A Forgetting-based Approach for Reasoning with Inconsistent Distributed Ontologies

Abstract

In the context of multiple distributed ontologies, we are often confronted with the problem of dealing with inconsistency. In this paper, we propose an approach for reasoning with inconsistent distributed ontologies based on concept forgetting. We firstly define concept forgetting in description logics. We then adapt the notions of recoveries and preferred recoveries in propositional logic to description logics. Two consequence relations are then defined based on the preferred recoveries.

Introduction

Ontologies play a central role for the success of the Semantic Web as they provide shared vocabularies for different domains, such as Medicine. Web ontologies in practical applications are usually distributed and dynamic. One of the major challenges in managing these distributed and dynamic ontologies is to handle potential inconsistencies introduced by integrating multiple distributed ontologies.

For inconsistency handling in single, centralized ontologies, several approaches are known, see e.g. [ 10 ]. There are two main ways to deal with inconsistent ontologies [ 10 ]. One way is trying to live with the inconsistency and to apply a non-standard reasoning method to obtain meaningful answers [ 11, 14 ]. For example, a general framework for reasoning with inconsistent ontologies based on concept relevance was proposed in [ 11 ]. The second way to deal with logical contradictions is to resolve logical modeling errors whenever a logical problem is encountered. For example, several methods have been proposed to debug erroneous terminologies and have them repaired when inconsistencies are detected [ 17, 16 ].

At the same time, formalisms for modular ontologies – such as Distributed Description Logics (DDL) [ 4 ] and E -Connections [ 15 ] – provide an opportunity to handle multiple and distributed ontologies. Our approach is based on DDLs, where a special interpretation, called a hole, is proposed to interpret locally inconsistent ontologies [ 4, 19 ]. However, this approach is not fine-grained to deal with inconsistency that arises due to several connected local sources.

In [ 12 ], an approach based on variable forgetting is proposed to resolve inconsistency in a propositional belief base. The basic idea of this approach is to restore consistency of a belief base by forgetting propositional variables in some formulas of the belief base. It has been shown that this approach provides a general and flexible framework that encompasses several previous approaches for handling inconsistency as specific cases, such as coherence-based approaches for reasoning from preferred consistent subsets and approaches for merging multiple sources of information.

In this paper we define an approach for handling inconsistencies in distributed ontologies based on concept forgetting. We first define concept forgetting in description logics. We then adapt the notion of recoveries and preferred recoveries in [ 12 ] to description logics. Based on the preferred recoveries, two forgetting consequence relations are defined to draw nontrivial conclusions from inconsistent distributed ontologies. Our approach can generally be applied to any formalism for distributed ontologies. However, in this paper we present our approach in terms of DDL for two reasons. First, DDL has been proven to be a nice framework to represent distributed ontologies [ 2 ]. Second, there is some work which discusses inconsistency handling in DDL [ 19, 6 ], whilst relatively little work on dealing with inconsistency has been given for alternative approaches. By comparing our approach with existing approaches, we can show that our approach is indeed sufficiently powerful to handle inconsistency in distributed ontologies.

The rest of this paper is organized as follows. We first review DDL and variable forgetting in Section 2 and discuss related work in Section 5. We then introduce the notion of forgetting in description logic ALC in Section 3. We then adapt the notions of recoveries and preferred recoveries in propositional logic to DDL in Section 4. We finally conclude the paper and provide future work in Section 6. 2 2.1

Preliminaries

Distributed Description Logics

In this paper, we presume that the reader is familiar with Description Logics (DLs) and we refer to Chapter 2 of the DL handbook [ 1 ]. We consider the well-known DL ALC [ 18 ]. Let NC and NR be pairwise disjoint and countably infinite sets of concept names and role names respectively. We use the letters A and B for concept names, the letter R for role names. > and ? denote the top concept and the bottom concept respectively. The set of ALC concepts is the smallest set such that: (1) every concept name is a concept; (2) if C and D are concepts, R is a role name, then the following expressions are also concepts: :C (full negation), CuD (concept conjunction), CtD (concept disjunction), 8R:C (value restriction on role names) and 9R:C (existential restriction on role names). A DL knowledge base (or ontology)1 KB consists of a TBox T and an ABox A. A TBox is a finite set of terminology axioms which are of the form CvD, where C and D are concepts. An ABox is a finite set of assertions which is of the form C(a) or R(a; b), where C is a concept, R is a role, and a; b are individuals.

