Extending OWL with Explicit Dependency

Transcription

1 Extending OWL with Exlicit Deendency Jean-Paul Calbimonte, Fabio Porto Ecole Politechnique Fédéral de Lausanne EPFL Database Laboratory - Switzerland firstname.lastname@efl.ch Abstract. Functional Deendency has been extensively studied in database theory. It rovides an elegant formalism for secifying key constraints and is the basis for normalization theory used in Relational database design. Given its known axiomatization through logical imlications it is exected that the ontology community would be interested in investigating its alicability to concetual modeling. This aer investigates the extension of OWL ontologies with functional deendencies. In articular, we focus on key deendency and a new class referred to as exlicit deendency. The latter is a functional deendency in which the function that correlates the lefthand side to the right-hand side of the deendency is known. We extend current formalism to allow functions to aear in the consequent of a deendency declaration. A maing of the roosed formalism to Horn clause is secified and ermits FDs to be evaluated as logic rules. The ontology therein roduced may resent different behaviors deending on the interretation given to functional deendencies. Interesting enough is that exlicit deendency adds one extra ossible interretation, similar to views in databases. We have integrated the extended formalism into OWL and used SWRL to ma deendencies to logic rules. A rototye has been imlemented and illustrates the roosed aroach.. Introduction Data deendency has been introduced as a general model for a large class of database constraints that augments the exressivity of simle database models []. Functional deendencies (FD) comose a articularly interesting data deendency [] with which one can design correct Relational database schemas. This is due to the fact that FDs elegantly model the tye of relationshis existing between attributes of a relation, used, for instance, in defining rimary keys and in avoiding redundant data reresentation by alying normalization rules. The semantics exressed through functional deendencies are equally relevant when secifying a concetual schema. As a matter of fact, it has been observed [3] that in data-centric alications users exect ontologies to offer mechanisms similar to those found in the database area that guarantee the correctness of entered data. There has been quite a few works investigating the adotion of FD in ontologies [4,5,6,7]. In these works, functional deendency (including keys) is incororated into variants of descrition logic languages and an analysis of decidability of the new language is rovided. Albeit the differences in exressivity of the different FD aroaches, the semantics of the introduced FD construct remain essentially invariant. A semi-formal definition can be stated as: Given a concet C with sets of roerties X and Y, we say that X determines Y () (X Y) iff given two instances I, I of C with two bindings (x, x ) of X and (y, y ) of Y, where x i, y i I i, then if x = x imlies y = y As an examle of functional deendency that follows the semantics described in () and models flight information, one may secify that for a given flight number and date a single ilot can be allocated: Flight: hasnumber, hasdate haspilot () The semantics in () is suitable for secifying keys of concets and to constrain the cardinality of deendent roerties. In fact, OWL offers the Functional (resectively InverseFunctional) roerty construct that restricts the maximal cardinality of a single roerty to [8]. In this aer we extend the class of data deendencies that can be exressed through functional deendencies. Consider for instance the rule requiring ilots of a flight to hold assorts issued by the same country as the one reresented by the airline comany flag. This rule could be stated as:

2 hasairline.hasflag(x) and haspilot.haspassort.issuedby (y) -> x=y (3) Observe that differently to the exression in (), the one in (3) imlies an exlicit functional relationshi between the instances in the range of redicates hasflag and isissuedby. Thus, in addition to secifying constraint cardinality, the rule in (3) secifies the value the deendent roerty shall hold as a function of the determining roerty value. We refer to this class of functional deendency as Exlicit Deendency (ED), emhasizing the exlicit secification of the deendency function. In other to accommodate ED with keys and traditional functional deendency definitions, we extend current formalism by allowing functions to aear in the consequent of a FD. As roosed in the database literature [], a Horn clause reresentation is adoted, allowing