Consequence-Driven Reasoning for Horn SHIQ Ontologies
Yevgeny Kazakov
Oxford University Computing Laboratory
1 Introduction
Ontologiesare formal vocabularies of terms describing specific subjects like chemical elements, genes, or animal species. The terms in ontologies are “defined” by means of relationships with other terms of the ontology usingontology languages. Ontology languages based on Description Logics (DLs) [1], such as OWL [2], are becoming increasingly popular among ontology developers thanks to the availability ofontology reasoners, which provide automated support for visualization, debugging, and querying of ontologies.Classificationis a central reasoning service provided by ontology reasoners.
The goal of classification is to compute a hierarchical relation between classes. The class hierarchy is used to browse ontologies in ontology editors.
Most of the currently-available ontology reasoners are based on so-calledmodel building proceduressuch as tableau [3] and hyper-tableau [4] calculi. Such procedures classify an input ontology, in general, by iterating over all necessary pairs of classes, and trying to build a model of the ontology that violates the subsumption relation between them. Recent investigations of tractable DLs such asEL++[5] led to the discovery of another type of reasoning procedures. Instead of enumerating pairs of classes and building counter-models, the procedure derives subsumption relations explicitly using special inference rules. The advantage is that subsumption relations are computed “all at once” in a goal-directed way without costly enumerations.
TheEL++language sacrifices many commonly-used constructors, such as inverse roles and functional restrictions, in order to ensure that reasoning is tractable. In this paper we describe a reasoning procedure, which utilizes the core reasoning technique of EL++, but works for the larger class of so-called HornSHIQontologies. HornSHIQ ontologies are ontologies expressed in the DLSHIQ, which do not contain “non- deterministic” constructors, such as positive disjunction:AvBtC. HornSHIQhas originally drawn attention due to its tractabledata complexity[6]. Our paper demonstrates that HornSHIQontologies can have a more efficient classification procedure.
This paper is organised as follows. In the preliminaries we introduce HornSHIQ and a completion-based procedure for a fragment ofEL++. In Section 3 we discuss how to extend this procedure for new constructors in HornSHIQ. In Section 4 we present our reasoning procedure and prove soundness, completeness, and termination. Finally, in Section 5 we present an empirical comparison of a prototypical implementation of our procedure with other reasoners, which demonstrates a substantial performance improvement.
This paper is an adapted version of [7].
Name Syntax Semantics Concepts:
atomic concept A AI (given)
top concept > ∆I
bottom concept ⊥ ∅
negation ¬C ∆I\CI
conjunction CuD CI∩DI
disjunction CtD CI∪DI
existential restriction ∃R.C {x|RI(x, CI)6=∅}
universal restriction ∀R.C {x|RI(x, ∆I\CI) =∅}
min cardinality >nS.C {x| ||SI(x, CI)|| ≥n}
max cardinality 6mS.C {x| ||SI(x, CI)|| ≤m}
Axioms:
role inclusion R1vR2 RI1 ⊆RI2
role transitivity Tra(R) RI◦RI⊆RI concept inclusion CvD CI⊆DI
Table 1.The syntax and semantics ofSHIQ
2 Preliminaries
In this section we define the description logicsSHIQandEL+, as well as their frag- ments HornSHIQandELH. To simplify the presentation and save space, we will not consider ABox assertions in this paper.
2.1 The Description LogicSHIQ
A description logicvocabularyconsists of countably infinite setsNCofatomic concepts, andNRofatomic roles. ASHIQroleis eitherr ∈ NRor aninverse roler− with r∈NR. We denote byR−theinverse of a roleRdefined byR−:=r−whenR=r andR−:=rwhenR=r−.
The syntax and semantics ofSHIQis summarized in Table 1. The set ofSHIQ conceptsis recursively defined using the constructors given in the upper part of Table 1, whereA∈NC,C,Dare concepts,R,Sroles, andn,mpositive integers. Aterminology orontologyis a setO of axioms specified in the lower part of Table 1. A roleR is transitive (inO)ifTra(R)∈ OorTra(R−)∈ O. Given an ontologyO, letR1vO R2 be the smallest transitive reflexive relation between roles such thatR1 v R2 ∈ O impliesR1vO R2andR−1 vOR−2. For concepts of the form>nS.Cand6mS.Cin Othe roleSmust besimple, that is,RvO Sholds for no transitive roleR.
