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D. Resolution-based approximate reasoning for OWL DL

Resolution-Based Approximate Reasoning

for OWL DL

Pascal Hitzler and Denny Vrandeˇc i′c

AIFB,Universit¨a t Karlsruhe,Germany

{hitzler,vrandecic}@aifb.uni-karlsruhe.de

Abstract.We propose a new technique for approximate ABox rea-

soning with OWL DL ontologies.Essentially,we obtain substantially

improved reasoning performance by disregarding non-Horn features of

OWL DL.Our approach comes as a side-product of recent research

results concerning a new transformation of OWL DL ontologies into

negation-free disjunctive datalog[1,2,3,4],and rests on the idea of per-

forming standard resolution over disjunctive rules by treating them as

if they were non-disjunctive ones.We analyse our reasoning approach

by means of non-monotonic reasoning techniques,and present an imple-

mentation,called Screech.

1Introduction

Knowledge representation and reasoning on the Semantic Web is done by means of ontologies.While the quest for suitable ontology languages is still ongoing, OWL[5]has been established as a core standard.It comes in three?avours, as OWL Full,OWL DL and OWL Lite,where OWL Full contains OWL DL, which in turn contains OWL Lite.The latter two coincide semantically with certain description logics[6]and can thus be considered fragments of?rst-order predicate logic.

OWL ontologies can be understood to consist of two parts,one intensional, the other extensional.In description logics terminology,the intensional part con-sists of a TBox and an RBox,and contains knowledge about concepts(called classes)and the complex relations between them(called roles).The extensional part consists of an ABox,and contains knowledge about entities and how they relate to the classes and roles from the intensional part.For the Semantic Web, TBox and RBox shall provide background vocabulary,while(annotated)web-pages etc.constitute ABoxes which are interlinked with intensional knowledge. The Semantic Web thus envisions a distributed knowledge source,built from OWL ontologies and intertwining the knowledge like the World Wide Web in-terconnects websites.

The authors acknowledge support by the German Federal Ministry of Education and Research(BMBF)under the SmartWeb project,and by the European Commission under contract IST-2003-506826SEKT and under the KnowledgeWeb Network of Excellence.The expressed content is the view of the authors but not necessarily the view of any of the projects as a whole.

Y.Gil et al.(Eds.):ISWC2005,LNCS3729,pp.383–397,2005.

c Springer-Verlag Berlin Heidelberg2005

384P.Hitzler and D.Vrandeˇc i′c

With an estimated25million active websites today and correspondingly more webpages,it is apparent that reasoning on the Semantic Web will have to deal with very large https://www.doczj.com/doc/2a4117938.html,plexity of ABox reasoning—also called data com-plexity—thus measures complexity in terms of ABox size only,while considering the intensional part of the ontology to be of constant size.For the di?erent OWL variants,data complexity is at least NP-hard,which indicates that it will not scale well in general[7].Methods are therefore being sought to cope with large ABoxes in an approximate manner.

The approach which we propose is based on the fact that data complex-ity is polynomial for non-disjunctive datalog.We utilise recent research results [1,2,3,4]which allow the transformation of OWL DL ontologies into disjunc-tive datalog.Rather than doing(expensive)exact reasoning over the resulting disjunctive datalog knowledge base,we do approximate reasoning by treating disjunctive rules as if they were non-disjunctive ones.The resulting reasoning procedure is complete,but may be unsound in cases.Its data complexity is polynomial.We are also able to give a characterization of the resulting ap-proximate inference by means of standard methods from logic programming semantics.

This paper is structured as follows.In Section2,we?rst discuss the general rationale behind approximate reasoning,and how it relates to other reasoning frameworks.We then recall formal terminology and notation for OWL DL,and shortly review datalog and SLD-resolution.Then,in Section4,we explain how OWL DL ontologies can be transformed into disjunctive datalog.In Section5 we introduce the new approximate SLD-resolution procedure which we propose. The presentation of our implementation Screech in Section6is followed by an Example in Section7,and an experimental evaluation in Section8.We conclude and discuss future work in Section9.

