Complete Classification of Complex ALCHO Ontologies Using a Hybrid Reasoning Approach

Complete Classification of Complex ALCH O Ontologies using a Hybrid Reasoning Approach

Weihong Song, Bruce Spencer, and Weichang Du

Faculty of Computer Science, University of New Brunswick, Fredericton, Canada {song.weihong, bspencer, wdu}@unb.ca

Abstract. Consequence-based reasoners are typically significantly faster than tableau-based reasoners for ontology classification. However, for more expres- sive DL languages likeALCH O, consequence-based reasoners are not appli- cable, but tableau-based reasoners can sometimes require an unacceptably long time for large and complex ontologies. This paper presents a weakening and strengthening approach for classification ofALCH Oontologies, using a hybrid of consequence- and tableau-based reasoning. We approximate the original ontol- ogyO_oby a weakened versionO_wand a strengthened versionO_s, both are in a less expressive DLALCHand classified by a consequence-based main reasoner. The classification fromO_wis sound but possibly incomplete with respect toO_o, while that fromO_sis complete but possibly unsound. The additional subsumptions de- rived fromO_smay be unsound so are further verified by a tableau-based assistant reasoner. A prototype classifier called WSClassifier is implemented based on this hybrid approach. The experiments results show that for classifying many large and complexALCH Oontologies, WSClassifier’s performance is significantly faster than tableau-based reasoners.

1 Introduction

The target of classification is to calculate all the subsumption relationships between atomic concepts implied by the input ontology. (Hyper)Tableau-based [9,13] and consequence-based reasoners are two of the mainstream reasoners for ontology clas- sification. Current (hyper)tableau-based reasoners such as HermiT [13], FaCT++[20], Pellet [18] and RacerPro [8], are able to classify ontologies in very expressive DLs.

However, despite various optimizations having been applied, classifying certain exist- ing large and complex ontologies is still a challenge for these reasoners, such as various versions of Galen and FMA ontologies. We regard an ontology complex if it is highly cyclic. In contrast to the (hyper)tableau-based reasoners, consequence-based reasoners [10,11,17] are typically very fast but support less expressive DLs. They are variations of so-called completion-based approaches proposed for the OWL EL family [3,5]. So far the most expressive languages that are supported by consequence-based reasoners are Horn-SH IQ[10] andALCH[17].

In this paper we introduce a hybrid reasoning approach for classification of on- tologies in the DL ALCH O, using a consequence-based main reasoner MR and a tableau-basedassistant reasonerAR. MRprovides sound and complete classification over the DLALCHwhich is less expressive thanALCH O, whileARprovides sound

and complete classification overALCH O. SupposeMRreasoning is much faster than AR.We try to classify an ontologyO_ousingMRto do the major work, andARto do auxiliary work. We produce a weakened versionO_wby removing fromO_othe nominal axioms that are beyondALCH, and a strengthened versionO_sby adding toO_wa set of strengthening axiomsO⁺inALCH that compensate for the removed axioms.O_sand O_ware inALCH and are classified by MRproducingH_wandH_s, correspondingly.

H_w,H_oandH_sare classification results ofO_w,O_oandO_s, respectively. Subsumptions inH_w are sound but may not be complete w.r.tO_o, whereas subsumptions inH_s are complete but may not be sound. Unsound subsumptions inH_s\ H_ware detected byAR and filtered out. Those that remain are added toH_w resulting in the sound and complete classification ofO_o. We call this approach weakening and strengthening (WS).

We have implemented a prototype WSClassifier by applying the proposed WS approach and evaluated it. Our previous study on WS approach and application to ALCH OIontologies [19] did not guarantee the completeness of classification. The contribution of this paper is to improve the algorithms and theoretically prove our clas- sification result onALCH Oontologies is sound(trivial) and complete.

The rest of the paper is organized as follows: Section 2 gives an overview of our hybrid classification procedure forALCH O ontologies and prove its completeness.

Section 3 introduces computation of strengthening axioms and Section 4 contains the related work. Section 5 is our partial empirical results and conclusion. Finally, we put all the proofs of lemmas and theorems in Appendix B and complete evaluation in Ap- pendix C.

2 Hybrid Classification of Ontologies

The syntax and semantics ofALCH Ofollows DL conventions. In this paper, we use A,B,E,F for atomic concepts,C,Dfor concepts,R,S for roles,a,bfor individuals, H,Kfor conjunctions of atomic concepts, andM,Nfor disjunctions.

