Recursive algorithms to repair prioritized and inconsistent dl-lite knowledge base

12/05/2019 ∙ by Ghassen Hamdi, et al. ∙ University of Biskra 0

The inconsistency in prioritized knowledge base is because the assertions (ABoxes) come from several sources with different levels of reliability. We introduce the handling of this inconsistency problem to query inconsistent DL-Lite knowledge bases. In the literature, firstly, repair all the inconsistent assertions of the DL-Lite's inconsistent knowledge base. Then, interrogate it. However, our algorithm, on proceeds directly with an interrogation of the knowledge base in order to recover an exhaustive list of answers to a given query. In a second time, to repair the answers of this list. The novelty of our article is the proposition of a recurring function that calculates the rank of coherence in order to manage the inconsistencies in the set of responses. This strategy allowed us to reduce execution time compared to existing algorithms. The experimental study as well as the analysis of the results, which we carried out, showed that our algorithm is much more productive than the other algorithms since it gives the greatest number of answers while remaining the best from the point of view of the execution time. Finally, as shown in our experimental studies, they allow an efficient handling of inconsistency. Such facts make all the repairs suitable for DL-Lite

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1. Introduction

Description Logics (DLs) are formal frameworks for representing and reasoning with ontologies. The DL knowledge base consists of: TBox as a terminological base represents the conceptual knowledge of a particular domain and ABox as an assertional base contains facts or assertions concerning particular individuals [1].

Recently, Ontology Based Data Access (OBDA) [2, 3, 4] is collecting great attention as a new paradigm in which structured knowledge or the ontological view (i.e.,  stored in a TBox) is used to provide better exploitation of assertions (i.e.,  stored as an ABox) when querying them. A crucially important service provided by an OBDA system is query answering, which aims to calculate the answers of a query posed in terms of ontologies.

In many applications, assertions are provided by several potentially conflicting sources with different levels of reliability. In addition, a given source may have different sets of uncertain assertions that together form a prioritized or a stratified assertional base (i.e ABox). In the case, where the data is provided from multiple and unreliable sources that may be inconsistent. A major problem arises in the context of DL-Lite query answering is how to deal with the case of inconsistency between ABox and TBox. Indeed, the TBox is generally verified and validated as long as the ABox will typically be larger and may have several modifications and thus may be in contradiction with the TBox.

Many works in the context of OBDA, inspired by database approaches (e.g.[5, 6, 7]) or propositional logic approaches (e.g.[8, 9, 10]), deal with the problem of querying inconsistent DL KBs by proposing several inconsistency-tolerant inferences, called semantics, and were introduced for the lightweight description logic DL-Lite e.g.[11]. Among these semantics, one can quote the AR and IAR semantics [12] which are the most known and the most studied. These two semantics are based on the notion of maximal assertional repair, which is based on the notion of repair in the database domain or the maximal consistent subsets in the propositional logic setting [13, 14].

The main contribution of this article is to develop a recursive function in order to deal with the inconsistency of the answers to a query with respect to the TBox. Our experiments focus on the running time, the precision calculation, the recall, the F-measure properties and the productivity of these inferences strategies after running a query.

The rest of this article is organized as follows: next Section 6 presents the related works. Section 2 provides a brief refresher on DL-Lite and querying multiple prioritized sources. Section 3 explains the notion of inconsistency tolerant reasoning for assertions associated with the answers profile (Conflicts set repair answers, Free set answers, reparis answers, and consistency rank). Section 4 introduces our proposed strategies of answers profile repairs. Section 5 presents the experimental studies and Section 7 concludes the paper.

2. Syntax and Semantics of Prioritized DL-Lite Knowledg Base

Let , and , three pairwise disjoint sets of atomic concepts, atomic roles and individuals respectively. Let , . Let also ’’, ’’ and ’’ three connectors are used to define complex concepts and roles. DL-Lite concepts are defined as follows:

A DL-Lite KB is a pair = where is called the TBox and is called the ABox. A TBox includes a finite set of inclusion axioms on concepts and on roles respectively of the form inclusion assertions: (resp.  negative inclusion assertions ) means that concept is included in concept (resp.  concept is not included in the concept ) and (resp.) means that role is included in the role (resp.  role is not included in role ).

The ABox contains a finite set of assertions (facts) of the form and where , and . The semantics is given in terms of interpretations = which consist of an non-empty domain and an interpretation function that assigns to each an element , to each a subset and to each an . A TBox is said to be incoherent if there exists a concept s.t :, we have =. A DL-Lite KB is said to be inconsistent if it does not admit any model.