The semantics ofSHIQis defined using interpretations. Aninterpretationis a pair I = (∆I,·I)where ∆I is a non-empty set called the domain of the interpretation and·I is theinterpretation function, which assigns to everyA∈NCa setAI ⊆∆I, and to everyr ∈ NR a relationrI ⊆ ∆I×∆I. The interpretation is extended to roles by(r−)I:={hx, yi | hy, xi ∈rI}and to concepts according to the right column of Table 1, whereρ(x, U)forρ ∈ ∆I ×∆I, U ⊆ ∆I, andx ∈ ∆I denotes the set{y | hx, yi ∈ρ∧y∈U}, and||V||denotes the cardinality of a setV ⊆∆I. An
interpretationIsatisfies an axiomα(writtenI |=α) if the respective condition to the right of the axiom in Table 1 holds;Iis amodel of an ontologyO(writtenI |=O) ifI satisfies every axiom inO. We say thatαis a(logical) consequenceofO, oris entailed byO(writtenO |=α) if every model ofOsatisfiesα.
Classificationis a key reasoning problem for description logics and ontologies, which requires to compute allsubsumptionsAvBentailed byObetween atomic concepts.
2.2 The Horn Fragment ofSHIQ
Positiveandnegative polaritiesof occurrences ofSHIQ concepts in concepts and axioms are defined as follows:
– Coccurs positively inC;
– Coccurs positively (negatively) in concepts of the forms¬C2,C1uD1,C1tD1,
∃R.C1,∀R.C1,>nS.C1,6mS.C2, andC2vD1 ifC occurs positively (nega- tively) inC1or inD1, or negatively (positively) inC2(when applicable).
A conceptCoccurspositively(negatively) in an ontologyOifCoccurs positively (negatively) in some axiom ofO. Note that it is possible for a concept to occur positively and negatively at the same time in an axiom or an ontology.
ASHIQontologyOisHornif:
– no concept of the formCtDor6mR.Cwithm >1occurs positively inO;
– no concept of the form¬C,∀R.C,>nR.Cwithn >1, or6mR.C occurs nega- tively inO.
2.3 Structural Transformation
Structural transformation is used to simplify the axioms of the ontology preserving its
“structure”. Given aSHIQontologyO, for every (sub-)conceptCinOwe introduce a fresh atomic conceptACand define a functionst(C)by:
– st(A) =A; st(>) =>; st(⊥) =⊥; st(¬C) =¬AC; – st(CuD) =ACuAD; st(CtD) =ACtAD; – st(∃R.C) =∃R.AC; st(∀R.C) =∀R.AC;
– st(>nR.C) =>nR.AC; st(6mR.C) =6mR.AC.
The result of applyingstructural transformationtoOis an ontologyO0that contains all role inclusion and role transitivity axioms inOin addition to the following axioms:
– ACvst(C)for everyCoccurring positively inO – st(C)vACfor everyCoccurring negatively inO – ACvADfor every concept inclusionCvD∈ O
It can be shown that structural transformation preserves the logical consequences of ontological axioms.
Proposition 1. LetO0be obtained fromOby structural transformation. Then for every axiomαcontaining no introduced atomic conceptsAC, we haveO |=αiffO0|=α.
IR1 AvA IR2Av > CR1 AvB
AvC :BvC∈ O
CR2AvB AvC
AvD :BuCvD∈ O CR3 AvB
Av ∃r.C :Bv ∃r.C∈ O
CR4Av ∃r.B
Av ∃s.B :rvs∈ O CR5 Av ∃r.B BvC
AvD :∃r.CvD∈ O Table 2.The Completion Rules forELH
2.4 The Description LogicEL+
The description logicEL+ is one of the few description logics for which standard reasoning problems are decidable in polynomial time.EL+ allows only for concepts constructed from atomic conceptsAand the top concept>using conjunctionCuD and existential restriction∃r.C, whereris an atomic role. The axioms ofEL+can be either concept inclusionsCvDorcomplex role inclusionsof the formr1◦ · · · ◦rnvs.
The last are interpreted inIasrI1 ◦ · · · ◦rIn ⊆sI, where◦is a composition of binary relations. For the purpose of this paper, we consider a common fragment ofEL+and SHIQcalledELH, which allows only for (simple) role inclusions of the formrvs. It can be easily seen that everyELHontology is a HornSHIQontology since it cannot contain concepts of the form¬C,CtD,∀R.C,>nR.C, or6mR.C.