2Non-classical Reasoning—Common Grounds

The sophisticated reasoning tasks required when dealing with expressive knowl-edge representation languages like those based on description logics are known to be of high computational complexity.In the face of ever increasing data quan-tities to be processed,new methods are needed to obtain usable systems.As the high computational complexity of the reasoning tasks is unavoidable,the method of choice for obtaining scalable systems is to use approximate reasoning techniques.In a nutshell,approximate reasoning rests on the idea of decreas-ing the complexity of a problem by imposing controlled changes on either the language used or the inference operation used for the deduction.The resulting lower complexity and consequent speed-up thus comes at the price of unsound-ness or incompleteness(or both),but in a controlled and well-understood manner which allows to assess the quality of the deduction made by the approximate rea-soner.So-called anytime algorithms develop the idea a bit further and guarantee convergence to exact answers given enough time,while providing approximate results during the reasoning process.

Resolution-Based Approximate Reasoning for OWL DL385

Fig.1.Semantic view on approximate reasoning

A semantic perspective on approximate reasoning is depicted in Figure1. When a theory is being considered,classical reasoning may be of high com-putational complexity and thus be unsuitable for time-critical tasks.By taking di?erent models into account than the classical ones,the complexity of reasoning can be reduced.The resulting approximate inference may be incomplete or un-sound with respect to classical inference,but in a controlled and well-understood manner,which makes the inferences suitable for further use.

Similar situations occur in the context of other sophisticated reasoning tech-niques.For non-monotonic reasoning,for example,a subset of the classical mod-els is usually considered,which is selected by means of e.g.additional syntax constructs or by rede?ning the semantics of existing ones.Non-monotonic rea-soning thus allows to arrive at conclusions which cannot be derived using classical reasoning:It is complete,but unsound,and can be described as supraclassical[8]. The rationale in this case is to model aspects of human commonsense reasoning like jumping to conclusions,again in a controlled and well-understood manner. Complexity considerations are often treated as secondary in this context.

Paraconsistent reasoning—or reasoning with inconsistency—can be ap-proached from a similar perspective.While inconsistent knowledge bases have no classical models,paraconsistent reasoning strives to identify suitable models to be assigned to the knowledge base nevertheless,in order to allow the infer-ence of meaningful consequences.As such,paraconsistent reasoning is sound,but incomplete with respect to classical logic,and can thus be termed subclassical.

Table1summarizes our discussion.While the table can certainly be extended further taking other forms of reasoning into account,we restrict ourselves to the mentioned examples,as the main goal of this paper is to present an approxi-mate reasoning method for OWL DL,and not a comparative theory of reasoning https://www.doczj.com/doc/2a4117938.html,parision of non-classical reasoning approaches

reasoning approach focus models taken into account typical complexity classical all classical models high

non-monotonic commonsense some classical models very high paraconsistent inconsistency more than the classical models high

approximate performance variable low

386P.Hitzler and D.Vrandeˇc i′c

approaches.We have included this discussion because it explains the general ra-tionale behind our approximate reasoning method,and will help us in analyzing it.Indeed,in all reasoning paradigms mentioned,it is important to obtain a clear understanding of the inference relation computed.This can be done by semantic analyses,i.e.by characterizations of the models taken into account.From the general perspective described in this section,it will later come as no surprise to the reader that we will analyze our approximate reasoning methods by means of standard techniques from non-monotonic reasoning.Indeed,in our particular case the models taken into account for approximate reasoning will turn out to be a subset of the classical models,as in non-monotonic reasoning.

3Preliminaries

3.1OWL DL Syntax and Semantics

OWL DL is a syntactic variant of the SHOIN(D)description logic[9].Hence, although several XML and RDF syntaxes for OWL DL exist,it will be convenient to use the traditional description logic notation since it is more compact,and we recall the notation below.For the correspondence between this notation and various OWL DL syntaxes,see[9].