The prioritized profile is a multiset of prioritized or stratified ABoxes denoted by where is a standard DL-Lite TBox and is a prioritized ABox profile. It is assumed that each ABox is consistent with the ontology (TBox). In this case, each sets are called layers or strata, which each layer contains a set of assertions with the same level of priority and they are considered more reliable than those present in a layer when . Accordingly, contains the most important assertions as long as contains the least important ones.

A query is a first-order logic formula, denoted =, where =(,…,) are free variables, is the arity of and atoms of are of the form or with and and , are terms, i.e., constants of or variables. When is of the form where are bound variables called existentially quantified variables, and is a conjunction of atoms of the form or with and and , are terms, then is said to be a conjunctive query (CQ). An answer to a CQ over a KB is a non empty set of tuples such that .

Let be a query, we consider a various sets of answers to a query regarding the prioritized profile called the profile of sets of answers where each is the set of answers to w.r.t. for , defined as follows : . Certainty, when there is no answer to the query with respect to , = .

3. Inconsistency Tolerant Reasoning for Assertions Associated with the Answers Profile

In this section, we have assumed that each ABox is consistent with the TBox. Coping with inconsistency can be done by first computing the set of consistent subsets of assertions associated to a set of answers to a given query, called repairs answers.

3.1. Conflicts set Answers

The conflicts sets answers represent a minimal inconsistent subset of the assertions associated to such that is inconsistent.

Let be a prioritized DL-Lite KB with: , be a query, a set of answers to a query with respect to , is the set of assertions associated to such that . A subset is said to be a conflicts sets answers iff is inconsistent and , is consistent.

3.2. Free Set Answers

We denote by the set of assertions belong to that are not responsible for conflicts in .

Let be a prioritized DL-Lite KB with: , be a query, a set of answers to a query with respect to , is the set of assertions associated to such that . A free assertion is said to be free if and only if . This notion of free elements is formerly proposed by [17] in a propositional logic setting.

3.3. Repairs Answers

A subset is said to be a repair answers if is consistent and is said to be a maximally inclusion-based repair answers of , denoted by , if is consistent and is inconsistent. According to this definition of , adding any assertion from to implies the inconsistency of . Furthermore, the maximality in is used in the sense of set inclusion. We denote by the set of of with respect to . The definition of is similar to that defined in [12]. Using the concept of repair answers, the treatment of inconsistency in flat DL-Lite KBs can be done by applying standard query answering either using the whole set of repairs answers (universal entailment or AR-entailment [12]) or only using one repair answers. A repair answers is defined as follows:

Let be a prioritized DL-Lite KB with: , be a query, a set of answers to a query with respect to , is the set of assertions associated to such that .

3.4. Consistency Rank

Generally, the checking of consistency degree and several inference services can be done with standard DLs reasoning services through consistent subsets of DL knowledge base has been explained in [18] and [19]. Clearly, the computing of inconsistency degree comes down to perform a dichotomie search in standard DL, and it is closely related to the method proposed in [21] for computing inconsistency degrees of a possibilistic propositional knowledge base.

This notion of consistency rank defined is inspired by the degree of inconsistency used in the possibilistic logic where the degrees are encoded using values in the unit interval . It is counterpart of the algorithm proposed in [21] (resp.[18]) in the propositional logic (resp.  description logic) setting.

Let be a prioritized DL-Lite KB with: , be a query, a set of answers to a query with respect to , is the set of assertions associated to such that . The consistency rank of , denoted by : is defined as follows:

We propose the recursive Function 1 which calculate the consistency rank of . Then, we will use it in all our proposed strategies of computing a consistent assertions associated to the answers. Formally, this recursive function is faster than that sequential one proposed in [21].

0:  Inconsistent assertions associated to
0:  Consistency rank
1:  if  is consistent then
2:     return  
3:  else
4:     
5:     
6:     
7:     if  is consistent then
8:        
9:     else
10:        
11:     end if
12:  end if
Algorithm 1 CnsRank

4. Assertions Associated with the Answers Profile Repairs

This section proposes three repairs to cope with inconsistent answers that seen as a set of facts. The input of these approaches is a prioritized DL-Lite KB with the prioritized profile, a query and the profile of sets of answers. On other hand, the output of our approaches is a consistent assertions associated to the sets of these answers (repair answers).