In [5], a polynomial time classification procedure has been presented for the de- scription logicEL++, which extendsEL+with the bottom concept⊥, nominals, and
“safe” concrete domains. The procedure uses a number of so-calledcompletion rulesthat derive new concept inclusions. In Table 2 we list the completion rules relevant toELH.
The rules are applied to anormalisedELHontologyOthat is obtained from the input ontology by structural transformation and simplification. Structural transformation for ELHproduces only axioms of the form(1)AvB,(2)AuBvC,(3)AvBuC, (4)Av ∃r.B,(5)∃r.AvB, and(6)rvs, whereA,B, andCare atomic concepts or>, andr,satomic roles. AxiomsAvBuCof form(3)are then replaced with a pair of axiomsAvBandAvCof form(1).
The completion rules in Table 2 derive new axioms of the formAvBandCv ∃r.D from already derived axioms using axioms inOas side conditions, whereA,B,C, and Dare atomic concepts or>, andr,satomic roles.1In [5] it was shown that the rules
IR1–CR5arecomplete for classification, that is, a concept subsumptionAvBis entailed byOif and only if it is derivable by these rules.
3 From ELH to Horn SHIQ
HornSHIQextendsELHby allowing many new constructors, notably, inverse roles, functionality restrictions, and positive occurrences of universal restrictions. These con-
1In [5], instead of deriving new axiomsAvBandCv ∃r.D, the completion rules addB to the setS(A)and a pair(C, D)to the setR(r). Thus the setS(A)represents the set of super-concepts ofAand the setR(r)represents the pairs ofr-existentially related concepts.
structors have been disallowed inELHandEL++for complexity reasons—in [5, 8] it was shown that adding any of these constructors results in a complexity increase from PTimetoExpTime. In this section we give an informal explanation for this complexity increase and outline how the completion-based procedure forELHcan be extended to handle the new HornSHIQconstructors.
3.1 Inverse Roles and Universal Restrictions
According to the definition in Section 2.2, HornSHIQallows for the usage of inverse roles and positive occurrences of universal restrictions. In fact, positive occurrences of universal restrictions can be expressed by means of inverse roles using the equivalence:
Av ∀R.B ⇔ ∃R−.AvB. (1) The increase in the complexity of reasoning with inverse roles and universal restric- tions, can be partially explained by a new kind of interaction between these constructors.
Axioms of the formAv ∃R.B andCv ∀S.D can interact in two ways. First, the following inference (2) is possible:
Av ∃R.B Bv ∀R−.D
AvD (2)
It is easy to see using equivalence (1) that (2) is an instance ofCR5in Table 2. A new kind of interaction is inference (3):
Av ∃R.B Cv ∀R.D
AuCv ∃R.(BuD) (3)
In contrast to (2), and all inference rules in Table 2, which have only conclusions of the formAvBorAv ∃R.B, repeated applications of (3) can produce axioms of the form dAiv ∃R.d
Bj, whered
Aiandd
Bjare arbitrary conjunctions of atomic concepts.
The number of such axioms is no longer polynomially bounded. Note, however, that there are at most exponentially many (semantically non-equivalent) axioms of this form since the number of different concepts in the conjunctions is always bounded. We will use this property to establish an exponential upper bound for our procedure.
3.2 Functional Restrictions on Roles
HornSHIQallows for expressing a functional restriction for a simple roleS using the constructor61S.>. Since, for a functional roleS, the axiomC v ∃S.Dimplies Cv ∀S.D, the following analogs of inferences (3) and (2) are possible:
Av ∃S.B Cv ∃S.D Av61S.>
AuCv ∃S.(BuD) (4)
Av ∃S.B B v ∃S−.C Bv61S−.>
AvC (5)
In Section 4, we generalize inferences (2)–(5) and present a sound and complete system of inference rules for HornSHIQontologies, which derives only consequence axioms of the formd
AivBandd
Aiv ∃R.(d Bj).
4 The Calculus for Horn SHIQ Ontologies
In this section we present a consequence-based procedure for classifying HornSHIQ ontologies. The procedure consists of a preprocessing stage that applies structural transformation, simplifications, and elimination of transitivity, and asaturation stage that applies inferences to derive new axioms.
4.1 Normalization
From now on we useA(i),B(i)to denote atomic concepts,R(i)roles, andS(i)simple roles. We say that a conceptC issimpleif it is of the form ⊥,A,∃R.A,∀R.A, or 61S.A. We denote byd
Aia (possibly empty) conjunction of atomic concepts. The empty conjunction is abbreviated to>.