We indeed assume that the reader is familiar with OWL and thus with SHOIN(D),as space restrictions forbid to reintroduce them,but recall that SHOIN(D)supports reasoning with concrete datatypes,such as strings or inte-

gers[10].Recall also that the description logic syntax for concepts in SHOIN(D) is de?ned as follows,where A is an atomic concept,R is an abstract role,S is an abstract simple role,T(i)are concrete roles,d is a concrete domain predi-cate,a i and c i are abstract and concrete individuals,respectively,and n is a non-negative integer:

C→A|?C|C1 C2|C1 C2|?R.C|?R.C|≥n S|≤n S|{a1,...,a n}| |≥n T|≤n T|?T1,...,T n.D|?T1,...,T n.D

D→d|{c1,...,c n}

The SHIQ(D)description logic is obtained from SHOIN(D)by disallowing nominal concepts of the form{a1,...,a n}and{c1,...,c n},and by allowing qual-i?ed number restrictions of the form≥n S.C and≤n S.C,for C a SHIQ(D) concept and S a simple role.

As description logics,SHOIN(D),i.e.OWL DL,and SHIQ(D)inherit their semantics from?rst-order logic by the standard translations known e.g. from[11],which we do not repeat here.

3.2Datalog and SLD-Resolution

A(de?nite or negation-free)disjunctive logic program P consists of a?nite set of clauses or rules of the form

?x1...?x n.(H1∨···∨H m←A1∧···∧A k),

Resolution-Based Approximate Reasoning for OWL DL387 commonly written as

H1∨···∨H m←A1,...,A k,

where x1,...,x n are exactly all variables occuring in H1∨···∨H m←A1∧···∧A k,and all H i and A j are atoms over some given?rst-order language Σ.The disjunction H1∨···∨H m is called the rule head,and the conjunction A1∧···∧A k is called the rule body.The set of all ground instances of atoms de?ned overΣis called the Herbrand base of P and is denoted by B P.The set of all ground instances of rules in P is denoted by ground(P).A rule is said to be non-disjunctive if m=1.It is called a fact if k=0.We abstract from the order of the atoms in the heads respectively bodies;it is not important for our results.A disjunctive logic program is called a(disjunctive)datalog program if it does not contain function symbols.

Note that we do not consider logic programs to come with one speci?c se-mantics.Some people for example associate datalog with the minimal model semantics only.For our treatment,datalog and logic programs are de?ned via syntax only.We do not specify a speci?c semantics because in the following we will discuss di?erent semantics for logic programs in their relation to proof procedures.One of the semantics we will consider is the semantics coming from interpreting logic programs as a set of?rst order formulas,and in this case we use|=to denote entailment in classical?rst-order predicate logic.

SLD-resolution(see e.g.[12])is an e?cient top-down query-answering tech-nique for programs consisting of non-disjunctive rules,and has been imple-mented and successfully applied in standard Prolog systems.1In this framework, a ground atom can be derived from a program if and only if it is true in the least (and thus in all)Herbrand models of the program.

In the following,we mean by a conjunctive query simply a conjunction B1∧···∧B n of atoms.The query is called ground if it does not contain any variables.

Given a conjunctive query B1∧···∧B n,an SLD-resolution step on the atom B i with a non-disjunctive rule H←A1,...,A k produces a conjunctive query

B1θ∧···∧B i?1θ∧A1θ∧···∧A kθ∧B i+1θ∧···∧B nθ

whereθis the most general uni?er of B i and H.An SLD-refutation of a con-junctive query B1∧···∧B n in a non-disjunctive program P is a?nite sequence of conjunctive queries Q0,...,Q n,where(i)Q0=B1∧···∧B n,(ii)each Q i with i>0is obtained from Q i?1by an SLD-resolution step with some rule from P on some literal B i,and(iii)Q n= ,i.e.the conjunctive query Q n does not contain any literals.If an SLD-refutation of B1∧···∧B n in P exists,we write P B1∧···∧B n.