4.1. Possibilistic-Based Repairs Answers

Possibility theory [20] and possibilistic logic [21] are natural frameworks to deal with uncertain, incomplete, qualitative and prioritized information. One of the interesting aspects of possibilistic KBs is the ability of reasoning with partially inconsistent knowledge [22]. As shown in [23], the entailment in possibilistic DL-Lite, an adaptation of DL-Lite entailment within a possibility theory setting, is based on the selection of one consistent, (not necessarily maximal) subset of . The subset is formed by assertions with priority levels that are less or equal to .

More formally, is the repair answers of defined by . If is consistent with the TBox then we simply let . The Algorithm 2 returns the possibilistic-based repair answers.

0:  Inconsistent assertions associated to
0:  Consistent assertions
1:   /*Calculate recursively the consistency rank*/
2:  return  
Algorithm 2

This algorithm requires inconsistency tests on a set of assertions associated to answers to a query with respect to . It returns the possibilistic based repair answers in polynomial time.

The possibilistic conclusions are considered intact since our algorithm stops in the first assertions associated to an answer where inconsistency is introduced. Hence, only the assertions having a degree strictly less or equal than the one the consistency degree are taken into account. However, assertions with priority levels strictly greater than the consistency degree are simply inhibited despite being unaffected by any conflict. To overcome this limitation, we improve possibilistic based repair answers to Linear-based repair answers.

4.2. Linear-Based Repairs Answers

In order to recover the assertions inhibited by possibilistic-based repair answers, we propose a new way corresponds to the use of linear-based repair answers from .

Let be a prioritized DL-Lite KB with:
, be a query, a set of answers to a query with respect to , is the set of assertions associated to such that . The linear-based repair answers of , denoted by: is defined as follows:

(1)

is obtained by discarding the set of assertions when it conflicts with the previous set . The following Algorithm 3 implements the .

0:  Inconsistent assertions associated to
0:  Consistent assertions
1:   /*Calculate recursively the consistency rank*/
2:  
3:  for  to  do
4:     if  is consistent then
5:        
6:     end if
7:  end for
8:  return  
Algorithm 3

The time complexity of computing is in . In fact, according to Algorithm 3, the computational complexity of computing needs executions to verify the consistency of the set of assertions .

4.3. Non-defeated Repair Answers

This new inference makes also to get a preferred repair answers. It consists in determining among the union of the set of assertions associated to the answers to a given query with respect to the ABox Profile , the set of free elements.

Let be a prioritized DL-Lite KB with: , be a query, a set of answers to a query with respect to , is the set of assertions associated to such that . We define The non-defeated repair answers, denoted by: as follows:

Namely, .

The non-defeated repair is computed in polynomial time in DL-Lite but its computation is hard in propositional logic setting. In what follows, we present Algorithm 4 which computes the non-defeated repair answers. The complexity of this algorithm is where is the number of answers to a query w.r.t.the profile .

0:  Inconsistent assertions associated to
0:  Consistent assertions
1:   /*Calculate recursively the consistency rank*/
2:  
3:  for  to  do
4:     
5:  end for
6:  return  
Algorithm 4

In the following example, we show that the productivity of our repairs answers strategies applied on prioritized KB after runing querying is more than the repairs strategies applied on the whole KB before querying. We mean by productivity, the number of answers returned by applying each algorithm.

Let be a prioritized DL-Lite knowledge base. Then we have:

and such that:

= ,
= ,
= ,
= ,
=

Let we have the following query (giving all concepts which are on relation with ):
. We get the following set of assertions :

,
,
,
,

One can check that the set of conflictsis answers:

However, the set of conflicts is of KB (according to [15, 25]):

In addition, the set of elements for the assertion is:

While that, the set elements for is:

According to Algorithm 1, the consistency rank of equal 2. While, the consistency rank of equal 1. Moreover, we have the the following free sets of :

,
,
,

Using the definitions of possibilistic-based repair answers, linear-based repair answers and non-defeated repair answers, we have:

,
,

Now, we have the the following free sets of :

,
,
,

Although, if we use the definitions of possibilistic-based repair, linear-based repair and non-defeated repair proposed in ([15, 25]) directly on (before querying), we have:

,
,

By applying the same query on these repairs, we have:

= ,
= ,
=

Clearly, , , .

Next section is an experimental study which based on recursive programming (Consistency Rank function) about the running time of our approach.

