Revision and Reasoning on Revision Rules in DL-based Knowledge Bases

Transcription

1 Revision and Reasoning on Revision Rules in DL-based Knowledge Bases Chan Le Duc and Nhan Le Thanh Laboatoie I3S, Univesité de Nice - Sophia Antipolis, Fance cleduc@i3s.unice.f, Nhan.Le-Thanh@unice.f Abstact. Knowledge of evey application aea is heteogeneous and it develops in time. Theefoe, a combination of seveal fomalisms on which evision opeations ae defined, is equied fo knowledge epesentation of such a hybid system. In this pape, we will intoduce definition of evision opeations in DL-based knowledge bases and show how evision opeations ae computed fo Desciption Logics with existential estictions. Fom this, we can define evision ules whose consequent is a evision opeation. Such ules allow us to epesent context ules in tanslation between DL-based ontologies. Finally, we will intoduce a easoning on the hybid fomalism combining the evision ules and Desciption Logics. 1 Intoduction and Motivation Desciption Logics can be used as a fomalism fo ontology design in an application domain. In ode to educe the size of ontologies, diffeent ontologies fo diffeent subdomains o use pofiles ae deived fom a shaed common ontology. These deived ontologies shae atomic concepts, atomic oles and defined basic concepts in the common ontology. The shaed concepts can be edefined in a deived ontology with the aim of fitting an adaptable context. This is necessay to detemine the moe sufficient meaning of a shaed concept in the context of cuent uses. Owing to context ules, the edefining can be pefomed in un-time. This mecanism is specified in ebxml 1 famewok fo electonic commece [9]. Context ule defined in ebxml can be descibed as follows. Antecedent of a context ule is a pedicate which epesents context condition; consequent of ule is an opeation which tanslates a concept definition into anothe. Fo this eason, we need anothe fomalism to captue the semantics of such ontologies. It is a specific poduction ule, namely evision ule, whose consequent is an opeation allowing one to change a concept definition with the aim of shaing compehension about this concept. Moeove, since the definition of a concept in knowledge base can be modified by the application of a evision ule, evision opeations must be defined on DL-based teminology. A pofound study and a famewok fo evision opeations given in [2] togethe povide a good backgound fo the extension of these opeations to moe expessive DL languages. Nevetheless, thee ae seveal diffeent viewpoints concening 1 Electonic Business extensible Makup Language

2 the evision of DL-based knowledge base which ague both fo and against the pesevation of subsumption elationships in teminology. One viewpoint states that subsumption elationships ae deived knowledges and that they may be changed when concept definitions ae modified. Anothe viewpoint consides subsumption elationships as liteal and invaiant knowledges. That is, the subsumption elationships have neve been changed despite a change in concept definitions. It is obvious that teminology expansion cannot eplace teminology evision in this viewpoint. In fact, apat fom modifying a concept definition the popagation of this modification in all teminology is also equied. The following example shows the necessity of teminology evision and evision ules. Intoductoy Example. This example is extacted fom an exchange scenaio in ebxml [9]. Assume that thee ae two ontologies which ae deived fom a shaed basic ontology. In the the fist deived ontology that is defined by an Ameican acto, the following concept is intoduced: AntiPollutantPod := Poduct esist.noise. This concept is diffeently defined in the second deived ontology by an Euopean acto as follows: AntiPollutantPod := Poduct esist.fie esist.noise. To shae the compehension about the concept AntiPollutantPod when these actos exchange a document containing this concept, ebxml intoduces the following geometical context ule allowing us to change the definition of concept: 1 : soldby(x,y 1 ) boughtby(x,z 1 ) associate(y 1,Y 2 ) associate(z 1,Z 2 ) Euopean(Y 2 ) Ameican(Z 2 ) TELL(AntiPollutantPod, esist.fie); This ule states that if a poduct X is sold by a company Y 1 who has an Euopean associate, the poduct X is bought by a company Z 1 who has an Ameican associate, then the concept AntiPollutantPod should be intepeted as a poduct capable of esisting Fie. Moeove, we now have two vesions of the concept PodFoImpot as follows: In the Ameican acto s ontology: PodFoImpot := AntiPollutantPod contol.toxicity esist.heat In the Euopean acto s ontology: PodFoImpot := AntiPollutantPod contol.toxicity Simila to the ule 1, a shaed industial context ule is intoduced as follows : 2 : ConstuctionCo(Y 1 ) soldby(x,y 1 ) AntiPollutantPod(X) FORGET(PodFoImpot, esist.heat); This