Lemma 1. Every HornSHIQontologyOcan be transformed into an ontologyO0 containing only axioms of the forms: (n1)d
Ai vC, (n2)R1vR2, and (n3)Tra(R), whereCis a simple concept. The transformation preserves (non)entailment of axioms αover the signature ofOand can be performed in polynomial time assuming unary coding of numbers.
Proof. By applying structural transformation toO, we obtain an ontologyO0containing only concept inclusions of the formA1 vA2,A vst(C1), andst(C2)vA, where C1occurs positively inOandC2occurs negatively inO. SinceOis a HornSHIQ ontology,C1can only be of the form>,⊥,A,¬C,CuD,∃R.C,∀R.C,>nS.C, or 61S.C, andC2only of the form>,⊥,A,CuD,CtD,∃R.C, or>1R.C.
Concept inclusions of the formAvst(C1)that are not of form (n1), are transformed to form (n1) as follows:
– Avst(¬C) =¬AC ⇒ AuACv ⊥;
– Avst(>nS.C) =>nS.AC ⇒ Av ∃S.Bi,BivAC,1≤i≤n,BiuBj v ⊥, 1≤i < j≤n, whereBiare fresh atomic concepts.
Concept inclusions of the formst(C2)vAthat are not of form (n1) are transformed to form (n1) as follows:
– st(CtD) =ACtADvA ⇒ ACvA,ADvA;
– st(∃R.C) =∃R.ACvA ⇒ ACv ∀R−.A;
– st(>1S.C) =>1S.ACvA ⇒ ACv ∀S−.A.
It is easy to show using Proposition 1, thatO0 |=αiffO |=αfor every axiomα
containing no new symbols. ut
4.2 Elimination of Transitivity
After normalization, we apply a well-known technique, which allows the elimination of transitivity axioms. Transitivity axioms of form (n3) in Lemma 1 can interact only with axiomsd
Aiv ∀R.Bof form (n1) through role inclusions (n3). The transformation
introduces a triple of axioms (6) for every axiomd
Aiv ∀R.Bof form (n1) and every transitive sub-roleTofR, whereBT is a fresh atomic concept:
dAiv ∀T.BT BT v ∀T.BT BT vB (6) Intuitively,BT is used to propagateBto all elements reachable from elements ind
Ai
via aT-chain.
Lemma 2. LetObe consisting of axioms of forms (n1)–(n3) in Lemma 1, andO0be obtained fromOby adding axioms(6)as described above and removing all transitivity axioms (n3). Then for every axiomαthat does not contain conceptsBT and non-simple roles, we haveO |=αiffO0|=α.
Proof (sketch).It is sufficient to show that(i)every modelIofOcan be turned into a modelJ ofO0by (re-)interpreting conceptsBT, and conversely,(ii)every modelJ ofO0can be turned into a modelIofOby (re-)interpreting non-simple roles. Both of these transformations preserve the interpretation of axiomsαthat do not contain these symbols.
For proving(i), we interpretBT in J as(B u ∀T.B)I. Trivially, the last two axioms in (6) are satisfied inJ. The first axiom in (6) is satisfied inJ sinceJ satisfies dAiv ∀R.BandTis a transitive sub-role ofR.J satisfies the remaining axioms in O0since they do not contain the new conceptsBT.
For proving(ii), we interpret every non-simple roleRinIas the union ofRJ and the transitive closures ofTJ for every transitive sub-roleT ofR(TJ is not necessarily a transitive relation sinceO0does not contain the transitivity axioms). It is easy to show thatI satisfies all axioms (n1)–(n3). In particular,I satisfies the axioms of the form dAiv ∀R.BsinceIremains to be a model of axioms (6). ut
4.3 The Inference Rules
Our procedure for HornSHIQontologies works by saturating the input axioms under the rules in Table 3. The rules are applied to a preprocessed ontology containing axioms of form (n1) and (n2) in Lemma 1, and produce axioms of the formM v C and M v ∃R.N, whereMandN are conjunctions of atomic concepts, andCis a simple concept.
RulesI1andI2produce the initial (tautological) concept inclusions for arbitraryM andA, which are then extended using axioms inOby ruleR1. RulesR2–R6are specific to the kinds of simple concepts implied by the conjunctions. RuleR2propagates the entailment of bottom backwards over existential restrictions. RulesR3andR4deal with universal restrictions and generalize respectively inferences (3) and (2) in Section 3.1.