Distributed Description Logics (DDL) [ 4 ] adopt a linking mechanism. It is a welldeveloped logical language for representing both ontologies and context [ 5 ]. In DDL, the distributed knowledge base (DKB), D = hfKBigi2I ; Bi consists a set of local knowledge bases fKBigi2j in ALC and bridge rules B = fBij gi6=j2I that represent the correspondences between their terminologies. The semantic mappings between modules KBi and KBj are represented in terms of axioms of the following form: 1 In this paper, we consider a DL knowledge base as an ontology and use them interchangeably. – INTO: i : C v! j : D hfKBigi2I ; Bi if

– ONTO: i : C w! j : D The semantics of DKB is given by the interpretation J = (fIgi2I ; frij gi6=j2I ) that consists of standard DL interpretations Ii of KBi, and binary relations rij Ii

Ij . We use rij (X) to denote [x2X fy 2 Ij : (x; y) 2 rij g for any X Ii . A distributed interpretation J d-satisfies (denoted as J j=d) the element of the DKB D = – Ii j=d AvB for all A v B in KBi – J j=d i : C v! j : D, if rij (CIi ) – J j=d i : C w! j : D, if rij (CIi ) DIj (ONTO) – J j=d D, if for every i; j2I, J j=d KBi and J j=d Bij ,

DIj (INTO)

where Ii and Ij are local interpretations of KBi and KBj , respectively, C; D are concepts, rij (called domain relation) is a relation that represents an interpretation of Bij . Finally, D j=d i : C v D if for every J, J j=d D implies J j=d i : C v D. More details of the semantics of DDL can be found in [ 4 ].

A DKB is inconsistent if there is no interpretation J which satisfies all KBi, where i 2 I. Given a DKB D = hfKBigi2I ; Bi, a DKB D0 = hfKB0igi2I ; B0i is a sub-base of D, denoted D0vD, iff, KBi KBi for all i2I and B0 Bi.

0 i 2.2

Variable forgetting

Let L be a propositional language constructed from a finite alphabet P of propositional symbols, the Boolean constants > (true) and ? (false), using the usual operators : (not), _ (or) and ^ (and). The classical consequence relation is denoted by `. Given a propositional formula , V ar( ) denotes the set of propositional variables occuring in . x 0 (resp. x 1) denotes the formula obtained by replacing in every occurence of variable x by ? (resp. >).

Variable forgetting, which is also known as projection, is originally defined in [ 13 ] and is applied to resolve inconsistency in [ 12 ].

Definition 1. Let be a formula over P and V P be a set of propositional symbols. The forgetting of V in , denoted as Forget(V; ), is a formula over P , defined inductively as follows: – Forget(?; ) ; – Forget(fxg; ) x 0 _ x 1; – Forget(fxg[V; ) Forget(V; Forget(fxg; )).

Forget(V; ) is the logically strongest consequence of which is independent of V . By forgetting a set of variables in a formula we get a formula which is inferentially weaker than , i.e. ` Forget(V; ). For example, Forget(fag; :a^b) b and Forget(fag; a_b) >.

Forgetting in Description Logics

In this section, we define forgetting in a TBox of a DL knowledge base. The notion of forgetting can be applied to any description logic. Here, a concept name plays the same role as a variable in propositional logic. In particular, we define the forgetting of a set of concept names in a concept. We use Con(C) to denote the concept names in a concept C. CA > (resp. CA ?) denotes the concept obtained by replacing in C every occurrence of concept name A by > (resp. ?).