One of the fundamental results in logic programming states that A∈B P can be proven by SLD-resolution if and only if A is a logical consequence of P,i.e. if and only if A is true in the least Herbrand model of P:

1Like SWI or XSB Prolog,https://www.doczj.com/doc/2a4117938.html,,https://www.doczj.com/doc/2a4117938.html,.

388P.Hitzler and D.Vrandeˇc i′c

Theorem1([12]).For a ground conjunctive query B1∧···∧B n and a non-disjunctive program P,P B1∧···∧B n if and only if P|=B1∧···∧B n.In other words,entailment of ground conjunctive queries under SLD-resolution is entailment in predicate logic.

SLD-resolution also allows to derive answers to non-ground queries:For a con-junctive(and not necessarily ground)query Q there exist an SLD-refutation if and only if P|=?x1...?x n.Q,where x1,...,x n are the variables occuring in Q.By keeping track of the most general uni?ers used in the process,it is also possible to obtain bindings for(some of)the x i in the form of(answer)substitu-tionsθ,such that P|=?y1...?y k(Qθ),where the y i are exactly those variables occurring in Qθ.In order to keep our exhibition focused,we will only deal with ground queries.

4Reducing OWL DL Knowledge Bases to Disjunctive Datalog Programs

We utilise recent research results about the transformation of OWL DL ontolo-gies into disjunctive datalog,and perform approximate reasoning by transform-ing the disjunctive database into a non-disjunctive one.The transformation is based on the fact that OWL DL is a subset of?rst-order logic.OWL axioms can thus be translated directly into logical formulas and transformed into clausal form using any of the standard algorithms.The resulting clauses can be repre-sented as disjunctive datalog rules which do not contain negation.

Note,however,that due to possible skolemization steps in the clausal form translation,the resulting datalog rules may contain function symbols.In general, datalog with function symbols is undecidable,but since we obtain the datalog program by a translation from OWL DL,which is decidable,inferencing over the resulting program must be decidable.Standard datalog engines,however,do in general not terminate in the presence of function symbols.To cope with this problem,a sophisticated method has been presented in[2,3]which allows to get rid of the function symbols without loosing ABox consequences.As a result,we obtain a function-and negation-free disjunctive datalog program,which can be dealt with using standard techniques.

There is one other catch:The approach presented in[2,3]does not yet allow to deal with nominals,i.e.it supports only SHIQ(D)instead of SHOIN(D) (the latter is the description logic coinciding with OWL DL).We remark that to date—and to the best of our knowledge—no reasoning algorithms for SHOIN(D)have been implemented.We will return to a possible treatment of nominals in our approach later.

The translation algorithm is schematically depicted in Figure2.It transforms a SHIQ(D)knowledge base KB into a disjunctive datalog program DD(KB). The steps of the algorithm are as follows.(1)Transitivity axioms are removed by adding axioms of a form similar to?S.C ?S.(?S.C)for transitive roles S.

(2)The knowledge base is translated into clausal form by standard transforma-tions based on?rst-order predicate logic.This introduces function symbols due

Resolution-Based Approximate Reasoning for OWL DL389

SHIQ(D)

KB

Disjunctive

Program

DD(KB) Fig.2.Algorithm for Reducing SHIQ(D)to Datalog Programs

to necessary skolemization steps.(3)The TBox of the knowledge base is par-tially saturated by adding logical consequences.This is the crucial step of the algorithm.(4)The saturation from step(3)now allows to remove all function symbols which were introduced in step(2).Some additional axioms are added to ensure that the algorithm remains sound and complete.(5)The knowledge base is translated into disjunctive datalog clauses;this step is now straightforward.