5. Experimental Evaluation

We implemented our proposed algorithms in Java for computing a repair answering in prioritized assertional bases under inconsistent KBs. The parses DL-Lite KBs expressed in OWL2-QL function syntax and a SQLite database engine. We used a part of benchmark 111Available at: https://code.google.com/p/combo-obda/ we considered the LUBM ontology (i.e., TBox) [31], and we generated by the Extended University Data Generator (EUDG) an ABox contains assertions and we split them into 5 strata. These ABoxes with respectively 50, 200 and 500 conflict sets. We ran the proposed algorithms in [25] and our developed algorithms for computing repairs before and after launching a instance, ground and conjunctive query.

We are interested in the basic performance metrics used for evaluate our algorithms. In our case, our system classifies the assertions of the ABox into two classes: consistent and inconsistent. Consistent assertions are placed by the system in the positive class, and inconsistent assertions are placed by the system in the negative class.

When our algorithm classification is correct, the assertions are retrieved. However, if the algorithm makes a mistake, the assertions are not retrieved. We can compute the four following performance indices:

  • CR is the number of consistent retrieved assertions after applying the repairs.

  • CNR is the number of consistent not retrieved assertions after applying the repairs.

  • IR is the number of inconsistent retrieved assertions before applying the repairs.

  • INR is the number of inconsistent not retrieved assertions before applying the repairs.

In the following, we present the precision, recall and F-measure measures ([52, 53]), which we will use to evaluate the performance of our algorithms:

  • Precision (P) is the ratio of the number of consistent assertions retrieved by the total number of retrieved assertions.

    The principle of precision measure: when we ask a query on our ABox, we wish that the assertions proposed as answer correspond to our expectations. All retrieved irrelevant assertions constitute what is called ”the noise”. Precision, is opposed by assertional noise. If it is high, this indicates that few unnecessary assertions are offered by the system and that the system can be considered ”precise”.

    (2)
  • Recall (R) is the ratio of the number of consistent assertions retrieved by the total number of consistent assertions.

    The principle of recall measure: when we ask a query on our ABox, we wish to see appearing all assertions that could answer our need of information. If this correspondence between the questioning of the user and the number of assertions presented is important then the recall rate is high. Conversely, if the system have many interesting assertions but they do not appear in the list of answers, we speak of ”silence”. Silence opposes recall.

    (3)
  • F-measure (F) is the harmonic average of the precision P and the recall R:

    (4)

The following table shows the precision, recall and F-measure measures for our algorithms, after launching a instance, ground and conjunctive query on our ABox.

Conflict Query
size type
50 Instance 84.03 94.33 88.88 84.67 94.59 89.35 88.41 96.02 92.05

Ground 79.78 92.59 85.70 80.80 93.02 86.48 87.24 95.58 91.21

Conjunctive 75.94 90.90 82.74 77.38 91.54 83.86 78.65 92.10 84.84
200 Instance 55.55 83.33 66.66 55.55 83.33 66.66 60.97 86.20 71.42

Ground 52.94 81.81 64.28 52.94 81.81 64.28 55.55 83.33 66.66

Conjunctive 42.85 75 54.53 42.85 75 54.53 51.51 80.95 62.95
500 Instance 24.24 48.97 32.42 24.24 48.97 32.42 36.97 63.76 46.80

Ground 21.05 44.44 28.56 21.05 44.44 28.56 34.78 61.53 44.44

Conjunctive 21.05 44.44 28.56 21.05 44.44 28.56 32.43 59.01 41.85

Table 1. Experimental evaluation of proposed inferences expressed in %

All retrieved inconsistent assertions constitute what is called ”the noise”. Precision, is opposed by assertional noise. If it is high, this indicates that few unnecessary assertions are offered by the system and that the system can be considered ”precise”.

The precision measure obtained in the previous table show that the non-defeated algorithm is more precise than the two others. Also, when increasing the number of conflicts, the precision measure decreases in the three algorithms. Hence, this measure is influenced by the number of conflicts in the ABox.

Now, if the correspondence between the questioning of the user and the number of assertions presented is important, then the recall rate is high. Conversely, if the system have many interesting assertions but they do not appear in the list of answers, we speak of ”silence” (silence opposes recall).

Similarly, according to the results of the previous table, we note that the recall measure of the non-defeated algorithm is higher than the other algorithms. In addition, when increasing the number of conflicts, the recall measure decreases with the three algorithms. Thus, this measure is influenced by the number of conflicts in the ABox.

Now, we are interested in the time taken to compute our proposed algorithms of repairs. For this aim, we generated and splitted the ABoxs respectively into 3 strata, 5 strata and then 7 strata. These ABoxes contain respectively 50, 200 and 500 conflict sets. The results of this exprentation are shown in Table 2.