A straightforward way to forget a concept name A in a concept C can be given as follows: Forget(fAg; C)=CA >tCA ?. However, this definition suffers from a problem. That is, by forgetting a concept name A in a concept C in a TBox T , we do not necessarily obtain a concept which is logically weaker than C, i.e. T j= C v Forget(fAg; C), a very important property of forgetting. The main reason for this problem is that the DL concept constructors include value restrictions on role names and existential restriction on role names. Let us look at an example. Suppose C = 8R:(Au8R0::A) and we want to forget A. Since CA > = 8R:(>u8R0:?) = 8R:(8R0:?) and CA ? = 8R:(?u8R0:>) = 8R:?, we have Forget(fAg; C) = (8R:(8R0:?)) t 8R:?. In this case, we do not necessarily have T j= C v Forget(fAg; C).

We consider the following definition of forgetting.

Definition 2. Let C be a (complex) concept and A a concept name. Suppose C is in negation normal form, i.e. negation can only appear in the front of a concept name. The forgetting of A in C, denoted Forget(fAg; C), is a concept obtained by replacing every occurrence of A and :A by >.

In Definition 2, it is very important that the concept C is transformed into its negation normal form before we apply the forgetting operator to it. Otherwise, we cannot ensure that CvForget(fAg; C). For example, suppose C = :(A1uB1) and we want to forget A1. If we replace A1 by > without transforming C into its negation normal form, then C0 = :B1, which is subsumed by C.

We give a running example to illustrate the notion of forgetting.

Example 1. Let us consider a TBox

T = fProfessortPostDoctortResearchAssistantvEmployee, PhDStudentv ResearchAssistant, PhDSupervisor Professoru9isSupervisorof:PhDStudentg. Then, Forget(fProfessorg; ProfessortPostDoctortResearchAssistant) = > and Forget(fProfessorg; Professoru9isSupervisorof:PhDStudent) = 9isSupervisorof: PhDStudent.

The following proposition tells us that the forgetting of A in C indeed results in a concept which is a super concept of C.

Proposition 1. Let C be a concept in a TBox T and A be a concept name. Then T j= CvForget(fAg; C).

We now define the forgetting of a set of concept names in a concept.

Definition 3. Let C be a concept, and let SN be a finite set of concept names. The forgetting of SN in C, denoted Forget(SN ; C), is defined inductively as follows: – Forget(?; C)=C; – Forget(fAg[SN ; C) = Forget(SN ; Forget(fAg; C)).

Example 2. (Example 1 Continued) Let C = Professoru9isSupervisorof:PhDStudent. Suppose we want to forget Professor and PhDStudent in C, i.e., SN = fProfessor; PhDStudentg, then we have Forget(SN ; C) = 9isSupervisorof:>.

Unlike the forgetting notion in propositional theories, we do not have a corresponding model theoretic semantics for concept forgetting. However, we have the following properties which precisely characterize our notion of concept forgetting. Proposition 2. Let C, D and E be concepts, and let A and A0 be two concept names. We then have 1. If C = A or :A, then Forget(fAg; C) = >; 2. If C = DuE (or DtE), where A2Con(D) and A62Con(E), then Forget(fAg;

C) = Forget(fAg; D)uE (or Forget(fAg; C) = Forget(fAg; D)tE). 3. If C = AtE, then Forget(fAg; C) = >.

The proof of Proposition 2 is clear by the definition of forgetting. Item 1 says that if C is a concept name or full negation of a concept name, then by forgetting it we get a top concept. Item 2 shows that forgetting a concept name A is a concept which is a conjunction (or disjunction) will not influence the conjunct (or disjunct) which is irrelevant to the concept name. Item 3 is a disjunction of the form AtE, then by forgetting A we get a top concept.