It shall be noted that the details of the crucial step(3)are very sophisticated. They guarantee that the removal of function symbols in step(4)is at all possible. Step(3)is of exponential complexity,however for the ABox reasoning task which we focus on in this paper,Step(3)can in principle be performed o?ine,as this step is independent of the ABox–but note that this o?ine computation may still be di?cult if the TBox is large,which is a seperate issue and deserves further in-depth studies which are outside the scope of this paper.A full presentation of the translation with correctness proofs is technically involved and lengthy, and space restrictions forbid to go into further detail;we refer the interested reader to[2,3].In[1]full proofs are given which show amongst other things that KB is unsatis?able if and only if DD(KB)is unsatis?able.This su?ces for reasoning over KB as reasoning tasks can be transformed into unsatis?ability checks.

5Approximate Resolution

While approximate reasoning methods for propositional and?rst-order logic have been proposed(see e.g.[13,14,15,16,17,18]),they have hardly been applied in the context of Semantic Web technologies.The few exceptions are reported e.g. in[19,20,21]—to the best of our knowledge,this list is exhaustive.The success of the approaches is mixed.[21]reports on an analysis indicating that straight-forward adaptations of methods proposed by[14]do not su?ce.[20]reports good results but is not an approximate reasoning method in the more narrow sense as the reasoning performed is exact,and thus does not address the complexity problems underlying OWL DL reasoning.[19]deals with approximating queries, while we focus on ABox reasoning.We will now present a novel approach based on the translation of OWL DL to disjunctive datalog,as presented earlier.

5.1Approximate SLD-Resolution

Having obtained the translated knowledge base in the form of a disjunctive datalog program,ABox reasoning remains NP-hard,and thus untractable.If the datalog program is non-disjunctive,though,reasoning is polynomial in the size of

390P.Hitzler and D.Vrandeˇc i′c

the ABox.We therefore propose the following approximate reasoning technique in order to facilitate this insight.Given a conjunctive query B1∧···∧B n, an approximate SLD-resolution step on the atom B i with a disjunctive rule H1∨···∨H m←A1,...,A k is a conjunctive query

B1θ∧···∧B i?1θ∧A1θ∧···∧A kθ∧B i+1θ∧···∧B nθ

such thatθis the most general uni?er of B i and some H j.Approximate SLD-refutation is de?ned analogously to SLD-refutation,where approximate SLD-resolution steps are used instead of(usual)SLD-resolution steps.

It is necessary to pursue the question what notion of entailment underlies the approximate reasoning technique we propose.Following the spirit of the obervations from Section2,we want to identify the set of models which underly the inference relation provided by approximate SLD-resolution.For this purpose, we need the following notion,which is derived from standard notions in non-monotonic reasoning over logic programs.

De?nition1(cf.[22,23,24]).A model M of a disjunctive program P is called well-supported if there exists a function l:B P→N such that for each A∈M there exists a rule A∨H1∨···∨H m←A1,...,A k in ground(P)with M|=A i and l(A)>l(A i)for all i and k.

De?nition1is a straightforward adaptation of the notion of well-supported model for non-disjunctive programs,as given in[23].For non-disjunctive(and negation-free)programs,the well-supported models are exactly the minimal ones,but this is not in general the case for disjunctive programs:Just consider the program consisting of the single rule p∨q←.Then{p,q}is a well-supported model,but is not minimal.

Lifted appropriately to(non-disjunctive)programs with negation,the well-supported models coincide with the well-known stable models.This was shown in [23]and studied in-depth in[24,25].Stable models[26]provide the base for the most popular non-monotonic reasoning paradigm called Answer Set Program-ming,of which the two most prominent implementations are Dlv and Smodels [27,28].Our results thus stand within this well-established tradition.

It is apparent that A∈B P is entailed by a(disjunctive)program P by approximate SLD-resolution if and only if it is true in at least one well-supported model of P.This is called brave reasoning with well-supported models.A formal proof of the following proposition is omitted for space restrictions. Proposition1.Entailment of ground conjunctive queries under approximate SLD-resolution is brave reasoning with well-supported models.

As an example,consider the(propositional)program consisting of the two rules p∨q←and r←p∧q.Its minimal models are{q}and{p},so r is not bravely entailed by reasoning with minimal models.However all of{q},{p},{p,q}and {p,q,r}are well-supported models,so r is bravely entailed by reasoning with well-supported models.