natural maing of FD formal definition into logic rules. The ontology obtained by adding FDs may induce different behaviors according to the interretation given to FDs [3]. Basically, the business rules imlied by FDs can be interreted as constraints, as in databases, or as art of the theory (TBox) affecting reasoning. Interesting enough, EDs allow further interretation, similar to views in databases. The extended FD secification has been integrated into OWL and a maing to SWRL [9] language allows rules to be evaluated and results to be integrated back into OWL. An initial rototye has been imlemented validating the roosed extensions. The remaining of this aer is organized as follows. Section discusses related work. Next, Section 3 motivates the need for functional deendencies in ontologies. A formal framework for extending DL ontologies with functional deendencies is resented in Section 4. Section 5 discusses the extension of OWL with FD and Section 6 describes our rototye imlementation. Finally, Section 7 concludes.. Related Work Functional deendency has been extensively studied in databases with the main intention of adding more semantics to database schemas []. Imortant results of FD in databases include the secification of key deendency within a relation and referential integrity between relations, known as inclusion deendency. Although inclusion deendency is laced in the realm of ontology languages, key deendency has long being neglected. Most of the work in DL concerning FD exlores alternatives to integrating the new feature into the language and analyze the decidability and comlexity of satisfiability of axioms under the extended language. Calvanese et al. [4] extend DLR with multile-keys (keys constituted by more than one attribute) for concets and n-ary relations. Comonents of keys are simle roerties that cannot ma into a ath of relationshis [0]. Keys are in fact modeled as cardinality constraints over roles in concet constructors and relations without requiring further extension to the language, which ermitted to kee the satisfiability of concets in EXPTIME. Borgida and Wedell [0] extend the Classic descrition logics with functional deendencies and show how subsumtion and views can be comuted using descrition grahs even in the resence of FDs. An interesting oint raised by the authors is the integration of FD definitions with the TBox. In articular, the extension of DL through extra constructs for FD reresentation may fail to rovide any interesting deduction based on associated concets relationshis. In order to circumvent this roblem the FD interretation is secified with reference to the domain making it valid for any subsumtion assertion related to the FD definition. In our aroach, as FDs are maed into SWRL rules, the antecedent formula is comosed of conjunctions of roles forming aths with a common root concet. In this context, any ABox instances matching the redicates fires the rule, including those related to the involved terms through subsumtion relationshi with the resective classes. The reresentation of views in Classic was also insiring, as described in Section 4. Lutz and Milicic [6] roose an extension of descrition logics ALC(D) with concrete domains to suort functional deendencies with aths in the antecedents and consequents. A set of functional deendencies over a concet C forms a key-box. The work shows that for large class of ractical scenario key-boxes, comuting the satisfiability of concets is decidable. Our formalization for FDs equally considers aths in comosing the left and right hand side of its imlication with the difference of not distinguishing between concrete and abstract features. Toman and Weddell [7] investigate the decidability of a terminology in DLFD when FDs are art of concet constructors using aths. It is shown that decidability is maintained with ath functional deendencies iff the latter is restricted to aear in the right-hand side of monotone concet

3 constructors. Our results do not consider FD in concet constructor; rather we add a new constructor for FDs and integrate it to the TBox through logic rules derived from the FD formulation. Motik and colleagues [3] elegantly discuss the role of constraints in ontologies and trace an interesting comarison to constraints in databases. Indeed, the very same business rule may be interreted as terminological axiom or as a constraint. The authors extend the definition of a knowledge base to admit a secial constraint comonent that adverts against non-comliance to the rules without being art of the terminology. The constraint comonent is one of the elements of our concetual model with very similar behavior. Finally, Ludascher et al [] introduced a distinguishable witness redicate for holding instances not conforming to secified constraints. The aroach we adoted for integrating constraints enforcement to our framework was insired by the aforementioned roosal. 