Conflict
Strata Before querying ([15, 25]) After querying
size level
50 3 10.93 11.09 45.45 2.83 2.88 11.50

5 10.96 32.65 67.43 2.85 8.22 17.50

7 16.36 32.92 94.26 4.10 8.25 23.58
200 3 12.20 28.17 47.90 3.06 3.80 11.96

5 12.47 33.09 78.18 3.12 8.25 19.55

7 16.53 38.53 96.99 4.17 9.65 24.25
500 3 13.86 14.41 50.68 3.45 7.07 12.77

5 18.12 29.98 93.36 4.55 8.27 23.50

7 19.89 43.41 99.43 4.98 10.90 25.00


Table 2. Runing time of our repairs before and after querying (in seconds)

As expected, the existing strategies which based on access to the whole Knowledge Base take more running time than our proposed strategies which which handle only with the set of answer of a specific request. Precisely, the computing of our repairs after querying requires in most of our experiments, less time than its computing before querying. However, the time needed for computing the non-defeated repair answers increases with the size of conflicts in the ABox. Finally, according to the results obtained, we conclude that non-defeated algorithm is the most performing followed by linear and possibilistic algorithms.

Now, we are interested to evaluate the productivity of our repair algorithms before and after querying. We mean by productivity, the assertions that are preserved from the ABox (resp.  answers) in order to restore the consistency of the KB (resp.  answers)


Conflict
Query Before querying After querying
size type
50 Instance 15 16 25 20 21 29

Ground 14 15 24 15 16 26

Conjunctive 10 12 14 12 13 14
200 Instance 18 19 22 20 20 25

Ground 17 18 20 18 18 20

Conjunctive 10 10 15 12 12 17
500 Instance 11 11 20 12 12 22

Ground 11 11 18 10 10 20

Conjunctive 10 10 17 10 10 18


Table 3. Productivity of applying our repair algorithms before and after querying (expressed in %)
Figure 1. Productivity of applying our repair algorithms before querying
Figure 2. Productivity of applying our repair algorithms after querying

From Table 3, figure 1 and figure 2, the productivity of possibilistic-based repair answers is very cautious comparing to the other strategies. Namely, for a given ABox and a given number of strata, possibilistic-based repair answers has the largest number of dropped elements. This similarly holds for the linear-based repair answers when there exists at least a conflict in each strata. Hence, it is obvious that the size of conflicting elements in the assertional bases is one of main parameters that influence the productivity of the repairs answers. The non-defeated repair answers gives a significant number of consistent answers compared to the other strategies.

Table 1 shows that the productivity of , and are more productive than , and respectively. We note that the computing on our proposed algorithms requires a polynomial running time.

6. Related Works

The principal inspiration for the present paper comes from a line of research in inconsistency-handling. Inconsistency is defined with respect to some assertions that contradict the terminology. Typically, a TBox is usually verified and validated while the assertions can be provided in large quantities by various and unreliable sources and may contradict the TBox. Moreover, it is often too expensive to manually check and validate all the assertions. This is why it is very important in OBDA (Ontology-based Data Access) to reason in the presence of inconsistency. Many works (e.g.[12, 45]), basically inspired by database approaches (e.g.[6]), tried to deal with inconsistency in DL-Lite by adapting several inconsistency-tolerant inference methods. In many applications, assertions are often provided by several and potentially conflicting sources having different reliability levels.

Moreover, a given source may provide different sets of uncertain assertions with different confidence levels. Gathering such sets of assertions gives a prioritized or a stratified assertional base. The role of priorities in handling inconsistency is very important and it is largely studied in the literature within propositional logic setting (e.g.[36, 37]). Several works (e.g.[27, 28, 29]) studied the notion of priority when querying inconsistent databases or DL KBs. Unfortunately, in the OBDA setting, there are only few works, such as the one given in [46] for dealing with reasoning under prioritized DL-Lite ABox.

A recent line of work studies the inconsistency in lightweight ontologies. For instance, the authors in [11, 12, 45, 47, 48] investigate the problem of inconsistency in KBs by computing a set of consistent subsets of assertions called repairs, which recovers the consistency with respect to the ontology, and then using them to answer the queries. Moreover, the authors propose in [15, 25] polynomials algorithms for select a single preferred repair from a prioritized inconsistent DL-Lite KB to allow an efficient query answering once the repair is computed. Particularly, the authors in [15] propose a new approach based on the selection of only one preferred repair. However, in [25], the authors propose a sequential inference strategies based on the selection of one consistent assertional base. The authors in [49, 50, 51] propound a new algorithm which makes easy to query answering without access to Web databases.