Likewise, rulesR5andR6with 7 premises, deal with qualified functional restrictions and generalize inferences (4) and (5). Note that in contrast to (5), ruleR6has two conclusions.
RulesR3–R6 assume that the closureR1 vO R2 of role inclusions of form (n2) is computed. The inference rulesI1–R6are applied exhaustively until no new axiom can be derived. The resulting ontologyO0is calledthe saturationofO.
I1 MuAvA I2 M v >
R1 MvA1 . . . MvAn
MvC :
n
l
i=1
AivC∈ O R2M v ∃R.N N v ⊥ Mv ⊥
R3 Mv ∃R1.N M v ∀R2.A
M v ∃R1.(NuA) :R1vOR2 R4M v ∃R1.N Nv ∀R2.A
MvA :R1 vOR−2
R5
Mv ∃R1.N1 N1vB Mv ∃R2.N2 N2vB Mv61S.B
M v ∃R1.(N1uN2) :R1 vOS
R2 vOS R6
M v ∃R1.N1 M vB N1v ∃R2.(N2uA) N1v61S.B N2uAvB
M vA M v ∃R−2.N1
:R1vOS− R2vOS Table 3.Saturation Rules for HornSHIQOntologies
4.4 Soundness and Completeness
In this section we demonstrate that our procedure is sound and complete for classification, that is, it derives all subsumptions between atomic concepts in the input ontology.
Theorem 1. LetObe an ontology containing only axioms of the form (n1) and (n2) in Lemma 1 andO0the saturation ofOunder the rules in Table 3. Then(i)O |=M vB iffM v ⊥ ∈ O0orM vB∈ O0, and(ii)O |=M v ∃R.NiffM v ⊥ ∈ O0or there existsM v ∃R1.N1∈ O0such thatR1vO RandN1vB∈ O0for every conjunctB inN.
Proof (sketch).It is easy to show that every axiom inO0 is entailed byOsince the inference rules in Table 3 derive only logical consequences of the axioms in their premises. In other words, the system of rules is sound. Therefore the “if” direction of points(i)and(ii)are trivial.
In order to prove the “only if” direction of points(i) and(ii), we construct a canonical modelofO0. LetΣbe the set of all conjunctionsM such thatM v ⊥∈ O/ 0. IfΣis empty, thenM v ⊥ ∈ O0for everyM, and so,(i)and(ii)hold. In the remainder of the proof, we assume thatΣis not empty.
LetI = (∆I,·I)be such that∆I =Σ+is the set of all finite non-empty words overΣ, and·Idefined as follows:
– AI={wM |M vA∈ O0, w∈Σ∗}
– The roles are interpreted with smallest relations satisfying all role inclusions such that ifM v ∃R.N∈ O0andM v ∃R.N0∈ O/ 0for every super-conjunctN0ofN, then for everyw∈Σ∗, eitherhwM, wi ∈RI andw∈AIfor everyA ∈N, or otherwisehwM, wM Ni ∈RI.
Intuitively, the interpretationIis a forest where nodes are labeled with non-contradictory conjunctionsM ∈Σand every node has a successor node for every non-contradictory conjunctionN ∈ Σ. Roles can connect only successive nodes in such a way that all axiomsM v ∃R.N ∈ O0 with maximalN are satisfied: if for a node marked with
M the connection to its predecessor does not satisfy this axiom, it is connected to its successor marked withN. RuleR3ensures thatNis not contradictory ifMis not. Note thathwM, wM Ni ∈RI impliesM v ∃R1.N ∈ O0for someR1 vO R. Below we demonstrate thatIis a model ofO(and consequently ofO0).
Claim 1 wM ∈Σ+andM vC∈ O0implywM ∈CI. – ForC=⊥, ifM v ⊥ ∈ O0thenwM6∈Σ+.
– ForC=Athe claim follows from the definition ofAI.
– ForC=∃R.B, takeM v ∃R.N∈ O0with the largestN containingB(it exists sinceM v ∃R.B∈ O0). By definition ofRI, we havewM ∈(∃R.N)I⊆CI. – ForC=∀R.B, we show that:(i)hwM, wM Ni ∈RIimplieswM N ∈BI, and
(ii)hwN M, wNi ∈RIimplieswN ∈BI.