Resolution-Based Approximate Reasoning for OWL DL391 There is an alternative way of formalizing approximate SLD-resolution using a modi?ed notion of split program[29].Given a rule

H1∨···∨H m←A1,...,A k,

the derived split rules are de?ned as:

H1←A1,...,A k...H m←A1,...,A k.

For a given disjunctive program P its split program P is de?ned as the collection of all split rules derived from rules in P.Approximate SLD-resolution on P is obviously identical to SLD-resolution over P .

Minimal models are well-supported,as can be seen from the following result which was obtained along the lines of research laid out in[24,25].

Theorem2([30]).Let P be a disjunctive program.Then a model M of P is a minimal model of P if and only if there exists a function l:B P→N such that for each A which is true in M there exists a rule A∨H1∨···∨H m←A1,...,A k in ground(P)with M|=A i,M|=H k and l(A)>l(A i)for all i and k.

We hence have the following result,noting that P|=Q for any ground con-junctive query Q and program P if and only if Q is true in all minimal models of P.

Proposition2.Let P be a(possibly disjunctive)program and Q be a ground conjunctive query with P|=Q.Then there exists an approximate SLD-refutation for Q.

We remark that for negation-free disjunctive programs minimal models again coincide with answer sets[26],as in the currently evolving Answer Set Program-ming Systems,as already mentioned.

5.2Approximate Resolution for OWL DL

Our proposal is based on the idea of converting a given OWL DL knowledge base into a function-free de?nite disjunctive logic program,and then to apply approximate resolution for ABox reasoning.

In order to be able to deal with all of OWL DL,we need to add a pre-processing step to get rid of nominals,i.e.we need to compile SHOIN(D) ontologies to SHIQ(D).We can do this by Language Weakening as follows:For every occurrence of{o1,...,o n},where n∈N and the o i are abstract or con-crete individuals,replace{o1,...,o n}by some new concept name D,and add ABox assertions D(o1),...,D(o n)to the knowledge base.Note that the trans-formation just given does in general not yield a logically equivalent knowledge base,so some information is lost in the process.Putting all the pieces together, we propose the following subsequent steps for approximate ABox reasoning for OWL DL.

392P.Hitzler and D.Vrandeˇc i′c

1.Apply Language Weakening as just mentioned in order to obtain a SHIQ(D)

knowledge base.

2.Apply transformations as in Section4in order to obtain a negation-free

disjunctive datalog program.

3.Apply approximate SLD-resolution for query-answering.

The?rst two steps can be considered to be preprocessing steps for setting up the intensional part of the database.ABox reasoning is then done in the last step. From our discussions,we can conclude the following properties of approximate ABox reasoning for SHIQ(D).

–It is complete with respect to?rst-order predicate logic semantics.

–It is sound and complete wrt.brave reasoning with well-supported models.–Data complexity of our approach is polynomial.

6Screech OWL

A preliminary implementation of our approach is available as the Screech OWL approximate reasoner.2It is part of the KAON2OWL tools.3

KAON24is the KArlsruhe ONtology framework,which includes a fast OWL reasoner based on the transformation algorithms mentioned in Section4,and also includes many other features helpful to work with ontologies.Among the KAON2 OWL tools,deo performs the language weakening step described in Section5.2 in order to obtain a SHIQ(D)knowledge base.As KAON2implements the sophisticated translation algorithms described in Section4,we can convert an OWL ontology into a disjunctive datalog program,e.g.by using the dlpconvert KAON2OWL tool with the-x switch.

Screech then accesses the results of the translation through the KAON2 API,creates the corresponding split programs and serialises them as Horn logic programs in Edinburgh Prolog syntax.The result can be fed to any Prolog interpreter—or other logic programming engine—,which in turn can be used to perform ABox reasoning and inferencing over the knowledge base.