ule states that if a constuction company (ConstuctionCo) sells someone a poduct X which is AntiPollutantPod, the concept PodFoImpot should be intepeted as a poduct without the popety to esist the heat i.e. we have to delete the popety to esist the heat fom the definition of PodFoImpot. Futhemoe, the Ameican and Euopean actos shae a basic ontology which contains the following concepts: AmeicanCo:= associate.ameican; EuopeanCo:= associate.euopean; Fie:=Heat Cabonization; and pimitive concepts and oles. Assume that the actos use the same ABox as follows:{soldby(a,b); boughtby(a,c); AntiPollutantPod(a); EuopeanCo(b); AmeicanCo(c); Pollutant(Noise)}. Fom the assetions in the ABox and the definitions of concepts EuopeanCo and AmeicanCo, we obtain the following assetions: associate(a,b ), Euopean(b ), as-

3 sociate(a,c ), Ameican(c ). Thus, the ule 1 becomes active. It is obvious that the modification of concept definition AntiPollutantPod pefomed by 1 can lead to a modification of concept definition PodFoImpot. Futhemoe, assume that now the assetion ConstuctionCo(b) is added to the ABox. Hence, the ule 2 becomes active. Howeve, the assetion AntiPollutantPod(a) no longe may be veified following application of the ule 1 since a new facto ( esist.fie) is added to the definition of concept AntiPollutantPod. By consequent, we have two choices to make fo ule application ode: ( 1, 2 ) o ( 2, 1 ). It is obvious that the teminologies obtained fom these two choices ae diffeent. Howeve, if the application ode ( 2, 1 ) is chosen, the deived ontologies obtained become neae each othe since both of ules 2, 1 ae applied. Thee ae at least two poblems that aise fom this example: how to compute the evision opeations (TELL, FORGET) and how to eason on context ules so that the numbe of ules applied in an ode is maximum. By what follows, we will popose an appoach to these poblems. The second section of the pape will pesent AGM famewok and pojection of this famewok onto DL-based knowledge base evision. The next section will concentate on the computing of evision opeations fo the languages: EL (which contains only conjunction and existential estiction) and FLE (which is obtained fom EL by adding value estiction) by using the tee epesentation of FLE-concept deciptions developed in [6]. The section 3 will intoduce evision ules and easoning on DL-based knowledge base with evision ules. The last section will discuss extensions of the evision opeations to moe expessive DL languages which contain numbe estictions and disjunction. 2 DL-based Knowledge Base Revision 2.1 Fom AGM Famewok to DL-based Knowledge Base Revision If we conside knowledge base as a set of popositions (belief), efuting a poposition o accepting a new poposition coesponds espectively to contaction o evision opeation which allows us to add new knowledge to o to delete obsolete knowledge fom knowledge base (KB). Such a modification can geneate some contadictions in the new KB. In ode to guaantee consistency of the KB, some popositions must be deleted fom this. The authos of the wok in [1] (called AGM famewok) poposed a set of postulates that evision (expansion) o contaction opeations should satisify. Fom this set of postulates and some hypothesises concening elationships between evision and contaction, the authos also gave pecise definitions fo evision and contaction opeations on a set of popositions. Howeve, in a DL-based knowledge base, if evision opeation is consideed as addition of a new concept to teminology (TBox), evision and expansion opeations ae identical since addition of a concept to teminology does not cause any inconsistency. Unfotunately, modification of a concept definition in teminology cannot always be tanslated into teminology expansion since this modification should be popagated in all teminology. To show this claim, we need to fomulate evision and contaction

4 opeations on a TBox. In fact, if we use a DL language which allows us to both nomalize concept desciptions and ewite concept definitions as conjunctions of simple concepts, these opeations can be witten as follows: TELL(T, C, Exp) := C Exp FORGET(T, C Exp, Exp) := C whee T is a TBox, C is concept to be evised and Exp is a simple concept. In a TBox, a concept definition can depend on othe concept definitions. Fo a static KB on which evision opeatos do not exist, we can tansfom an acycle TBox into the equivalent unfolded TBox (which have the same models) in which all ight sides do not contain defined concept names. This tansfomation is pefomed unde the hypothesis which states that ight side of a concept definition detemines necessay and sufficient condition fo left side of the concept definition (defined concept name). Nevetheless, this can lead to an incompatibility if evision opeatos ae defined in KB. Fo example: Let a TBox T 1 ={A := P B; B := Q R} and its unfolded TBox: T 2 ={A := P Q R; B := Q R}. If we apply FORGET(T 1, B, Q) and FORGET(T 2, B, Q), then T 1 and T 2 thus obtained ae no longe equivalent. It is obvious that what we want to obtain afte the evision is T 1 since concept A is still defined as a conjunction of P and B. This means that the dependence should be peseved. As a esult, if TBox is unfolded, some opeations have to be pefomed to popagate modification of a concept definition in the entie TBox. By contast, if TBox is folded the concepts which depend on the concept to be evised ae automatically changed. A pojection of AGM famewok onto teminology evision was caied out by B. Nebel in [2]. The following pinciples intepeted fo teminology fom the postulates of AGM famewok ae the most poblematic. (i) Success. This pinciple equies that any evision opeation must be successful, i.e following TELL(T, C, Exp) the new knowledge (Exp) must be subsumed by concept C; and following FORGET(T, C Exp, Exp) concept C must no longe be subsumed by the deleted knowledge (Exp). (ii) Minimal Change. This pinciple equies that evision opeation should cause minimal distubance of KB. This means that the new definition obtained of the concept to be evised is the neaest definition, semantically speaking, fom the oiginal definition of this concept. (iii) Recovey. This pinciple ensues that knowledge deleted by opeation FORGET should be ecoveed by opeation TELL i.e TELL(T, FORGET(T, C, Exp), Exp) is subsumed by C. Howeve, the appoach in [2] does not take into account semantic aspects (e.g subsumption elationships) in a sufficient way. Thefoe, this appoach cannot simultaneously satisfy the pinciples pesented and is not extensible to DL-languages which contain existential estictions. In what follows, we popose anothe appoach in languages EL and FLE fo evision opeations allowing to each these pinciples. Syntax, semantics and subsumption notion of these languages can be found in [8]. Futhemoe, the notion lcs (least common subsume) which is used in this pape, is defined and computed in [6]. Because of space limitation, we cannot pesent all the technical details in this pape. In the emainde of the pape, if thee is not any confusion, we can omit the fist paamete of evision opeations.

5 2.2 Revision Opeations We begin by intoducing definition of FORGET and S-FORGET (stand fo SEMI- FORGET) opeations fo a concept in TBox T which uses a L-language without the bottom-concept. Definition 1. (FORGET) Let C be a concept in TBox T and Exp a L-concept desciption without conjunction at the top-level, Exp. 1. If C Exp, FORGET(C, Exp) := C; 2. If C Exp, FORGET(C, Exp) is a concept desciption which has the following popeties: (a) C FORGET(C, Exp) ; (b) FORGET(C, Exp) Exp ; (c) If D is a L-concept desciption so that C D and D Exp, then FORGET(C, Exp) D. The S-FORGET opeation is simila to FORGET except that the condition 2.(c) is eplaced by: If D is a L-concept desciption so that C D and D Exp, then D S- FORGET(C, Exp). The conditions 2.(a) and 2.(b) guaantee the success pinciple while the condition 2.(c) ensues the minimal change pinciple. We emak that fo language EL, FORGET always exists and S-FORGET(C, Exp) FORGET(C, Exp). Nevetheless, fo language FLE, S-FORGET(C, Exp) is not necessaily unique and FORGET(C, Exp) may not exist. Fo example, let C :=.(A 1 A 2 ).(A 1 A 2 A 3 ), Exp 1 :=.A 1, and Exp 2 :=.(A 1 A 2 ). We have: FORGET(C, Exp 1 ) =.A 2.(A 2 A 3 ). Nevetheless, we have: S- FORGET(C,Exp 2 ) {.A 2.(A 2 A 3 ),.A 1.(A 1 A 3 ),.(A 1 A 3 ).(A 2 A 3 )}. The simplicity of evision expession Exp (i.e. Exp does not allow fo conjunction at the top-level) in the Definition FORGET is equied since knowledge added to o deleted fom concept to be evised is consideed as a simple chaacteistic of the concept. On the othe hand, the simplicity of evision expession Exp whee Exp is a value estiction helps to guaantee the uniqueness of S-FORGET. This implies the existence of FORGET in language FLE. As mentioned in the pevious section, we will use the tee epesentation pesented in [6] fo FLE-concept desciptions to compute the evision opeations. A FLE-descip tion tee is of the fom G = (V, E, v 0, l) whee v 0 is the oot, edges vw, v w E ae labeled by oles N R (set of pimitive oles), nodes v V ae labeled by sets of pimitive concepts fom N C (set of concept names). The empty label coesponds to the top-concept. Note that we have to nomalize FLE-concept desciptions befoe tanslating them to desciption tees. Convesely, each FLE-desciption tee can be tanslated into a FLE-concept desciption C G and C G C. Theoem 1. Let C be a defined concept in a EL-TBox. Let Exp be an EL-concept desciption without conjunction at the top-level and Exp. We have: FORGET(C, Exp) always exists and thee exists a polynomial algoithm fo computing it.