We now consider the weakening of a TBox T by forgetting a concept name A. A first thought would be that we simply forget A in every concept appearing in T . However, this approach may cause some problems. Suppose a terminology axiom CvD is in T such that C = AuB, and A62Con(D) and A62Con(B). By forgetting A in C we get a new axiom BvD. It is easy to check that BvD infers CvD whilst the converse does not always hold. Therefore, CvD is not weakened. On the contrary, it is reinforced. To solve this problem, we can equivalently transform every axiom of the form CvD into >v:CtD. According to Proposition 1, we have the following corollary.

Corollary 1. Let be a terminology axiom of the form > v C and A a concept name. Suppose A2Con(C), then >vC j= >vForget(fAg; C), but not vice versa. Corollary 1 tells us that an axiom is weakened by forgetting a concept name in the right-side concept.

We give the notion of forgetting in a TBox.

Definition 4. Let T be a TBox. Let be a terminology axiom in T which has been transformed into the form > v C and SN a finite set of concept names. The forgetting of SN in is defined as Forget(SN ; ) = >vForget(SN ; C) and the forgetting of SN in T is defined as Forget(SN ; T ) = fForget(SN ; ) : 2T ; Sig( )\SN 6= ?g, where Sig( ) denotes the signature of .

By Corollary 1, it is clear that we have that every axiom in Forget(SN ; T ) can be inferred from T , for any SN .

Example 3. (Example 1 Continued) Suppose we want to forget Professor in T . We need some pre-processing. First, ProfessortPostDoctortResearchAssistantv Employee is transformed into > v (:Professor u :PostDoctor u :ResearchAssistant) t Employee. Second, we split PhDSupervisor Professoru9isSupervisorof:PhDStudent into PhDSupervisor v Professoru9isSupervisorof:PhDStudent and

Professoru9isSupervisorof:PhDStudent v PhDSupervisor, then transform them into > v :PhDSupervisor t (Professor u 9isSupervisorof:PhDStudent) and > v :Professor t :9isSupervisorof:PhDStudent t PhDSupervisor. Therefore, we have Forget(fProfessorg; T ) = f> v (:PostDoctor u :ResearchAssistant) t Employee; > v :PhDSupervisor t 9isSupervisorof:PhDStudent

PhDStudentvResearchAssistantg: 4

Recoveries for Distributed DL Knowledge Bases

In this section, we apply the forgetting-based approach for resolving inconsistency in a propositional knowledge base [ 12 ] to deal with inconsistency in a DKB by forgetting concepts in TBoxes of local ontologies. We do not consider inconsistency which is caused only by inter-ABoxes as this kind of inconsistency may be better addressed by performing some kind of weakening on the ABoxes of local ontologies. Dealing with this problem is also important, but it is out of scope for this paper. 4.1

Recoveries

In [ 12 ], two key notions of the approach are forgetting context and recovery vector. The former one is a collection of sets of variables to be forgotten in each formula from the knowledge base, and the latter one is a vector of subsets of the set of all the variables appearing in the knowledge base, which satisfies the constraints in the forgetting context. Forgetting of the variables in the recovery vector will result in a consistent knowledge base. We adapt the definitions of forgetting context and recovery vector to DDL.

In the following, we assume that I = [n] where n2N. The forgetting context in DDL is defined as follows.

Definition 5. Given a DKB D = hfKBigi2I ; Bi, where KBi = hRi; Ti; Aii, a forgetting context CD for it is a triple hF; G; Hi where: – F = hFiii2I such that Fk NC with k2I, and for each i2I, Fi is the set of concept names that can be forgotten in Ti. – G = hGiii2I such that for each i, Gi = f(Ai; A0i) : Ai; A0i2CN (Ti)g, where CN (Ti) is the set of all concept names in Ti and a pair (Ai; A0i) in Gi means that A0i is meaningful only if Ai exists; – H NC f1; :::; ng NC f1; :::; ng is a set of quadruples (A; i; B; j) such that A2Fi, B2Fj and i6=j, which means that if A is forgotten in Fi, then B must be forgotten as well in Fj .

