={I∈M od(T ):@I 0 ∈M od(T ),I 0 /A M in I}
π
from n distinct sources, and let T be a DL-Lite TBox rep- In general, merging two DL-Lite normalized possibility
resenting the integrity constraints to be satisfied. Let us as- distributions may lead to a sub-normalized possibility distri-
sume that π1 , ..., πn are possibility distributions provided by bution. The normalization process comes down to set the
n sources of information that share the same domain of inter- degrees of interpretations in M od(∆min T (A)) to 1.
pretations (namely ∆I1 = ... = ∆In ), and that all possibility From a syntactic point of view, the min-based merging
distributions use the same scale to represent uncertainty. We operator, denoted by ∆min T (A) is the union of all ABox.
suppose that each ABox is consistent with T , namely each Namely:
∆min
T (A)=hT , A1 ∪ A2 ∪ . . . ∪ An i. Definition 5 (Compatible ranking scale). Let A =
The aggregation of ABoxs is not guaranteed to be consis- {A1 , ..., An } where Ai = {(fij , WAi (fij ))}. Then a rank-
tent. Namely, the resulting knowledge base T , ∆min (A) ing R is defined by:
T
may be inconsistent. To restore the consistency of the result- R: A1 ∪ ... ∪ An → ]0, 1]
ing knowledge base a normalization step is required. The (fij , WBi (fij )) 7→ R(fij )
following definition gives the formal logical representation A ranking R is said to be compatible with WA1 , ..., WAn
of the normalized knowledge base. if and only if:
Definition 4. Let T be a TBox and ∆min (A) be the aggre- ∀Ai ∈ A, ∀f, WAi (f )), (f 0 , WAi (f 0 )) ∈ Ai ,
T
gation of A1 , ...An , n ABox using classical min-based op- WAi (f ) ≤ WAi (f 0 ) ⇐⇒ R(f ) ≤ R(f 0 ).
erator. Let x=∆minT (A). Then, the normalized knowledge Definition 5 is basically the adaptation of the one given
base, denoted ∆minT (K) is such that: in (Benferhat, Lagrue, and Rossit 2007) for the context of
∆min (K)={(fij , W (fij )):(f, W (fij )∈∆min (A) and DL-Lite.
T T
W (fij ) > x}i Example 3 (continued). Let us consider again the follow-
Example 2 (continued). Let us continue with the TBox ing set of ABoxes to be linked to T given in Example
T ={A v B, B v ¬C} presented in Example 1 while 2: A1 ={(A(a), .6),(C(b), .5)}, A2 ={(C(a), .4), (B(b), .8),
assuming that the certainty degree of each axioms is set (A(b), .7)}. The following table gives examples of ranking
to 1. Let us consider the following set of ABoxes to be scales.
linked to T : A1 ={(A(a), .6), (C(b), .5)}, A2 ={(C(a), .4),
fij WAi (fij ) R1 (fij ) R2 (fij ) R3 (fij )
(B(b), .8), (A(b), .7)}. We have:
A1 A(a) .6 .5 .4 .6
C(b) .5 .2 .7 .5
I .I πA1 πA 2 ∆min
T (A) A2 C(a) .4 .3 .3 .4
I1 A={a},B={a},C={b} 1 .2 .2 B(b) .8 .7 .6 .8
I2 A={},B={},C={a,b} .4 .2 .4 A(b) .7 .4 .2 .7
I3 A={a,b},B={a,b},C={} .5 .6 .5
I4 A={b},B={b},C={a} .4 1 .4 Table 3: Examples of ranking scales
Table 2: Example of merging of possibility distributions us- The scaling R1 is a compatible one, because it preserves
ing min-based operator the order inside each ABox. However, the scaling R2 is not
a compatible one since it inverses priorities inside A1 and
One can check that the resulting possibility distribution A2 .
(∆min
T (A)) is sub-normalized. To normalize ∆min T (A), According to Example 3, it is obvious that a compatible
it is enough to set I3 = .5 to 1. At syntactic level, ranking scale is not unique. Let us denote by R(A) the set
we have ∆minT (A)=hT , {(A(a), .6), (C(b), .5), (C(a), .4), of compatible scaling associated with A = {A1 , ..., An }.