In our work, a recursive algorithm starts by querying all the whole knowledge base which will allow us to have an exhaustive list of all the possible answers. Then, if this list of answers is inconsistent, the algorithm repairs it. Hence, it do not any correction except if the set answers is inconsistent. However, existing algorithms start by repartion the knowledge base, then, querying this reparation with a sequential function.

7. Conclusion

We focused in this work on the problem of inconsistency answers over prioritized DL-Lite Knowledge Bese. For this purpose, we started out by giving some bases notions about: prioritized DL-Lite knowledge bese, inconsistency tolerant reasoning with the answers set. Then, we developed a recursive function to calculate the consistency rank in order to use it on our proposed algorithms. The main contribution of this paper is how to repair the set of answers instead of the whole Knowledge Base without increasing the computational complexity time? The experimental studies evaluated the productivity and rapidity of running of our proposed repairs answers using the basic performance metrics: Precision, Recall and F-measure and the running time.
A future work is to apply our approaches to query the closure of KB in presence of possible answers.

References

  • [1] Baader F, Calvanese D, McGuinness D, Daniele N, Nardi D, Patel-Schneider P F, The description logic handbook: theory, implementation, and applications, Cambridge University Press, New York, USA (2010).
  • [2] Lenzerini M, Ontology-Based Data Management, In: Proceedings of the 6th Alberto Mendelzon International Workshop on Foundations of Data Management, Volume 866, pp. 12–15, ACM, Glasgow, United Kingdom (2011).
  • [3] Poggi A, Lembo D, Calvanese D, De Giacomo G, Lenzerini M, Rosati R, Linking Data to Ontologies, Journal on Data Semantics, Volume 4900, page 133–173 (2008).
  • [4] Rodriguez-Muro M, Kontchakov R, Zakharyaschev M, Ontology-Based Data Access: Ontop of Databases, In: 12th International Semantic Web Conference, Volume 8218, pp. 558–573, Springer, Sydney, NSW, Australia (2013). Artale A, Calvanese D, Kontchakov R, Zakharyaschev M, The DL-Lite Family and Relations, CoRR - Computing Research Repository, Volume abs/1401.3487, (2014).
  • [5] Arenas M, Bertossi L E, Chomicki J, Consistent Query Answers in Inconsistent Databases, In: Proceedings of the Eighteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 68–79, ACM Press, Philadelphia, Pennsylvania, USA (1999).
  • [6] Bertossi L E, Database Repairing and Consistent Query Answering, Morgan & Claypool Publishers, (2011).
  • [7] Chomicki J, Consistent Query Answering: Five Easy Pieces, In: 11th International Conference of Database Theory, pp. 1–17, Springer, Barcelona, Spain (2007).
  • [8]

    Benferhat S, Cayrol C, Dubois D, Lang J, Prade H, Inconsistency Management and Prioritized Syntax-Based Entailment, In: Proceedings of the 13th International Joint Conference on Artificial Intelligence, pp. 640–647, Morgan Kaufmann, Chambery, France (1993).