6 The algoithm fo computing FORGET(C, Exp) is based heavily on the tee epesentation of EL-concept desciption. FORGET(C,Exp) is inductively computed as conjunction of all concept desciptions which ae not subsumed by Exp but subsume C. This conjunction too is not subsumed by Exp since the conjuncts ae EL-concept desciptions. We can show that the numbe of these conjuncts is polynomial in the size of C. In what follows, we suppose that concept to be evised C and evision expession Exp ae nomalized i.e. value ectictions ae popagated on existential estictions. (cf. [6]) We now ty to extend this esult to language FLE. This extension is not diect since FORGET may not exist despite the existence of S-FORGET. The eason is that the occuence of value estiction on the top-level causes unification of existential estictions. This unification comes fom the popagation of the value estiction on existential estictions (nomalization) when FORGET is built fom a conjunction of S-FORGET (we can show that the unique existence of FORGET(C, Exp) in language FLE will be guaanteed if value estiction does not occu on the top-level of C). Howeve, we can choose a concept desciption as FORGET fom the set of S-FORGET such that estiction of this concept desciption on language EL coincides with the esult obtained fom the computing pocedue fo FORGET in language EL. We can show that thee is a unique S-FORGET which satisfies this estiction. By consequent, thee ae two easons to choose this S-FORGET as FORGET: i) S-FORGET satisfies the minimal change pinciple (i.e. thee does not exist any concept desciption D which is infeio to S-FORGET(C, Exp) so that D subsumes C and D Exp) ii) the S-FORGET chosen is compatible with the FORGET defined in language EL. Similaly, the algoithm fo computing the concept desciption FORGET(C, Exp) defined as above in a FLE-TBox is based on the tee epesentation of FLE-concept desciption. The computing is inductively pefomed on the depth of the desciption tee G C. On each level of tee G C, the S-FORGET s ae computed. Next, conjunctions of these S-FORGET ae built so that unification of these S-FORGET s does not occu. Biefly, the algoithm can be descibed as follows: 1. If Exp is a value estiction, Exp is of the foms:....p whee P is a pimitif concept,o.....d whee D is a FLE-concept desciption (since Exp does not accept conjunction on the top-level). In this case, the existential estictions of FORGET(C, Exp) ae the existential estictions of C and the value estiction of FORGET(C, Exp) is inductively computed as follows:.forget(val (C), val (Exp)) whee val (C) is the expession unde the value estiction of C w..t ole. 2. If Exp is an existential estiction, fist the existential estictions of FORGET(C, Exp) ae inductively constucted fom.forget(d, E ) whee D, E ae expessions unde the existential estictions of C and Exp. The value estiction of FORGET(C, Exp) is computed as the least common subsume lcs of the expessions unde the existential estictions obtained and val (C). Futhemoe, the size of the esult tee FORGET is bounded by an exponential numbe in the size of input since the algoithm must compute lcs fo a polynomial numbe of concept desciptions. As a esult, we obtain the following theoem:

7 Theoem 2. Let C be a defined concept in a FLE-TBox. Let Exp be a FLE-concept desciption without conjunction at the top-level and Exp. We have: FORGET(C, Exp) always exists and thee exists an exponential algoithm fo computing it. To show how the algoithm to wok, we conside the following example. Let C :=.(A B.(A B).(A B C)), Exp :=.(A.A.(A B)). The computing pocedue fo FORGET(C, Exp) descibed above is illustated in Figue 1. In this example, FORGET(C, Exp) is computed as a conjunction of S-FORGET(C, G C G Exp G F ORGET(C,Exp) A, B A B A, B A, B A, B A, B A, B, C A A, B A, B A, B, C B A, B, C B B, C A A, C Fig. 1. Desciption Tees fo computing FORGET Exp). Fist, let v 1 be the -successo of the oot v 0 of the desciption tee G C and w 1 be the -successo of the oot w 0 of the desciption tee G Exp. The subtee G FORGET(C,Exp) (u 1 ) coesponding to existential estiction.