3. Preliminaries As we have seen, functional deendencies can be used to exress different tyes of relationshis between ontology elements. In classical FDs and the secial case of keys, the function that links the antecedent and consequent is a selection function, which is based on semantic interretations of the domain of interest. For instance consider the following examle in the flight ontology: the destination airort of a flight functionally determines the dearture gate. This constraint would ensure that assengers from two different flights with the same destination should ass through the same gate. In terms of DL, let s suose that we have defined the flights F and F, gates G, G and airort HEATHROW: hasdestination(f,heathrow) hasdestination(f,heathrow) deartsthrough(f,g) The ABox assertions above would imly that it is not ossible to have the following assertion: deartsthrough(f,g) It would indeed violate the FD. However the selection of G instead of G is rather arbitrary. It deends absolutely on the user s ercetion and interretation of reality. In fact if we change G by G for both flights, the FD constraint is resected anyway. This is understandable because the function that establishes a relationshi between antecedent and consequent is not exlicitly stated. But there are cases where we might want to formally reresent this function in the FD declaration. Consider this examle: the flight origin and destination functionally determine the rice of the ticket. In this case the ticket rice deends on a known function, which takes the flight origin and destination as arameters. For examle, we could calculate the rice based on the distance of the two airorts. Now we cannot arbitrarily choose between all ossible rice values, the function comutes it and all instances should agree on it: hasdestination(f,heathrow) hasdestination(f,heathrow) hasorigin(f,geneva) hasorigin(f,geneva) ticketforflight(ticket,f) ticketforflight(ticket,f) rice(t,300) rice(t,300) In the examle we see that two comletely different flights that agree on origin and destination (GENEVA and HEATHROW), also agree on the ticket rice, which is comuted by a function f. So we see the need of defining all these flavours of FDs in DL. In OWL-DL, only basic FDs are exressible thanks to the FunctionalProerty and the InverseFunctionalProerty. If a roerty P is declared as functional it satisfies the following rule for all instances x, y and z: P( x, y) P( x, z) y = z (4) And for the inverse case: P( y, x) P( z, x) y = z (5) It is easy to see that this is very limited in terms of exressivity, not even comound keys or FDs can be constructed. By using the key-box constructs introduced in [6] and Path-FD s of [7] we are able to

4 define quite comlex FD but we are required to modify the reasoners in order to check comliance of the theory to the FDs. Moreover, we are anyway unable to exress exlicit functions with FDs like in the ticket rice examle. It is well known that there is an intrinsic relation between Horn clauses and functional deendencies, as described in [] in the context of databases. So it seems natural to use Horn clauses for FDs in DL as well. First we analyze the general and common case of a classical FD. Consider the flight and gate examle described reviously. We can translate the FD as a Horn clause rule in the following way: hasdestination( x, y) hasdestination( x, y) deartsthrough( x, z ) hasdestination( x, z ) z = z It is interesting to see that similar rules can be constructed for the case of keys. For instance if we define that two assorts issued for the same country and the same assort numbers are the same, then we can exress this constraint with the following rule: issuedincountry( x, y) issuedincountry( x assnumber( x, z) assnumber( x, z), y) x = x (6) (7) Finally we can write a similar rule for the ticket rice examle: ticketforflight( t, x ) hasorigin( x, w) rice ( t, f ( w, z)) hasdestination( x, z) (8) Notice that f is a known deterministic function that takes two instances, the origin and destination, and returns a rice value. The examles above denote a requirement to a more exressive DL knowledge base. Some roosals suggest the extension of DL with functional deendency and keys but, to the best of our knowledge, no work has considered exlicit deendency. Section 4 describes a formal framework that accommodates: keys, functional and exlicit deendencies. 