Fi imposes the constraints over the concept names which can be forgotten in Ti. That is, if a concept name A62Fi, then forgetting A in Ti is forbidden. Gi contains the pairs of concept names in local TBox Ti such that the existence of the former is meaningful or significant only in presence of the latter. In other words, forgetting the latter imposes to forget the former. H imposes the constraints over inter-TBoxes. It forces that if A is forgotten in Ti then B should also be forgotten in Tj . The forgetting context is dependent on the application at hand. A simple example for forgetting context is a standard forgetting context which is given by Fi = NC for every i = 1; :::; n, Gi = ? for every i = 1; :::; n, and H = ?. Note that our approach does currently not include weakening of bridge rules. Such a weakening may be preferable in some situations (for example, to improve mappings created by different matching systems [ 6 ]), but its treatment is out of scope for this paper.

We next define a recovery vector in DDL. It is a vector of subsets of concept names of NC , which satisfies the constraints in the forgetting context.

Definition 6. Given a DKB D = hfKBigi2I ; Bi, where KBi = hRi; Ti; Aii, and a forgetting context CD, a recovery vector (or recovery for short) !V for D given CD is a vector !V = hV1; :::; Vni of subsets of NC s.t.: – for every i2I, Vi Con(Ti), – for every i2I, for any (A; A0)2Gi, if A2Vi then A02Vi, – for every (A; i; B; j)2H, A2Vi implies B2Vj , – Dj !V = hfhForget(Vi; Ti); Aiigi2I ; Bi is consistent.

Dj !V is called the cure of !V for D given CD. We denote the set of all recoveries for D given CD as RCD (D).

According to the definition of recovery, the cure of !V for D is always consistent if the recovery !V exists. When RCD (D) = ?, we say that D is not recoverable w:r:t: CD. This may happen when some concepts, which cannot be forgotten according to the forgetting context, must be forgotten to restore consistency.

We give an example to illustrate the definitions in this section.

Example 4. Suppose we have a

DKB D = hfKB1; KB2g; fB12; B21gi, where (1) KB1 = hR1; T1; A1i : – R1 = ?; – T1 = fProfessortPostDoctortResearchAssistantvEmployee,

PhDStudentvResearchAssistant,

PhDSupervisor Professoru9isSupervisorof:PhDStudentg; – A1 = fPhDStudent(M ary); PhDSupervisor(T om)g; (2) KB2 = hR2; T2; A2i : – R2 = ?; – T2=fProfessortLecturervAcademicSta ; StudentuAcademicSta v?; PhDSupervisor AcademicSta u9isSupervisorof:PhDStudent;

PhDStudentvStudentg; – A2 = fPhDStudent(J ohn)g; (3) B21 = f2 : AcademicSta

w! 1 : Employee; 2 : PhDStudent w! 1 : PhDStudent; 2 : :AcademicSta v! 1 : :Employee; 2 : :PhDStudent v! 1 : :PhDStudentg and B12 = ?. KB1 and KB2 are two University ontologies which are connected by bridge rules.

Let D0 = hfT1; T2g; fB12; B21gi. Since D0 j=d PhDStudentv? and PhDStudent (M ary)2A1, D is inconsistent. We have the following forgetting context CD for D: – F =hFiii=1;2, where Fi= fAcademicSta ; Employee; Student; Professor; Lecturer;

ResearchAssistant; PhDStudent; PhDSupervisorgi=1;2. – G=hGiii=1;2, where

G1=f(PhDStudent; PhDSupervisor)g; G2=f(PhDStudent; PhDSupervisor)g. – H = f(PhDStudent; 1; PhDStudent; 2)g.

For each local ontology, all the concept names in it can be forgotten. G1 is used to show that the concept PhD supervisor is not meaningful if the concept PhD student or professor is forgotten. G2 can be similarly explained. H ensures that the concepts PhD student, professor, student in KB1 are respectively equal to PhD student, professor, student in KB2 w:r:t: weakening from the point of view of KB2.