For completeness,we need to mention that in general support for concrete do-mains and other features like integrity constraints is not necessarily implemented in o?-the-shelf logic programming systems.In these cases,concrete domains etc. cannot be used.The KAON2OWL tool ded,3for example,performs a language weakening step by removing all concrete domains,and may come in handy in such situations.

7An Example

We demonstrate our approach by means of a simple OWL DL ontology.It con-tains only a class hierarchy and an ABox,and no roles,but this will su?ce to display the main issues.

2http://logic.aifb.uni-karlsruhe.de/screech

3http://www.aifb.uni-karlsruhe.de/WBS/dvr/owltools

4https://www.doczj.com/doc/2a4117938.html,

Resolution-Based Approximate Reasoning for OWL DL393

serbian croatian european

eucitizen european

german french beneluxian eucitizen

beneluxian≡luxembourgian dutch belgian serbian(ljiljana)serbian(nenad)german(pascal)french(julien)

croatian(boris)german(markus)german(stephan)croatian(denny)

indian(sudhir)belgian(saartje)german(rudi)german(york)

Fig.3.Example ontology

The ontology is shown in Figure3,and its intended meaning is self-explanatory. Note that the fourth line,

beneluxian≡luxembourgian dutch belgian,

translates into the four clauses

luxembourgian(x)∨dutch(x)∨belgian(x)←beneluxian(x),(1)

beneluxian(x)←luxembourgian(x),

beneluxian(x)←dutch,

and beneluxian(x)←belgian(x).

Thus,our approach changes the ontology by treating the disjunctions in line(1)as conjunctions.This change a?ects the soundness of the reasoning procedure.However,most of the ABox consequences which can be derived by approximate SLD-resolution are still correct.Indeed,there are only two derivable facts which do not follow from the knowledge base by classical reasoning,namely

dutch(saartje)and luxemburgian(saartje).

All other derivable facts are correct.

Screech translates the ontology from Figure3into the Prolog program listed in Figure4.As standard implementations of SLD-resolution do not use fair selection functions and also use depth-?rst search for higher e?ciency,they may sometimes fail to produce answers because they run into in?nite branches of the search tree.This occurs,for example,when using SWI-Prolog5.A reordering of the clauses may improve the results,but does not solve the problem entirely. More satisfactory performance can be obtained by using SLD-resolution with tabling,as implemented e.g.in the XSB Prolog system6.In this case,all desired consequences can be derived.

5https://www.doczj.com/doc/2a4117938.html,/

6https://www.doczj.com/doc/2a4117938.html,

394P.Hitzler and D.Vrandeˇc i′c

serbian(ljiljana).serbian(nenad).german(pascal).french(julien). croatian(boris).german(markus).german(stephan).croatian(denny). indian(sudhir).belgian(saartje).german(rudi).german(york). european(X):-serbian(X).

european(X):-croatian(X).

european(X):-eucitizen(X).

eucitizen(X):-german(X).

eucitizen(X):-french(X).

eucitizen(X):-beneluxian(X).

beneluxian(X):-luxembourgian(X).

beneluxian(X):-dutch(X).

beneluxian(X):-belgian(X).

dutch(X):-beneluxian(X).

luxembourgian(X):-beneluxian(X).

belgian(X):-beneluxian(X).

Fig.4.Example Screech output

8Experiments and Evaluation

An approximate reasoning procedure needs to be evaluated on real data from practical applications.Handcrafted examples are of only limited use as the ap-plicability of approximate methods depends on the structure inherent in the experimental data.

For our evaluation we have performed experiments with the OWL DL version of the GALEN Upper Ontology,7as it appears to be su?ciently natural and real-istic.As it is a TBox ontology only,we populated GALEN’s175classes randomly with500individuals.8GALEN does not contain nominals or concrete domains. GALEN has673axioms(the population added another500).The TBox trans-lation to disjunctive datalog took about2300ms,after which we obtained2687 disjunctive datalog rules containing267disjunctions within133rules.Among these were152integrity constraints(i.e.rules with empty head),which we re-moved for our experiment as they led to inconsistency of the database.9After splitting disjunctive rules,we arrived at2802Horn rules.