4. Formal Framework Section 3 illustrates that comlex keys and exlicit deendency definitions are relevant business rules that require an extended DL knowledge base to be exressed. The extension we roose is introduced by a new concet FD that is added to a DL knowledge base. In fact, instances of FD are rules descritions that are maed to logic rules using a rule language. We study the consequence of adding FDs as rules and rovide three interretations for their evaluation. As a result of these different interretations our concetual model distinguishes elements of the terminology and the assertion base therein roduced. 4.. Concetual Model Functional deendencies establish articular relationshis between elements of the knowledge base. Keys, for examle, induce equality of values between non-key roerties for instances identified to be the same. Similarly, exlicit deendencies relate values of deendent roerties to those of determining ones by means of an exlicit function. The referred relationshis are exressed by means of redicates in the head of FD rules that, as a result of rule evaluation, may roduce new facts. Thus, one may say that the facts roduced by running FD rules corresond to the subset of assertions in the knowledge base that agrees on the FD relationshi. In this resect, an imortant issue is to analyze the corresondence of such subset with the ABox without FD inferences. We roose three different interretations: Assertions usual ABox facts; Constraints witnesses [] facts about ABox assertions conflicting with the FD rule; Views facts whose maing to the FD rule defines a necessary and sufficient condition for belonging to the resulting head redicate [0];

5 In order to accommodate these different interretations we extend the concetual model roosed in [3] according to the following definition. Definition : An extended DL-FD knowledge base is a quintule (9) K=(T,A,C,C A,V, FD) such that T is a finite set of standard TBox axioms, A is a finite set of standard ABox assertions, C is a finite set of axioms secifying constraints (i.e. witnesses), C A is a finite set of assertion hurting some constraint in C and exressed as witnesses facts, V is a finite set of view definitions of tye V={v fdv,, v n fdv n }, where fdv i FDV FD, and FD is a finite set of functional deendency definition instances with FDa FDc FDv FDk = FD, where FDa is a finite set of assertion FDs, FDc is a finite set of constraint FDs, FDv is a finite set of views FDs, and FDk is a finite set of key FDs. In Definition, we have that elements of K are air wise disjoint. The constraint interretation corresonds to the behavior in which a ABox is searched for assertions not conforming to a FD rule. Note that redicates reflecting the FD constraint must be negated in the body of the rule. ABox assertions conforming to the aths in the FD rule body lead to the literals that match with the variables of a articular witness redicate in the rule head. Witnesses are secified as distinguished constraint concets in C. The roduced witness facts are added to the constraint assertion box C A. The view interretation secifies queries whose answers are comuted by the exlicit deendency function over determining roerty values. The view characterization defers from simle assertions in that the FD rule definition secifies necessary and sufficient conditions for ABox assertions to match with redicates in FD. In fact, comuting view results is only deendent on redicates on the body of FD rule. V comrehends view labels maed to corresonding FDv instances. Observe that the equivalence between a V instance and an FDv one allows querying from both FDv i or view evaluation. The consequent of an FDv rule is a redicate defined in T and having the resulting assertions due to the view evaluation added to the usual ABox. One may clearly observe that V FDk =. The assertion interretation assumes that the results of FDa evaluation are added to the set A. Predicates in the head of an FDa functional deendency are defined, as usual, in T. Thus, is exected that assertions corroborate to the comlete theory as exressed by T A. 4.. Formalizing Functional Deendencies A FD definition fd is comosed of the following elements: the antecedent A, consequent C, a root concet R and eventually a deterministic function f: fd = ( A, C, R, f ) (0) The antecedent is a list of aths. A ath u i is in turn comosed of roles r i : A = { u, u,..., u } () u i = { r i, C = { u}, r u = { s, s i,,..., r,..., s } l n i, ni } The consequent is defined by a single ath u, which is comosed of l roles u i. The root concet R is the starting oint of all aths in the antecedent and consequent. This ensures that all aths in the FD definition have a common source, which serves as a linking element between them.