In case(i)there isM v ∃R1.N ∈ O0withR1vOR. SinceM vC∈ O0andO0 is closed underR3, we haveN =NuB(sinceNis a maximal conjunction), and so,wM N ∈BI. In case(ii), there isN v ∃R1.M∈ O0withR1vO R−. Since M vC∈ O0andO0is closed underR4, we haveN vB∈ O0, and so,wN ∈BI. – ForC =61S.B, we show that:(i)hwM, wM Nii ∈SI andwM Ni ∈BIfor
i= 1,2impliesN1 =N2, and(ii)hwN M, wNi ∈ SI, andwN ∈BI implies hwN M, wN M N1i∈/ SIwheneverwN M N1∈BI.
In case(i)we have M v ∃Ri.Ni ∈ O0 whereRivO S and Ni v B ∈ O0, i= 1,2. SinceO0is closed underR5, we haveM v ∃Ri.(N1uN2)∈ O0,i= 1,2.
Since eachNi is maximal forM andRi,i = 1,2, we haveN1 = N1uN2 = N2. In case (ii), we have Nv ∃R.M ∈ O0, R vO S−, and N v B ∈ O0. Assume to the contrary thathwN M, wN M N1i ∈SIandwN M N1∈BI. Then hwN M, wN M N1i ∈RI1 for someR1vO Ssuch thatM v ∃R1.N1∈ O0, and N1 vB ∈ O0. SinceO0 is closed underR6, we haveN v ∃R1−.M ∈ O0 and NvA∈ Ofor every conjunctAinN1. HenceM v ∃R1.N1is already satisfied inwN M by its predecessor wN, and so the relation hwN M, wN M N1i ∈ RI1 should not have been created.
Now for every axiomd
AivCof form (n1) inO, ifwM∈AIi thenM vAi∈ O0, and so,M vC ∈ O0 byR1. By Claim 1 we then havewM ∈CI. Thus,I satisfies every axiomd
AivCinOof form (n1). The definition of the interpretation of roles inIensures that every role inclusion of form (n2) is also satisfied inI. Hence,Iis a model ofO.
Now we can finish the proof for the “only if” directions of points(i)and(ii). For proving(i), ifO |=M vB andM v ⊥ ∈ O/ 0 then for everywM ∈ Σ+, we have wM ∈BIsinceIis a model ofO. HenceM vB ∈ O0by definition ofI. For proving (ii), ifO |=M v ∃R.NandM v ⊥∈ O/ 0, then forM ∈Σ+, we haveM ∈(∃R.N)I.
SinceM ∈Σ+does not have a predecessor inΣ+, there existsM v ∃R1.N1∈ O0 such thatR1vO RandM N1∈BIfor every conjunctBinN. By definition ofI, this implies thatN1vB∈ O0for every conjunctBinN. ut Theorem 1, in conjunction with Lemmas 1 and 2, implies that our procedure based on the rules in Table 3 can be used, in particular, for computing all subsumptions A vB between atomic concepts implied by the input ontology. In other words, the
procedure is complete for classification of HornSHIQontologies. Moreover, since the number of axioms of the formM vCandM v ∃R.Nis at most exponential in the number of atomic concepts, the procedure is guaranteed to terminate in exponential time.
Since deciding concept subsumptions in HornSHIQontologies isExpTime-complete (inExpTimesince HornSHIQ is a fragment ofSHIQandExpTime-hard since, e.g., HornSHIQcontainsELwith functionality), this implies that our procedure is computationally optimal.
4.5 Practical Considerations
Theorem 1 actually implies a stronger result than just completeness of the calculus for classification. The rules in Table 3 can, in fact, be used to derive all subsumptions between conjunctions of atomic concepts and (existentially restricted conjunctions of) atomic concepts. For computing just a subset of these relations, e.g., for classification, our procedure can be significantly optimized by applying rulesI1and I2selectively depending on the goal subsumptions. Specifically,I1andI2can be restricted to produce conjunctionsN only when either(i)N is the left hand side of a goal subsumption (e.g., for classification,Nis an atomic concept from the input ontology), or(ii)an axiom of the formM v ∃R.Nhas been derived.
The above optimization not only reduces the number of inferences, but also yields a polynomial time classification procedure forELHontologies. Indeed, it is easy to see that forELHontologies the rulesR3,R5, andR6never apply. The remaining rules produce only polynomially many axioms of the formAvCandAv ∃R.Bsince those rules (includingI1andI2) never form conjunctions of several concepts.