(b.(a B).(A B C)) is obtained fom the tee G C by deleting P l(v 1 ) \ ((l(v 1 ) l(w 1 )) fom l(v 1 ). This means that this subtee coesponds to the least concept desciption so that l(u 1 ) l(v 1 ), l(w 1 ) l(u 1 ). The subtee G FORGET(C,Exp) (u 2 ) coesponding to existential estiction.(a B.B.(A B C)) is obtained by copying the existential estiction.(a B C) fom G C and by applying S-FORGET(.va(C GC(v 1)),.va(C GExp(w 1))) to obtain the subtee.b. Note that this S-FORGET contains only one element. The subtees G FORGET(C,Exp) (u 3 ) and G FORGET(C,Exp) (u 4 ) coesponding espectively to C 1 :=.(A B.B.(B C)) and C 2 :=.(A B.A.(A C)) ae computed as follows. Fist, S-FORGET(.C GC(v 1),.C GExp(w 1)) is applied to obtain two subtees E 1 :=.(B C) and E 2 :=.(A C). Second, lcs(ex(e i ), va(g C (v 1 ))) is computed fo i {1, 2} whee ex(e i ) denotes expession unde existential estiction of E i. Finally, C 1, C 2 ae built fom E 1, E 2 and.lcs(ex(e 1 ), va(g C (v 1 ))) =.B,.lcs(ex(E 2 ), va(g C (v 1 )))=.A. We now ty to define TELL opeation such that two opeations FORGET and TELL veify the ecovey pinciple. On the othe hand, TELL should espect the success and minimal change pinciples. Note that the satisfaction of the ecovey pinciple is not

8 tivial in this case. As an example, let C be a concept in EL-TBox and Exp be an ELconcept desciption as follows: C :=.(A B).(A C);Exp :=.A. We have: FORGET(C, Exp)=.B.C. Howeve, if the opeation TELL(C,Exp) is defined as the most concept desciption w..t subsumption so that TELL(C,Exp) C and TELL(C,Exp) Exp, then TELL(FORGET(C,Exp), Exp)=.B.C.A. Theefoe, C TELL(FORGET(C, Exp),Exp) which does not satisfy the ecovey pinciple. This implies that we cannot guaantee simutaneously the minimal change and ecovey pinciples fo opeation TELL. We will define opeation TELL with help of the opeation FORGET as follows. Definition 2. (TELL) Let C be a concept in TBox T and Exp a L-concept desciption without conjunction at the top-level, Exp. 1. If C, TELL(C, Exp) := Exp; 2. If C Exp, TELL(C, Exp) := C; 3. If C Exp, (a) If Exp is a pimitive concept,tell(c, Exp):= C Exp; (b) If Exp is a value estiction, TELL(C, Exp):= P PRIM(C) P.TELL(va (C), va (Exp)) C ex (C).C whee PRIM(C) denotes a set of pimitive concepts at the top-level of C; va (C) denotes expession unde the value estiction at the top-level of C w..t ole and ex (C) denotes a set of expessions unde the existential estictions at the top-level of C w..t ole ; (c) If Exp is an existential estiction, TELL(C,Exp) is a concept desciption which has the following popety: i. TELL(C,Exp) C, TELL(C, Exp) Exp and FORGET(TELL(C,Exp),Exp) C; ii. If D is a L-concept desciption so that D C, D Exp and FORGET(D,Exp) C, then TELL(C, Exp) D. In the definition TELL above, we must distinguish two cases whee expession Exp is a value o existential estiction. Indeed, opeations FORGET(C, Exp) and TELL(C, Exp) ae not symmetic if Exp is a value estiction. By what it means we can define FORGET, TELL such that FORGET(TELL(C, Exp),Exp) C if Exp is an existential estiction while we cannot do it if Exp is a value estiction. We can show that if thee exists TELL(C,Exp), then TELL(C,Exp) is unique modulo equivalence. Futhemoe, the Definition 2 fo TELL veifies success and ecovey pinciples but espects weakly the minimal change pinciple. In fact, the success pinciple is obviously satisfied. We will show that the ecovey pinciple is veified as well. Assume that C Exp. We have: FORGET(C,Exp) C. Thus, TELL(FORGET(C,Exp),Exp) TELL(C, Exp) C (fom definition of TELL and the uniqueness of TELL). Assume now that C Exp. We have: FORGET(C, Exp) Exp. Moeove, since C Exp, C FORGET(C, Exp) and FORGET(C, Exp) FORGET(C, Exp) (fom the uniqueness of FORGET), we can deduce that TELL(FORGET(C,Exp),Exp) C fom the condition 3.(c).ii) of Definition 2 (whee C denotes D and D denotes FORGET(C,Exp)). Fo the case 3.(b) in Definition 2, the