6 In case of having the deterministic function f defined, it takes as arameters the ranges of the last roles of the antecedent aths. And the result of f must be comarable to the range of the last role of the ath in the consequent. If the FD is a key FD, then the consequent is the instance of the root concet itself (Id). In the simle examle of the assort with a key FD, the fd definition would be comosed of the following antecedent, consequent and root concet: A = { u, u} C = { u} R = Passort () u = { issuedincountry} u = { assnumber} u = { Id} Or in a simlified way, we could write it as: Passort :{ issuedincountry},{ assnumber} { Id} (3) For the more comlex case of the ticket rice we would have: A = { u, u} C = { u} R = Ticket f = f (4) ticket u = { ticketforflight, hasorigin} u = { ticketforflight, hasdestination} u = { rice} Or in simlified notation: Ticket :{ ticketforflight, hasorigin}, ticket { ticketforflight, hasdestination} f { rice} (5) We can see that these abstract FD definitions can be maed to their corresonding Horn clause rule counterarts (as seen in exressions (6), (7) and (8)). In the general case the abstract FD definition in () can be translated to the following Horn rule: r ( r,,... i, ( ) r ) r ) r ( ( (,,, )... r )... r s ( ) s (, )... s i,, r (,,, i,,, i,, i,, )... r l, n i, ni ( l (, n ( ( i, ni,, n,, n, ), i, n ) i ) l, f (,,..., )) sl ( l, n, n i, ni The i,j and a elements are variables in the rule language. The variable a is the common root node linking all the aths in the antecedent and consequent of the FD. Notice that the formalization in (6) imlies a new assertion interretation. The head of the rule is the new information added to the theory. In the following sub-section we discuss the different tyes of interretations of functional deendencies within our framework., n, n ) (6) 4.3. FD Interretation As we have introduced in Section 4., deending on the kind of FD enforcement, its results can be interreted differently. We have identified three basic cases of FD interretations: as constraint, new assertions and views. To better understand the different usages of FD deendencies, consider the following examle, again in the context of the flight ontology : "The tax on a ticket rice functionally deends on the assenger age-grou, the dearture airort and the arrival airort". This rule may make sense if for instance a secial ticket tax is added to the rice deending of the age of the assenger and the airort

7 he is arriving to. The following grah will hel us to reresent the concets and roles involved in the aforementioned rule: Ticket ticketforflight Flight deartsfrom Airort haspassenger hasprice arrivesto Airort Passenger Price belongstogrou hastax AgeGrou Tax Fig.. Grah of concets and roles involved in the ticket rice tax examle. We can easily identify the following aths for the antecedent: {ticketforflight, deartsfrom} {ticketforfligh, arrivesto} {haspassenger, AgeGrou} And the single ath for the consequent (dashed arrows in Fig. ): {hasprice,hastax} In addition, we define f tax as the function that comutes the tax based on the dearture, arrival and age grou: tax = f tax (deartureairort,arrivalairort,agegrou) Our FD is then defined as: fd tax C = In the Horn-rule version: : ( A, C, Ticket, f { ticketforflight, dertsfrom}, A = haspassenger( has Pr ice( ) hastax(, f ( tax { ticketforflight, arrivesto}, { haspassenger, agegrou} { has Pr ice, hastax } ticketforflight( ticketforflight( 3,,,, ) agegrou(, ) ) arrivesto( )),, ) ) (7) ) deartsfrom(, ) (8), 3, 3, In this case, if FDs are interreted as assertions, the head of the rule exress a new fact, which is added to the ontology as a consequence of FD evaluation. In the current scenario, the hastax redicate secifies the value to be aid as comuted by the know function f alied over the determining roerties:,, 3, hastax(, f (,,, 3, )) (9) This assertion would be added to the A set of the knowledge base. The roerty hastax should exist in the T set of the knowledge base. If the FD above is treated as a constraint, then we need to change the rule in such a way that we add a witness in case the theory does not obey the FDs. The rules would have to include a negation in the last redicate of the consequent:,,