We then randomly selected classes and queried for their extension using the KAON2datalog engine,both for processing the disjunctive datalog program and for the split program.Some of the typical results are listed in Table2,which indicates a signi?cant speed-up of more than40%on average,while the vast ma-jority of the retrieved answers is correct.Note that we obtain signi?cant speed-up although the KAON2datalog engine is not optimized for Horn programs,but rather tuned to e?cient performance on de?nite disjunctive datalog.

7https://www.doczj.com/doc/2a4117938.html,/~rector/ontologies/simple-top-bio/

8Using the pop KAON2OWL tool.

9This is an expected e?ect.Removal of the integrity constraints does not destroy completeness of the approximate reasoning procedure.

Resolution-Based Approximate Reasoning for OWL DL395 Table2.Performance comparison for instance retrieval using disjunctive datalog(DD) vs.the corresponding split program(SPLIT),on the KAON2datalog engine.Instances indicates the number of instances retrieved using DD versus SPLIT,e.g.class Multi-ple contained9individuals,while the split program allowed to retrieve13(i.e.the9 correct individuals plus4incorrect ones).The full name of the class in the last row is Biological object that has left right symmetry.

Time(DD)Time(SPLIT)Instances Class Name

11036ms6489ms154/154Biological object

11026ms5959ms9/9Speci?ed set

11006ms6219ms9/13Multiple

11015ms5898ms16/16Probe structural part of heart

11036ms7711ms4/4Human red blood cell mature

11055ms5949ms24/58Biological object that...

The times were obtained with initial Java VM memory set to256MByte.Un-der memory restrictions,the speed-up is more signi?cant,which is probably caused by the necessity to allocate additional memory for the DD reasoning task.Corre-sponding?gures are given in Table3.Our experiments also indicate that Screech may be useful when hardware is limited,for example in portable devices.

9Conclusions and Further Work

In a nutshell,our proposed procedure approximates reasoning by disregarding non-Horn features of OWL DL ontologies.We argue that this is a reasonable approach to approximate reasoning with OWL DL in particular because many of the cur-rently existing ontologies rarely use language constructs that do not fall into the Horn fragment of OWL DL[31].So it can be projected that even in the future these constructs will play a minor role and thus should be the?rst to be tempered with in order to gain tractable reasoning.

Our approach provides ABox reasoning with polynomial time complexity. While it is complete,it is also unsound with respect to?rst-order logic.We have shown,however,that the inference underlying our approach can be characterized using standard methods from the area of non-monotonic reasoning.We have also presented our implementation Screech,and veri?ed the usefulness of our approach by means of experiments.

Table3.Performance comparison as in Table2,but with128MByte intial memory

Time(DD)Time(SPLIT)Instances Class Name

32997ms4817ms154/154Biological object

33028ms4947ms9/9Speci?ed set

32927ms4987ms9/13Multiple

32977ms4957ms16/16Probe structural part of heart

32987ms7350ms4/4Human red blood cell mature

32947ms4796ms24/58Biological object that...

396P.Hitzler and D.Vrandeˇc i′c

The checking whether a conjunctive query is a predicate logic consequence of a (negation-free)disjunctive logic program P amounts to checking whether the query is valid in all minimal models of P,i.e.corresponds to cautious reasoning with minimal models.Theorem2suggests how an anytime algorithm for this might be obtained:After performing approximate SLD-resolution,it remains to be checked whether there is any(ground instance of a)rule used in the refutation of the query, which has an atom A in its head besides the one used in the refutation and such that A is(cautiously)entailed by the program.Such an algorithm might then?rst?nd a brave proof of a query,and then substantiate this proof by subsequent calculations. Our approach may also be useful for the quick derivation of possible answers to a query,which may then be used for e?cient guidance of the search within a sound and complete OWL reasoner.These and other issues are currently under investigation. Acknowledgement.We are grateful for discussions with Boris Motik about his and our work.

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