Given the forgetting context CD, the following are two recoveries for D: – !V1 = hV1; V2i, where

V1 = fResearchAssistantg and V2 = ?: Forget(V1; T1) = fProfessortPostDoctorvEmployee; PhDSupervisor Professoru9isSupervisorof:PhDStudentg, Forget(V2; T2) = T2 It is easy to check that the DKB D1 = hfhForget(Vi; Ti); Aiigi=1;2; Bi is consistent. – !V2 = hV1; V2i, where

V1 = ? and V2 = fPhDStudent; PhDSupervisorg: Forget(V1; T1) = T1, Forget(V2; T2) = fProfessortLecturervAcademicSta ; StudentuAcademicSta v?g The DKB D2 = hfhForget(Vi; Ti); Aiigi=1;2; Bi is consistent.

CD.

However, the following vector is not a recovery for D given CD: !V = hV1; V2i, where V1 = fPhDStudentg and V2 = ?, because it does not satisfy the constraints of H in

Given a forgetting context, there may exist several different recoveries. In many cases, we are only interested in some of these recoveries, called preferred recoveries, which is defined by some preference relation on recoveries. For example, among all the recoveries, we may be only interested in those contain minimal number of concepts to forgotten. The notion of preferred recovery is originally defined in [ 12 ] in propositional logic. We adapt it to DDLs here. We need to define a preference relation on the set of all recoveries for a DKB D given a forgetting context CD for it. For any two recoveries !V and !V0 of length n, !V c !V0 means that Vi Vi0 for each i21; :::; n. Definition 7. Given a DKB D and a forgetting context CD for it, a preference relation on RCD (D) is a reflexive and transitive relation which satisfies the following property: 8 !V; !V02RCD (D); if !V c !V0; then !V !V0: As usual, we denote !V !V0 !V.

!V0 for !V !V0 and !V06 !V, and !V !V0 for !V !V0 and In Definition 7, a recovery !V is at least preferred to another one !V0 ( !V !V0) if each Vi in !V is a subset of Vi0 in !V0.

There are many ways to define a preference relation. For instance, we can prefer recoveries that lead to forget minimal sets of concepts w:r:t: the set containment. We can also ask the experts or users for such a preference relation. We adapt two specific preference relations in [ 12 ] as follows.

– binaricity preference relation

(Vi6=? , Vi06=?), then !V bin !V0. – ranking function based preference relation : suppose is a ranking function from RCD (D) to N such that (h?; :::; ?i) = 0, and 8 V ; !V02RCD (D), if !V ! then ( !V) ( !V0). The preference relation induced by is the total preordering defined as follows. For all !V; !V02RCD (D), !V !V0 if and only if ( !V) ( !V0). c !V0, bin: 8 !V; !V02RCD (D), if for every i = 1; :::; n, The ranking function based preference relation is defined by a ranking function from RCD (D) to N. A particular ranking function can be defined by card( !V) = j S !Vj, where S !V = V1[:::[Vn. We denote the preference relation based on card as card. A recovery !V is preferred to another one !V0 w:r:t: the preference relation card if and only if the cardinality of the union of the sets Vi (i = 1; :::; n) is smaller than that of the union of the sets Vi0 (i = 1; :::; n).

By Corollary 1, when a concept name is forgotten in a DL knowledge base, the resulting DL knowledge base is weaker than the original one w:r:t: the inference power. Given !V; !V02RCD (D), if !V c !V0 then Dj !V j= Dj !V0. Therefore, among all the recoveries from RCD (D), those which are minimal w:r:t: c lead to cures preserving as much information as possible given the forgetting context CD.

Next, we define two consequence relations. We denote by !VD;CD the set of all minimal recoveries from RCD (D) w:r:t: .