8 ticketforflight(, ) deartsfrom(,,, ) ticketforflight(, ) arrivesto(,,, ) haspassenger( 3, ) agegrou( 3,, 3, ) hasprice( ) hastax(, f (,,, 3, )) w tax ( fd tax,hastax(, f (,,, 3, ))) (0) Notice that the witness w tax indicates: the failing instance, the FD and the redicate that caused the failure. This witness would be added to the C A set of the knowledge base. Notice that the witness can grow in comlexity, similarly to excetion handling classes in OO modelling. The taxonomy of the witness classes is defined in the C set of the knowledge base. Finally, ED rules may behave like views in databases by secifying values of a deendent roerty as derived from the function evaluation over determining roerties. Views are secified in the V set and corresond to labels associated to FD clauses. In the head of a view FD, a redicate, secified as art of the T set, is filled with matching values resulting from rule evaluation. Observe that differently to the two other interretations, a user can exlicit invoke a view FD. Clause () illustrates a view FD for the examle (7). Note that syntactically it is equivalent to the one resented in (8). ticketforflight( ticketforflight( haspassenger( has Pr ice( ) hastax(, f ( 3,,,, ) deartsfrom(, ) () ) arrivesto( ) agegrou(,, 3, )) 3,,,,, 3,,, ) ) Having defined FDs formally and having exlained the different interretations we can find, we exlain in the next section how we extended OWL with functional deendencies. 5. Extending OWL with FD The formalism introduced in Section 4 is the basis for the integration of the extended FD aroach into OWL. In this section we discuss such aroach and use the SWRL language to exress logic rules therein roduced. 5.. OWL FD Definition In order to model the abstract FD definition resented in (6), an OWL Class FD has been secified. The FD Class, just like in the definition introduced at (6), has the following roerties: antecedent, consequent, definedfor and function. The antecedent roerty links FD instances to one or more Path instances. A Path is a Class that contains a list of roerty references called PartList. The PartList class is a extension of the generic rdf:list. In order to make this a list of roerties, the first roerty of this list can only accet rdf:proerty instances. An examle of a consequent roerty is shown below: <owlfd:antecedent> <owlfd:ath rdf:id="ath_"> () <owlfd:art> <owlfd:artlist> <rdf:first rdf:reference="#issuedforperson"/> <rdf:rest> <owlfd:artlist> <rdf:first rdf:reference="#worksfor"/> <rdf:rest> <owlfd:artlist> <rdf:first rdf:reference="#basedincountry"/> <rdf:rest rdf:resource="htt:// 999/0/-rdf-syntax-ns#nil"/>

9 </owlfd:artlist> </rdf:rest> </owlfd:artlist> </rdf:rest> </owlfd:artlist> </owlfd:art> </owlfd:ath> </owlfd:antecedent> The Path instance contained in the antecedent secified in () secifies the following ath in abstract notation: ath _ = { issuedforperson, worksfor, basedincountry} (3) The consequent roerty also links an FD instance to a Path, but to maximum. The ath of the consequent also contains a list of rdf:proerty references. The definedfor roerty of the FD class corresonds to the root concet defined in (0). In case of being a FD key, we use the sub-roerty keydefinedfor. The definedfor roerty links an FD instance with a reference to a Class, like in the following examle: <owlfd:definedfor rdf:resource="#passort" /> (4) Notice that the Passort class should have been reviously defined somewhere in the ontology. Finally the hasfunction roerty indicates the resource id of the function corresonding to f in (0). We can summarize the aforementioned characteristics of the FD class with its definition in OWL using a simlified syntax: FD owl:thing antecedent only Path antecedent min consequent only Path consequent max = definedfor exactly = hasfunction exactly The Path class and the PartList are defined as follows: Path owl:thing art some artlist art min PartList rdf:list rdf:first only rdf:proerty = rdf:first exactly rdf:rest only rdf:list = rdf:rest exactly In addition we have defined three subclasses of FD: FD FDc and FDv. These subclasses corresond to the abovementioned interretation tyes: assertions, constraints and views resectively: FDa FD FDc FD FDv FD Another subclass of FD is FDk, for the secial case of key functional deendencies: FDk FD (8) Once we have these classes and roerties defined, we can reuse them to define FD s in any OWL ontology. For instance consider the Passort FD examle in (7). We would instantiate our FDk class as follows: <owlfd:fdk rdf:id="fd_"> <owlfd:antecedent> (9) <owlfd:path rdf:id="ath_"> <owlfd:art> <owlfd:partlist> (5) (6) (7)