9 ecove pinciple can be shown by using the case 3.(c) and Definition 1. We popose an algoithm fo computing the opeation TELL specified in Definition 2. We only need to teat the case 3.(c) in Definition 2 (the othes cases ae obvious). Let (G + H) denote the desciption tee of TELL(C,Exp) whee G =(N C, E C, v 0, l C ) and H = (N Exp, E Exp, w 0, l Exp ). The tee (G + H) is computed by induction on the depth of the tees. The node (v 0, w 0 ) labeled by l G (v 0 ) l H (w 0 ) is the oot of (G + H). Fo each successo v of v 0 in G and each -successo w of w 0 in H, we obtain a successo (v, w) of u 0 in (G + H) whee l G+H (v, w) := l C (v) l D (w). The node (v, w) is the oof of subtee G(v) + α.h(w ) whee w is a α successo of w, α {, }. The successo u of u 0 in (G + H) is the oof of the subtee coesponding to the following concept desciption: lcs{c (G+H)(u) fo all -successo u of u 0 }. Theoem 3. Let C be a defined concept in a FLE-TBox. Let Exp be a FLE-concept desciption without conjunction at the top-level and Exp. TELL(C, Exp) always exists and thee exists an exponential algoithm fo computing it. Figue 2 shows the desciption tee obtained fom the computing TELL(FORGET(C, Exp), Exp) whee the desciption tee FORGET(C, Exp) is given in Figue 1. G F ORGET(C,Exp) G Exp + B A, B A, B A, B A A, B A, B, C B A, B, C B B, C A A, C A A, B G T ELL(FORGET(C,Exp),Exp) A, B A, B A, B, C Fig. 2. Desciption Tees fo computing TELL

10 2.3 Revision on ABox Accoding to the discussion in Section 2.1, expansion of a TBox does not equie any modification of the coesponding ABox. In fact, all assetions in the ABox ae always veified following an concept addition. Similaly, the opeation FORGET(C, Exp) which is defined in Section 2.2, modifies the definition of concept C so that if an individual veifies the definition of concept C, then this individual also veifies the new definition of C. This implies that no evision is necessay following execution of FOR- GET(C, Exp). By constast, the opeation TELL(C, Exp) which is defined in Section 2.2, needs to evise the ABox. In fact, thee may exist an individual which veifies concept C but does not veify the concept TELL(C, Exp). Let C be a concept definition modified by a evision opeation in TBox. Let (O,. I ) denote a model of KB = ( T, A ) whee T is a TBox and A is a coesponding ABox. Fo each assetion C I (d I ) whee d I O, we can use instance checking infeence, fo example in [4], in ode to know if the assetion TELL(C, Exp) I (d I ) is veified i.e = TELL(C, Exp) I (d I ). 3 Revision Rule and Reasoning on Revision Rules The ule, which is going to be defined below, is a specific fom of poduction ule, called evision ule. Indeed, the consequent of such a ule is a evision opeation. Geneic fom of evision ule in a KB = ( T, A, R ) is defined as follows : R : p TELL(C, Exp) o : p FORGET(C, Exp) whee p is a pedicate composed fom assetions in A with assetion conjunction constucto: p := C 1 (d 1 )... C n (d n ) whee C i (d i ) belongs to o is deduced fom A and the evision opeations TELL and FORGET ae defined in Section 2.2. Let R denote a set of evision ules which is consideed as the thid component of the KB = ( T, A, R ). Semantics of can be defined owing to the semantics of each component. In the DL-based KBs which allow fo ules whose antecedent and consequent ae concept desciptions (assetions, moe pecisely), TBox does not change and thee exists a unique extension of ABox following application of the ules [5]. Howeve, application of evision ules can change TBox and TBox obtained may be not unique, o this application may