  • [9] Benferhat S, Dubois D, Prade H, Some Syntactic Approaches to the Handling of Inconsistent Knowledge Bases: A Comparative Study Part 1: The Flat Case, Studia Logica, Volume 58, page 17–45 (1997).
  • [10] Nebel B, Base Revision Operations and Schemes: Semantics, Representation and Complexity, In: ECAI, pp. 341–345 (1994).
  • [11] Lembo D, Lenzerini M, Rosati R, Ruzzi M, Savo D F, Inconsistency-tolerant query answering in ontology-based data access, Journal of Web Semantics, Volume 33, page 3–29 (2015).
  • [12] Lembo D, Lenzerini M, Rosati R, Ruzzi M, Savo D F, Inconsistency-Tolerant Semantics for Description Logics, In: Fourth International Conference of Web Reasoning and Rule Systems RR, pp. 103–117, Springer, Bressanone/Brixen, Italy (2010).
  • [13] Rescher N, Manor R, On inference from inconsistent premisses, Theory and Decision, Volume 1, page 179-217 (1970).
  • [14] Brewka G, Preferred Subtheories: An Extended Logical Framework for Default Reasoning, In: Proceedings of the 11th International Joint Conference on Artificial Intelligence, pp. 1043–1048, Morgan Kaufmann publisher, Detroit, MI, USA (1989).
  • [15] Benferhat S, Bouraoui Z, Tabia K, How to Select One Preferred Assertional-Based Repair from Inconsistent and Prioritized DL-Lite Knowledge Bases?, In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, pp. 1450–1456, AAAI Press publisher, Buenos Aires, Argentina (2015).
  • [16] Hamdi G, Omri M N, Papini O, Benferhat S, Bouraoui Z, Querying DL-lite Knowledge Bases from Hidden Datasets, In: International Symposium on Artificial Intelligence and Mathematics, Fort Lauderdale, Florida, USA (2018).
  • [17] Benferhat S, Dubois D, Prade H, Representing Default Rules in Possibilistic Logic, In: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning, pp. 673–684. Morgan Kaufmann publisher, Cambridge, MA, USA (1992).
  • [18] Qi G, Jeff Z P, Qiu J, Extending description logics with uncertainty reasoning in possibilistic logic., In: European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty. Springer, Berlin, Heidelberg, 2007.
  • [19] Qi G, Jeff Z Pan, Qiu J, A Possibilistic Extension of Description Logics., Description Logics. 2007.
  • [20] Dubois D, Prade H, Possibility theory, Springer, Boston, MA (2007).
  • [21] Dubois D, Lang J, Prade H, Possibilistic logic, Oxford University press, New York, NY, USA (1994).
  • [22] Dubois D, Prade H, Epistemic Entrenchment and Possibilistic Logic, Volume 50, page 223–239 (1991).
  • [23] Benferhat S, Bouraoui Z, Loukil Z, Min-Based Fusion of Possibilistic DL-Lite Knowledge Bases, In: International Conferences on Web Intelligence, pp. 23–28, IEEE Computer Society, Atlanta, GA, USA (2013).
  • [24] Qi G, Ji Q, Pan J Z, Du J, Extending description logics with uncertainty reasoning in possibilistic logic, International Journal of Intelligent Systems, Volume 26, page 353–381 (2011).
  • [25] Telli A., Benferhat S, Bourahla M, Bouraoui Z, Polynomial Algorithms for Computing a Single Preferred Assertional-Based Repair, Kunstliche Intelligenz, Volume 31, page 15–30 (2017).
  • [26] Calvanese D, De Giacomo G, Lembo D, Lenzerini M, Rosati R, DL-Lite: Tractable Description Logics for Ontologies, In: Proceedings, The Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference, pp. 602–607, AAAI Press / The MIT Press, Pittsburgh, Pennsylvania, USA (2005).
  • [27] Martinez M V, Parisi F, Pugliese A, Simari G I, Subrahmanian V S, Inconsistency Management Policies, In: Principles of Knowledge Representation and Reasoning: Proceedings of the Eleventh International Conference, pp. 367–377, AAAI Press, Sydney, Australia (2008).
  • [28] Staworko S, Chomicki J, Marcinkowski J, Prioritized repairing and consistent query answering in relational databases, Annals of Mathematics and Artificial Intelligence, Volume 64, page 209–246 (2012).
  • [29] Du J, Qi G, Shen Y, Weight-based consistent query answering over inconsistent over inconsistent SHIQ knowledge base, Knowledge and Information Systems, Volume 34, page 335-371 (2013).
  • [30] Benferhat S, Dubois D, Prade H, Some syntactic approaches to the handling of inconsistent knowledge bases: a comparative study. Part 2: the prioritized case, Physica-Verlag, Heidelberg, Volume 24, page 473–511 (1998).
  • [31] Lutz C, Seylan I, Toman D, Wolter F, The combined approach to OBDA: taming role hierarchies using filters, In: 12th International Semantic Web Conference, pp. 314–330, Springer, Sydney, NSW, Australia (2013).
  • [32] Everaere P, Konieczny S, Marquis P, Disjunctive merging: Quota and Gmin merging operators, Artificial Intelligence Journal, Volume 174, page 824–849 (2010).