10 <rdf:first rdf:resource="#issuedforcountry" /> <rdf:rest rdf:resource="htt:// -rdf-syntax-ns#nil" /> </owlfd:partlist> </owlfd:art> </owlfd:path> </owlfd:antecedent> <owlfd:antecedent> <owlfd:path rdf:id="ath_"> <owlfd:art> <owlfd:partlist> <rdf:first rdf:resource="#assortnumber" /> <rdf:rest rdf:resource="htt:// -rdf-syntax-ns#nil" /> </owlfd:partlist> </owlfd:art> </owlfd:path> </owlfd:antecedent> <owlfd:keydefinedfor rdf:resource="#passort" /> </owlfd:fdk> 5.. Maing to SWRL One we have defined the FD using our OWL FD classes, we can build Horn clause rules following the maing schema as shown in (6). We used the Semantic Web Rule Language SWRL to exress such rules. It is easy to see that a simle scrit rogram can easily generate the necessary SWRL rules based in a FD definition. For instance the SWRL rule counterart to (9) would be: <swrl:im rdf:id="rule_fd_"> <swrl:body rdf:arsetye="collection"> (30) <swrl:individualproertyatom> <swrl:roertypredicate rdf:resource="#issuedforcountry"/> <swrl:argument rdf:resource="#"/> <swrl:argument rdf:resource="#c"/> </swrl:individualproertyatom> <swrl:individualproertyatom> <swrl:roertypredicate rdf:resource="#issuedforcountry"/> <swrl:argument rdf:resource="#"/> <swrl:argument rdf:resource="#c"/> </swrl:individualproertyatom> <swrl:individualproertyatom> <swrl:roertypredicate rdf:resource="#assortnumber"/> <swrl:argument rdf:resource="#"/> <swrl:argument rdf:resource="er"/> </swrl:individualproertyatom> <swrl:individualproertyatom> <swrl:roertypredicate rdf:resource="#assortnumber"/> <swrl:argument rdf:resource="#"/> <swrl:argument rdf:resource="#er"/> </swrl:individualproertyatom> </swrl:body> <swrl:head rdf:arsetye="collection"> <swrl:individualproertyatom> <swrl:argument rdf:resource="#"/> <swrl:argument rdf:resource="#"/> <swrl:roertypredicate rdf:resource="#sameas"/> </swrl:individualproertyatom> </swrl:head> </swrl:im> As exlained above, the SWRL rule indicates that the issuedforcountry and assortnumber are the keys of the Passort class. In the figure below, dotted arrows mark the key roerties.

11 Passort issuedforcountry Country forperson assortnumber Person PassNumber Fig.. FDk for the Passort Class. Notice that, in this articular case, the FD is a key FD. If two assorts have same instances for the roerties issuedforcountry and assortnumber, then both Passorts are the same. We can exress this equality with the owl:sameas roerty, and add it to the ontology. Once the SWRL rules are generated, a SWRL reasoner can evaluate them and roduce results. These results may eventually be added to the ABox or used for constraint checking. An OWL reasoner can then be alied to see if the ontology is consistent. If not, then some instances of the ABox do not comly with the FD s defined reviously and they can be signaled in a consistency reort. 6. Imlementation Having described our aroach for adding functional deendencies to OWL, we roceed now to describe a rototye imlementation demonstrating the alicability of our ideas. As it can be seen throughout this aer, we have worked with a simle ontology concerning flights. This ontology has been written in OWL. In the next subsections we will resent the main comonents of our rototye and the main stes of the imlementation workflow. 6.. Building Blocks In order to make ossible an imlementation of our constructs, we have ut together several comonents and tools, which are detailed below. First we have added an abstract extension of FD s for OWL, which we call owlfd. Then we have used SWRL to define FDs in terms of horn clauses: FD definitions owlfd SWRL FDs as rules OWL Taxonomy and assertions Fig. 3. OWL SWRL and FDs Now to generate the SWRL rules based on the FD definitions, we have written a Java rogram that arses and interrets the FD s and returns the corresonding rules. The SWRL evaluation is erformed by the Jess engine. For consistency checking we have used Pellet reasoner. 6.. Imlementation Workflow The workflow of our imlementation can be summarized in the following figure:

12 Java Program 3 4 Jess 5 6 Pellet Fig. 4. Workflow of FD construction and evaluation with SWRL. Using the abstract FD OWL extension described in the revious section, we have written FD definitions on the flight ontology (ste in fig. 4). These definitions define antecedents and consequents just like described in 6. Since our FD classes and roerties are secified in OWL, it is ossible to use a generic OWL editor like Protégé. In order to generate SWRL rules based on the FD definitions (ste in fig. 4); we have written a Java rogram that generates such rules following the maing we established in section 4. It is imortant to notice that our imlementation currently focuses on the identity function as the f comonent of a FD described in section 4. Once the rogram has generated the SWRL rules, a SWRL reasoner can evaluate them (ste 4 in fig. 4). We have used Jess with the JessTab extension of Protégé to evaluate the rules. Once the SWRL reasoner roduces new assertions these are added to the ontology as new knowledge. Notice that it could also be ossible to treat the FD as constraints and never add the results of the SWRL evaluation back to