(B(b), .8), (A(b), .7)}i. We have Inc(∆min T (A))=.5 and The set R(A) is non-empty and an immediate way to obtain
∆min
T (K)=T , {(A(a), .6), (B(b), .8), (A(b), .7)}. a ranking relation over A is to consider R(fij ) = WAi (fij )
In the next section, we investigate min-based merging un- (For instance, the scale R3 (fij ) given in Example 3). Note
der incommensurability assumption. that this ranking is compatible in the sense that it permits to
preserve the relative ordering between assertions of each Ai .
Using compatible scales Given a compatible scales R, we denote by AR i the as-
The min-based merging operator presented in the previous sertional base obtained from Ai by replacing each assertion
section is defined over the assumption that all the sources (fij , WAi (fij )) by (fij , R(fij )). Similarly, we denote by
providing the ABoxs use the same scale to encode uncer- AR the set obtained from A by replacing each Ai in A by
tainties between facts. In Example 2, when dealing with ARi .
assertions, we assumed that the weight attached to f ∈ Ai Now, given the set of all compatible scales R(A), dif-
can be compared to the weight associated with g ∈ Aj with ferent possibilities may exist in order to merge the ABoxs.
j 6= i. In this section, we drop this assumption and we sup- For instance, one can only select one scale to perform merg-
pose that sources are incommensurable. ing or one can consider all the compatible ranking in R(A),
We investigate a min-based fusion operator to merge in- etc. To avoid an arbitrary choice, we consider all compatible
commensurable DL-Lite assertional bases. To make sources rankings to perform merging.
using different scale commensurable, we use the notion of
"compatible scale" on existing scales used by each source. Semantics merging
A ranking scale is said to be compatible with all sources We first introduce the notion of preference between inter-
if it preserves original order relations between assertions of pretation according to the notion of compatible scales. An
each ABox. The new ranking, denoted by R, defines a new interpretation I is then said to be preferred to I 0 , if for each
ranking relations for each ABox to be merged. More for- compatible scale R, I is preferred to I 0 using Definition 2
R
mally, (namely, I /A 0
M in I ). More precisely,
Definition 6 (Ordering between interpretations). Let A = Using the set of all compatible scales may lead to a very
{A1 , ..., An } be a set of DL-Liteπ ABoxs and R(A) be the cautious merging operation. One way to get rid of in-
set of all compatible scalings associated with A. Let I, I 0 commensurability assumption is to use some normalization
be two interpretations. Then: function in the spirit of the ones used in clustering methods
R for gathering attributes having incommensurable domains.
I .1 < 1, .4 > .4 Consider now q1 (x) ← A(x) ∧ B(x) and q2 ← B(a),
I2 < .2, .1 > .1 < .6, .4 > .4 queries given in Example 5. One can check that < b > is an
I3 < .6, .8 > .6 < .8, .7 > .7 answer of q1 (x) from the and B(a) holds from the resulting
I4 < .2, 1 > .2 < .6, 1 > .6 knowledge bases.
Table 4: Two equivalent compatible scalings Conclusions
This paper proposed a min-based possibilistic merging op-
Note that in both compatible scalings R1 and R2 , I3 is eration of uncertain assertional facts under incommensura-
the preferred one. bility assumption. The idea is to reuse standard min-based
Once preferred models are computed, query answering merging, over a set of compatible scales. Future work in-
from a set of uncertain ABox under incommensurability as- cludes developing a syntactic counterpart of incommensu-
sumption, is defined as follows: rable merging operation. A natural question is whether one
can extend a polynomial time complexity algorithm, defined
Definition 7. Let A = A1 , ..., An be a set of ABoxes
for query answering from a standard uncertain ABox, to the
linked to the same TBox T . A query q is said to be
case where uncertainty scales are incommensurable.
consequence of A under incommensurability assumption if
∀I, I ∈ M od(∆minT (AR )), I |= q.
Acknowledgment
Example 5 (continued). From Example 4, we have
M od(∆min (AR ))={I3 } where AI3 = {a, b}, B I3 = This work has been supported by the French National Re-
T
search Agency. ASPIQ project ANR-12-BS02-0003
{a, b} and C I3 = {}. Let q1 (x) ← A(x) ∧ B(x) be a
conjunctive query. One can easily check that < b > is an
answer of q1 (x) using ∆min (AR ). Similarly, let B(a) be References
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