Definition 8. Given a DKB D and a forgetting context CD for it, let be a preference relation on RCD (D). Let be a DL axiom. Then is said to be skeptically inferred from D w:r:t: , denoted by D j=CD;Ske , if and only if Dj !V j= for all !V2 !VD;CD . is called a skeptical consequence of D.

A DL axiom is a skeptical consequence of a DKB D if and only if it can be inferred from all the most preferred recoveries.

Example 5. (Example 4 Continued) Suppose = card. Then !V1 is a preferred recovery because j S !V1j = 1 and no other recovery can have cardinality smaller than it. Other preferred recoveries are given as follows: – !V3 = hV1; V2i, where V1 = fEmployeeg and V2 = ?. We have

Forget(V3; T1) = fPhDStudentvResearchAssistant; PhDSupervisor Professoru9isSupervisorof:PhDStudentg and Forget(V3; T2) = T2. – !V4=hV1; V2i, where V1=? and V2=fAcademicSta g. We have Forget(V4; T1) = T1 and Forget(V4; T2)=fPhDStudentvStudent; PhDSupervisor vAcademicSta u9isSupervisorof:PhDStudentg. – !V5 = hV1; V2i, where V1 = ? and V2 = fStudentg. We have Forget(V5; T1) = T1 and Forget(V5; T2) = fProfessortSeniorLecturertLecturertResearchAssistant vAcademicSta g.

Since PhDSupervisor Professoru9isSupervisorof:PhDStudent is in Forget(Vi; Ti) for all i = 1; 3; 4; 5, we have that Dj !V j= Professor(T om), for all !V2 !VD;CD . That is, we can conclude that T om is a professor using the skeptical inference.

Suppose D = hfKBigi2I ; Bi is a DKB. A maximal consistent sub-base of D w:r:t terminology is a sub-base D0 = hfKB0igi2I ; Bi (with KB0 = hR0; T 0; A0i) of D which satisfies the following conditions: (1) D0 is consistent; (2) fTi0 : Ti0 6= ?g fTi : i = 1; :::; ng, R0 = R, and A0i = Ai; (3) there does not exist a sub-base D00 of D which satisfies (1) and (2), and fTi0 : Ti0 6= ?g fTi00 : Ti00 6= ?g.

Proposition 3. Let D be a DKB and MaxCons(D) the set of all maximal consistent sub-bases of D. Suppose CD is the standard forgetting context and the preference relation is a binaricity preference relation. Then for every DL axiom , D j=CD;Ske if and only if Di j= , for all Di2MaxCons(D).

Proposition 3 tells us that the skeptical consequence relation based on binaricity preference relation is the same as the consequence relation based on maximal consistent sub-bases w:r:t terminology.

In the case that many preferred recoveries exist (see Example 4), the skeptical consequence relation is rather weak (i.e., only few DL axioms can be inferred). Therefore, we propose another consequence relation.

Dj !V0 j= : . is called an argumentative consequence of D.

Definition 9. Given a DKB D and a forgetting context CD for it, let be a preference relation on RCD (D). Let be a DL axiom and : is its negation2. Then is said to be argumentatively inferred from D w:r:t: , denoted by D j=CD;Arg , if and only if there exists a !V2 !VD;CD such that Dj !V j= and there does not exist !V02 !VD;CD such that A DL axiom is an argumentative consequence if and only if it can be inferred from a most preferred recovery and its negation cannot be inferred from any most preferred recoveries. Therefore, the argumentative consequence relation will not result in contradictory conclusions. Note that our argumentative consequence relation is different from consequence relation defined in the logic-based framework for argumentation [ 3 ] because a most preferred recovery can be viewed as maximal recovered consistent information in the DKB whilst an argument in an argumentation theory is a minimal consistent subset that can infer a conclusion. Unlike an argument in the argumentation theory, each axiom in a most preferred recovered is self-justified. A weakness of the argumentative consequence relation is that it may be too adventurous because an argumentative consequence may not be inferred by other most preferred recoveries (also its negation will not be inferred). The relationship between this consequence relation and the skeptical consequence relation is summarized by the following proposition. Proposition 4. Given a DKB D and a forgetting context CD for it, suppose axiom. If D j=CD;Ske , then D j=CD;Arg . is a DL That is, every skeptical consequence of a DKB is an argumentative consequence. Example 6. (Example 5 Continued) It is easy to check that, for i = 3; 4; 5, Dj !Vi j= ResearchAssistant(M ary) because PhDStudent(M ary)2A1 and PhDStudent v ResearchAssistant2Forget(Vi; T1). Since ResearchAssistant62 Con(Forget(V1; T1)) and ! ResearchAssistant62Con(T2), we must have that Dj V 16j=:ResearchAssistant(M ary). Therefore, D j=CD;Arg ResearchAssistant(M ary). 5