not even be teminated. In this case, to guaantee the temination of the easoning, we need some constaints on the evision ules. Futhemoe, in ode to obtain the uniqueness we have to choose stategy fo a specific pupose. In this pape, ou appoach consists of maximizing the numbe of evision ules applied. The eason fo this choice is that each context ule application makes the diffeence between of the ontologies smalle. This is justified in the emak at the end of this section. Let be a evision ule of the fom: C 1 (d 1 )... C n (d n ) Rev(C, Exp). We denote Ant():={C 1,..., C n }, Cons():={C}, Rev() {FORGET, TELL} and Ex():={Exp}. A ule is called acyclic if Ex() does not depend on any concept C to be evised and C i does not depend on Cons() whee C i Ant(), i {1,..., n}.

11 Rule component R :={ 1,..., n } of a KB is called non-ecusive if thee does not exist any cycle of ules i.e if thee does not exist any subset { i1,..., il } R such that Ant( i2 ) depends on Cons( i1 ),..., Ant( i1 ) depends on Cons( il ). Fo a KB := ( T, A, R ) whee T is acyclic and R is non-ecusive, a subset of ules R R is called completion of if no longe changes following application of the ules in R in an ode. A completion R is called maximal (minimal) if thee is an application ode of the ules in R such that all applicable (active) ules must belong to R following this application ode of the ules; and the numbe of ules in R is maximal (minimal). In ode to intoduce teminaison condition fo extension pocedue on launched by ule application, we need the following notion. Let 1, 2 be two evision ules whee Cons( 1 ) = Cons( 2 ). The notation Ex( 1 ) Ex( 2 ) is intoduced as follows: 1. If both of Ex( 1 ) and Ex( 2 ) ae existential estictions, Ex( 1 ) Ex( 2 ) is defined as Ex( 1 ) Ex( 2 ) and Ex( 2 ) Ex( 1 ). 2. If Ex( 1 ) is an existential estiction and Ex( 2 ) is a value estiction whee Ex( 2 )=.... }{{}.E, then Ex( 1 ) Ex( 2 ) is defined as E.E and.e n E fo all E whee G E is a subtee at the level n of the desciption tee G Ex(1). 3. If both of Ex( 1 ) and Ex( 2 ) ae value estictions, Ex( 1 ) Ex( 2 ) is defined via 1. and 2. at a level m of the desciption tees. Theoem 4. Let := ( T, A, R ) be a KB whee T and R ae acyclic. Fo each couple of ules i, j R whee Cons( i ) = Cons( j ), assume that : Ex( i ) Ex( j ). 1. Fo all tansfomation seies of pefomed by application of ules 1,..., i,... R : 0, 1,..., i,... whee each tansfomation i 1 - i is caied out by ule i, thee exists n finite so that n = n+1 = If R is non-ecusive, thee exists a unique maximal completion. The poof of the pat 1. of Theoem 4 elies heavily the condition of the theoem which applies on the ule component, and the popeties of opeations FORGET and TELL. Indeed, if each concept C to be evised no longe changes following a finite numbe of ule applications on C, then all the tansfomation seies of is finite. Next, we can show that tansfomation seies obtained fom an application of a ule is equivalent to tansfomation seies obtained fom seveal applications of a ule whee REV () = TELL. Futhemoe, we can pove that thee exists q such that fom the step q application of any ule FORGET no longe is necessay. By consequent, we obtain that all the tansfomation seies of is finite since R is finite. In ode to pove the pat 2., we popose an algoithm fo computing a subset R M of ules. The constuction of this algoithm is based on dependance gaph coesponding to the ule component R. Each node of the gaph coesponds to a ule i R and an edge ( i, j ) of the gaph is defined if Ant( j ) depends on Cons( i ). (The dependance gaph is acyclic since R is non-ecusive). Each step of the algoithm chooses