  • [33] Konieczny S, Pérez R P, On the Frontier between Arbitration and Majority, In: Proceedings of the Eights International Conference on Principles and Knowledge Representation and Reasoning, pp. 109–120, Morgan Kaufmann, Toulouse, France (2002).
  • [34] Jinxin L, Mendelzon A O, Knowledge Base Merging by Majority, In: Dynamic Worlds From the Frame Problem to Knowledge Management, pp. 195–218, Springer Netherlands, Toulouse, France (2002).
  • [35] Revesz P Z, On the Semantics of Arbitration, International Journal of Automation and Computing, Volume 7, page 133–160 (1997).
  • [36] Baral C, Kraus S, Minker J, Combining Multiple Knowledge Bases, IEEE Transactions on Knowledge and Data Engineering., Volume 3, page 208–220 (1991).
  • [37] Benferhat S, Dubois D, Prade H, How to infer from inconsistent beliefs without revising?, In: IJCAI, pp. 1449–1457, Morgan Kaufmann (1995).
  • [38] Bloch I, Hunter A, Appriou A, Ayoun A, Benferhat S, Besnard P, Cholvy L, Cooke R M, Cuppens F, Dubois D, Fargier H, Grabisch M, Kruse R, Lang J, Moral S, Prade H, Saffiotti A, Smets P, Sossai C, Fusion: General concepts and characteristics, International Journal of Intelligent Systems, Volume 16, page 1107–1134 (2001).
  • [39] Benferhat S, Bouraoui Z, Lagrue S, Rossit J, Min-based Assertional Merging Approach for Prioritized DL-Lite Knowledge Bases, In: 8th International Scalable Uncertainty Management Conference, Volume 8720, pp. 8–21, Springer, Oxford, UK (2014).
  • [40] Moguillansky M, Falappa M A, A Non-Monotonic Description Logics Model for Merging Terminologies, Inteligencia Artificial, Revista Iberoamericana de Inteligencia Artificial, Volume 11, page 77–88 (2007).
  • [41] Wang Z, Wang K, Jin Y, Qi G, OntoMerge A System for Merging DL-Lite Ontologies, ICEUR Workshop Proceedings, Volume 969, page 16–27 (2014).
  • [42] Decker H, Basic Causes for the Inconsistency Tolerance of Query Answering and Integrity Checking, In: Database and Expert Systems Applications, DEXA, International Workshops, pp. 318–322, IEEE Computer Society, Bilbao, Spain (2010).
  • [43] ten Cate B, Halpert R L, Kolaitis P G, Practical Query Answering in Data Exchange Under Inconsistency-Tolerant Semantics, In: Proceedings of the 19th International Conference on Extending Database Technology, pp. 233–244, OpenProceedings.org, Bordeaux, France (2016).
  • [44] Baral C, Kraus S, Minker J, Subrahmanian V S, Combining Knowledge Bases Consisting of First Order Theories, In: 6th International Symposium of Methodologies for Intelligent Systems ISMIS, pp. 92–101, Springer, Charlotte, N.C., USA (1991).
  • [45] Bienvenu M, Rosati R, Tractable Approximations of Consistent Query Answering for Robust Ontology-based Data Access, In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, pp. 775–781, IJCAI/AAAI, Beijing, China (2013).
  • [46] Meghyn Bienvenu M, Camille B, and Francois G. Querying inconsistent description logic knowledge bases under preferred repair semantics. In Brodley and Stone (2014), pages 996-1002.
  • [47] Benferhat S, Bouraoui Z, Croitoru M, Papini O, Tabia K, Non-Objection Inference for Inconsistency-Tolerant Query Answering, In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence IJCAI, pp. 3684–3690, IJCAI/AAAI Press, New York, USA (2016).
  • [48] Baget J, Benferhat S, Bouraoui Z, Croitoru M, Mugnier M, Papini O, Rocher S, Tabia K, A General Modifier-Based Framework for Inconsistency-Tolerant Query Answering, In: Principles of Knowledge Representation and Reasoning: Proceedings of the Fifteenth International Conference KR, pp. 513–516, AAAI Press, Cape Town, South Africa (2016).
  • [49] Boughammoura R, Omri M N, Querying deep web data bases without accessing to data, In: 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery, pp. 597–603, IEEE, Guilin, China (2017).
  • [50] Boughammoura R, Omri M N, Hlaoua L, Information Retrieval from Deep Web Based on Visual Query Interpretation, International Journal of Information Research and Review IJIRR, Volume 2, page 45–59 (2012).
  • [51] Boughammoura R, Hlaoua L, Omri M N, G-Form: A Collaborative Design Approach to Regard Deep Web Form as Galaxy of Concepts, In: 12th International Conference of Cooperative Design, Visualization, and Engineering, pp. 170–174, Springer, Mallorca, Spain (2015).
  • [52]

    Manning C D, Schutze H, Foundations of statistical natural language processing, MIT Press, (2001).

  • [53] Raghavan V, Bollmann P, Jung G S, A Critical Investigation of Recall and Precision as Measures of Retrieval System Performance, ACM Transactions on Information Systems, Volume 7, page 205–229 (1989).