Related Work

To handle inconsistency in DDLs, some special interpretation, called a hole, is proposed to interpret locally inconsistent ontologies in DDLs [ 4, 19 ]. In DDL, a hole for an ontology is an interpretation I = h?; i, where the domain is empty. It is proposed to deal with inconsistency. By doing this, even if a local ontology is inconsistent, there still exists a global model for the distributed knowledge base. However, if a hole is applied to a local ontology, it is equivalent to say that this ontology and all related bridge rules are removed from the distributed knowledge bases (see discussions in [ 4, 19 ]). This all-or-nothing semantics is clearly not very advisable. Furthermore, when the inconsistency is caused by several connected local ontolgoies via mapping, it is not clear how to apply the hole interpretation to deal with the inconsistency. Although it is possible to translate an inconsistent DKB to a DL knowledge base and then apply the

2 The notion of axiom negation is defined in [9]

paraconsistent semantics given in [ 14 ] or [ 11 ] to infer non-trivial conclusions, this approach does not differentiate bridge rules and axioms in local ontologies and is different from our approach.

In [ 6 ], the authors deal with the problem of inconsistency in DDL by removing some bridge rules which are responsible for the inconsistency. The idea of their approach is similar to that of the approaches for debugging and repairing terminology in a centralized ontology. Our work takes a different view on inconsistency handling in DDL that the bridge rules are reliable and we weaken the axioms in local ontologies to restore consistency.

In [ 7 ], the notion of forgetting is given for a class of OWL/RDF(S) ontologies. The definition of forgetting is semantically meaningful. However, their definition of forgetting has some restriction on axioms in the ontologies. Moreover, the computational complexity of the algorithm for computing the forgetting literals may be exponential in the worst case [ 8 ]. In contrast, our definition of forgetting does not have any restriction on the axioms in an ontology and it can be computed in polynomial time. Although we do not have corresponding model-theoretical definition for our notion of forgetting, we justify our definition by analyzing its logical properties. Also, the focus of this paper is to apply the notion of forgetting to deal with inconsistency in DDL knowledge bases. 6

Conclusions and Future Work

In this paper, a forgetting-based approach for reasoning with inconsistent distributed ontologies was proposed. We first defined the notion of concept forgetting in DL ALC. Some properties of the concept forgetting were given. The definitions of recoveries and preferred recoveries in propositional logic setting [ 12 ] were adapted to DDL. Based on the preferred recoveries, two consequence relations on inconsistent D-KB were defined. One is called skeptical consequence and the other is called argumentative consequence. We showed that every skeptical consequence of a DKB is an argumentative consequence. Our definition of concept forgetting does not have a corresponding model theoretic semantics. One of the reasons that prevent us from defining concept forgetting by simply extending variable forgetting is that we allow value restrictions on role names and existential restriction on role names. We will consider a less expressive DL that allow neither value restrictions on role names nor existential restriction on role names and see if we can define a semantic meaningful notion of concept forgetting. We will also provide discuss complexity issues of our approach in another work.

Acknowledgments References

This work is partially supported by the EU under the IST project NeOn (IST-2006027595, http://www.neon-project.org/)

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