12 a ule i so that application of i does not make othe ules inactive if possible. Thus, the numbe of ules in R M obtained fom the algoithm is maximal and R M is unique. In geneal, thee may exist diffeent minimal completions in a KB whose components satisfy the conditions of Theoem 4. Accoding to Theoem 4, the ule application ode ( 2, 1 ) in the Intoductoy Example is chosen since if 1 is applied befoe, 2 becomes inactive. Thus, the application ode ( 2, 1 ) is a maximal completion. This choice captues the intuition because the diffeence between the TBoxes obtained fom ule application ode ( 2, 1 ) is less impotant than the diffeence between the TBoxes obtained fom the application ode ( 1, 2 ). Note that the application ode ( 1, 2 ) is not a maximal completion since ule 2 becomes inactive following application of the ule 1. 4 Conclusion and Futue Wok We have intoduced definition of evision opeations in knowledge bases using Descition Logics with existential estictions. These definitions esult fom a pojection of the AGM famewok onto DL-based teminology evision. Next, we have descibed a method fo computing the defined evision opeations. This method is lagely based on the tee epesentation of concept desciptions poposed in [6]. A moe in-depth study of the complexity of these opeatos is necessay fo a possible impovement in the computing. Fo example, the compact epesentation fo lcs poposed in [10] may allow us to educe the size of concept desciptions obtained fom computing evision opeations. Anothe question aises concening the extension of the evision opeatos to languages allowing fo disjunction o numbe estictions. Fist, thee is a way which allows us to extend evision opeations to languages including the bottom-concept and negated pimitive concepts (e.g. ALE). Indeed, opeation FORGET fo the bottom-concept should be consideed as a special case since FORGET(, Exp) cannot simultaneously satisfy both success and minimal change pinciples. Second, extension of evision opeations to languages obtained fom the addition of numbe estictions to language FLE is elatively diect since the stuctual chaacteization of subsumption fo these languages has been developed in [7]. Howeve, fo the languages allowing disjunction constucto, we have to develop a stuctual chaacteization of subsumption which allows us to compute concept desciptions defined in the evision opeations. In addition to extensions concening expessiveness of D.L languages used in K.B, we can allow fo Hon ules in the ule component. In this case, the obtained fomalism will be an extension of language CARIN [3]. This combination will open some inteesting quesions which deseve investigation: which inteactions will take place in the combining fomalism and how to extend the infeence sevices of CARIN in the combining fomalism. Refeences 1. C.E.Alchouón, P.Gädenfos, and D.Makinson, On the logic of theoy change: Patial meet functions fo contaction and evision, Jounal of symbolic Logic, (50), (1985).

13 2. B. Nebel, Reasoning and Revision in Hybid Repesentation Systems, Lectue Notes in Compute Science, Spinge-Velag, A.Y.Levy and M.C.Rousset, Combining Hon Rules and Desciption Logics in CARIN, Atificial Intelligence, Vol 104, F.M.Donini, M.Lenzeini, D.Nadi, and A.Schaef, Reasoning in Desciption Logics, CSLI Publications, F.M.Donini, M.Lenzeini, D.Nadi, W.Nutt, and A.Schaef, An Epistemic Opeato fo Desciption Logics, Atificial Intelligence, Vol 100, F.Baade, R.Küstes, and R.Molito, Computing least common subsume in Desciption Logics with existential estictions, Poc. of the 16th Int. Joint Conf. on Atificial Intelligence, IJCAI, R.Küstes, and R.Molito, Computing least common subsume in ALEN, Poc. of the 17th Int. Joint Conf. on Atificial Intelligence, IJCAI, F.Baade et al., The Desciption Logic Handbook-Theoy, Implementation and Applications, Cambidge Univesity Pess, ebxml, Coe Components, Document Assembly and Context Rules, see C.Le Duc and N.Le Thanh, On the Poblems of Repesenting Least Common Subsume and Computing Appoximation in DLs, Poc. of Int. Conf. on Desciption Logics, 2003.