5 Experimental Evaluation
The main goal for our consequence-based procedure for HornSHIQontologies, was to reason efficiently with medium-sized ontologies such as Galen [9]. Galen turned out to be very difficult for model-building reasoners since it contains a large number of existential dependencies between classes of the formAv ∃R.B. These dependencies are used, e.g., for expressing partonomy relations between anatomical units, for exam- ple:BasilarArteryv ∃isBranchOf.VertebralArtery. The dependency relation between classes induced by such axioms is very large and highly cyclic. Because of the numerous cycles, model building reasoners fail to produce (a finite representation of) a model since cycle detection mechanisms (blocking) do not appear to be effective in these cases.
The reasoner CEL forEL+ does not exhibit this problem since it does not build models but performs inferences according to Table 2. Axioms of the formAv ∃R.B are relatively harmless for CEL since these axioms alone do not result in any inference.
Unfortunately, CEL cannot handle the full version of Galen since it does not support functionality and inverse roles, which Galen extensively uses. Thus the main question of our evaluation was whether the reasoner implemented according to the rules in Table 3 can scale well for ontologies outside the polynomial DLEL+.
To evaluate the performance of our procedure, we implemented a prototype reasoner CB [10] using a simplified version of the rules in Table 3 for Horn SHIF ontolo- gies, which are sufficient for Galen. We compared the results with the model-building
FaCT++ Pellet HermiT CEL CB
GO 15.24 72.02 199.52 1.84 1.17
NCI 6.05 26.47 169.47 5.76 3.57
GalenA− 3166.74 133.25 91.98 3.27 0.26
GalenA 465.35 — 45.72 n/a 0.32
GalenB− — — — 189.12 4.07
GalenB — — — n/a 9.58
Snomed 650.37 — — 1185.70 49.44
Table 4.A comparison of classification times (in seconds): “—” means that the reasoner failed to return the result; “n/a” means that the test could not be performed.
reasoners FaCT++ v.1.2.1 [11], Pellet v.1.5 [12], and HermiT v.0.9.3 [13], and the completion-based reasoner CEL v.1.0b [14]. We ran the experiments on a PC with a 2GHz IntelrCoreTMDuo processor and 1.5GB RAM operated by Linux v.2.6.27 with Java VM v.1.6.0. We set time out of 3600 seconds and Java heap space of 1GB for Java-based reasoners.
We tested the classification of seven medical ontologies of various sizes and com- plexities. The Gene Ontology (GO) [15] and National Cancer Institute thesaurus (NCI) [16] are fairly large ontologies describing respectively 20465 and 27652 classes, but with a rather shallow structure inducing not many dependencies. We have considered four versions of Galen. GalenA with 2748 classes is based on an early version of Galen [17].
GalenB with 23136 classes is based on a more recent version of Galen [9]. GalenA−and GalenB−are obtained from GalenA and GalenB by removing functionality and inverses.
We have also included the results for the very large but simple ontology Snomed, which contains 389472 classes. The results of the experiments are summarized in Table 4.
The time measured for CB includes loading, preprocessing, classification of the ontology and computing the transitively reduced and alphabetically sorted concept hierarchy. CB spends most of the time on loading for GO and NCI and on classification for the remaining ontologies, and considerably outperforms the other tested reasoners.
In particular, CB is able to classify in just under 10 seconds the largest version of Galen, which could not be classified by the other reasoners.
It is worth commenting on the difference in the performance between CEL and CB.
Although the rules in Table 3, when restricted toEL+ontologies, are very similar to those in Table 2, there is a small but important difference. Namely, our procedure has no analogue of ruleCR4in Table 2, which unfolds role inclusions into existential restrictions.
Instead, the closure of role inclusions is computed separately and used in the premises of rulesR3-R6. This strategy appears to be more efficient in situations when the number of role inclusions is small in comparison to the number of existential restrictions (which is mostly the case), and could explain the test results.
References
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9. http://www.co-ode.org/galen/
10. http://code.google.com/p/cb-reasoner/
11. http://owl.man.ac.uk/factplusplus/
12. http://clarkparsia.com/pellet/
13. http://www.hermit-reasoner.com/
14. http://lat.inf.tu-dresden.de/systems/cel/
15. http://www.geneontology.org/
16. http://www.cancer.gov/
17. http://www.comlab.ox.ac.uk/people/yevgeny.kazakov/ontologies/
galen-ian-horrocks-original.owl.zip