the theory. At this oint it is ossible to use a standard OWL reasoner to check if the ontology is still consistent (ste 6 of fig. 4). We used Pellet to do so. In case of finding inconsistencies, it could be due to several reasons. For examle, the existent instances do not comly with the rules. But it could also be the case that the rules are wrong or badly defined. Since they are art of the knowledge base and their semantic interretation deends on the domain and context, we can only let the user know that inconsistencies have been found. A deeer analysis should be erformed by an exert in the domain of the concerned ontology Conclusions The extension of DL knowledge based with functional deendencies has been acknowledged as relevant in roducing more exressive ontologies. In this aer we investigate the extension of knowledge bases with three kinds of functional deendencies: classical, keys and exlicit. In fact, to the best of our knowledge, this is the first work in ontologies that exlores exlicit functional deendencies, which are those where an exlicit function relating deendent to determining roerties is known. We roose a formal framework to extend ontologies with these three functional deendencies and study the different behaviors that can be considered when running FD rules. We identified three main interretations: constraints, assertions and view and show how to integrate them within a common framework. The concetual reresentation is imlemented in OWL by a new FD

13 concet holding functional deendencies definitions. Moreover, a maing function translates FD instance definitions into SWRL rules, allowing inferences to roduce the desired FD behavior. The framework has been imlemented in an initial rototye under Protegé and using Jess as the rule execution engine. Currently, the maing of exlicit deendencies to SWRL only consider identify function. We intend as future work to exlore the extension of SWRL with other builtin functions and the ossibility of including user defined functions. Our aroach to extend the knowledge base with a new FD class has both advantages and disadvantages. The advantage is that it can be easily adoted without requiring any extension to the language. Furthermore, as the FD evaluation is done through SWRL it does not affect subsumtion reasoning in the TBox. It turns out that this same asect can be seen as a disadvantage as subsumtion cannot be exressed over constrained concets with FD. Another interesting issue that we leave for future investigation is the case of key FD with multivalued non-key attributes. In this scenario deciding on equality of sets seems not evident. Similarly, if roerties in the head of a FD are allowed to be multi-valued then existential quantification over the set is required. Finally, we intent to further exlore the view interretation, investigating how queries should be exressed and the ossible effects of integrating view results with ABox assertions. References. S. Abiteboul, R. Hull and V. Vianu. Foundations of Databases, Addison Wesley, Ronald Fagin. Horn Clauses and Database Deendencies. In Journal of the Association for comuting Machinery, Vol 9, no 4, ages , Boris Motik, Ian Horrocks, Ulrike Sattler. Bridging the Ga Between OWL and Relational Databases. In Proceedings of the 6 th international conference on World Wide Web, ages , Diego Calvanese, Giusee De Giacomo, Maurizio Lenzerini. Keys for free in Descrition Logics. In Proc. of the 000 International Worksho on Descrition Logics (DL'000), Diego Calvanese, Giusee De Giacomo, Maurizio Lenzerini. Identification Constraints and Functional Deendencies in Descrition Logics. In Proc. of IJCAI 00, ages , Carsten Lutz, Maja Milicic, Descrition Logics with Concrete Domains and Functional Deendencies. In Proceedings of the 6th Euroean Conference on Artificial Intelligence (ECAI-004), David Toman, Grant E. Weddell, On Keys and Functional Deendencies as First-Class Citizens in Descrition Logics. On IJCAR 006: , S. Bechhofer, F. V. Harmelen, J. Hendler, I. Horrocks, D. L. McGuinness, P.F. Patel-Schneider, L. A. Stein. OWL Web Ontology Language Reference, htt:// W3C Recommendation I. Horrocks, P. F. Patel-Schneider, H. Boley, S. Tabet, B. Grosof, M. Dean, SWRL A Semantic Web Rule Langguage Combining OWL and RuleML, W3C member submission, th May Alexander Borgida and Grant E. Weddell. Adding uniqueness constraints to descrition logics (reliminary reort). In Proceedings of the Fifth International Conference on Deductive and Object Oriented Databases, ages 85--0, B. Ludascher, Amarnath Gut M. E. Martone, Model-Based Mediation with Domain Mas, 7th International Conference on Data Engineering, Heidelberg, Germany, 00.