Algorithm 1 describes our hybrid procedure for classifyingO_o. It consists of three stages: (1) a normalization stage (line 1) during which the ontology is rewritten to simplify the forms of axioms in it; (2) a main classification stage (lines 2 to 8) in whichO_wandO_sare generated and classified using theMR;and (3) averification stage (lines 9 to 17) in which the subsumptions arising from just theO_sare verified usingAR.

We use notationsCandCoto denote the set of atomic concepts inO_obefore and after normalization, respectively, andC^>,⊥=C∪ {>,⊥}.

In the normalization stage, theALCH OontologyO_o is rewritten to contain only axioms of formsd

Ai vFBj,Av ∃R.B,∃R.A vB,Av ∀R.B,R vS, orN_a ≡ {a}. The procedure extends normalization in Frantiˇseket al.[17] by introducing for each {a}anominal placeholder N_aand adding anominal axiom N_a ≡ {a}. We writeNPfor the set of all nominal placeholders after normalization. The transformation preserves subsumptions inO_o. We assume all ontologies are normalized in the rest of the paper.

In the verification stage, there are some cases we hand over the classification work toAR:(1)N_a v ⊥ ∈ H_s; (2)E v ⊥ ∈ H_sbutO_o 6|=E v ⊥, then for everyF ∈C^>,⊥, EvF ∈ H_s, while likely only few of them are inH_w, thusH_s\ H_wmay be huge; (3) the fractionk H_s\ H_w k/kC kis greater than a thresholdd. In the latter two cases,

Algorithm 1:HybridClassify(O_o) Input: AnALCH OontologyO_o Output: The classification result ofO_o

1 normalizeO_o;

2 O_w← O_owith nominal axiomsN_a={a}removed; /* Weakening */

3 H_w← MR.classify(O_w); /* Classify the weakened ontology. */

4 O⁺← getNominalStrAx(O_w,NP,C^>,⊥); /* from Algorithm 4 */

5 remove allEvFfromH_wwherehE,Fi<C^>,⊥×C^>,⊥;

6 ifO⁺=∅then returnH_w;

7 O_s← O_w∪ O⁺;

8 H_s←MR.classify(O_s); /* Classify the strengthened ontology. */

9 ifN_av ⊥ ∈ H_sfor some N_a∈NPthen returnAR.classify(O_o);

10 remove allEvFfromH_swherehE,Fi<C^>,⊥×C^>,⊥;

11 if k H_s\ H_wk/kCk>dthen returnAR.classify(O_o);

12 H_ws← H_w;

13 foreachEv ⊥ ∈ H_s\ H_wdo

14 ifAR.isSatisfiable(O_o,E)then returnAR.classify(O_o);

15 elseaddEv ⊥intoH_ws;

16 foreachEvF∈ H_s\ H_wwhere F,⊥do

17 ifnotAR.isSatisfiable(O_o,Eu ¬F)then addEvFintoH_ws;

18 returnH_ws

the estimated work for the stage is more than usingARto classifyO_o. For (3) we set d=1.5 in our implementation based on the experiments in [7].

In the main classification stage, the major work is to generate theALCHontologies O_w andO_s.O_wis produced by simply removing all the nominal axioms of the form N_a ≡ {a}fromO_o. SinceO_w ⊆ O_o,O_o |= O_w and soH_w ⊆ H_o, i.e. the classification result ofO_wis sound w.r.t.O_o.

Example 1. Consider the following normalized ontologyO_o,ex which we will use as a running example:

(1) AvC (2) Av ∃R.E (3) EvNa (4)C v ∀R.D (5)DvG (6) Av ∃S.Na (7)Na≡ {a}

:::::: (8) ∃S.DvF

Classification result ofO_o,exisH_o,ex={AvF,AvC,EvN_a,DvG}, whereAvFis implied by axioms (1) – (4),(6) – (8). The weakened versionO_w,exofO_o,exis obtained by removing nominal axiom (7). And its classification resultH_w,ex={AvC,EvN_a,DvG}.

We can see thatA vF, which requires (7) to imply, is missing inH_w,ex. We will see later how we add strengthening axioms to getAvFinH_s,ex.

The most difficult part of the procedure is to findO_swhich entails no fewer sub- sumptions thanO_o. A sufficient condition is that for anyA,B ∈ C^>,⊥ such thatO_s 6|= AvB, there exists a modelIofO_ssatisfyingAu ¬B, and it can be transformed to a modelI⁰ofO_osatisfyingAu ¬B. SinceO_sis obtained fromO_wby adding strengthen- ing axioms, every modelIofO_ssatisfies all axioms inO_oexcept possibly the nominal

axioms, which require the interpretation of eachN_a∈NPto have exactly one instance, whereas for an arbitrary modelIofO_s,N_a^Icould have zero or multiple instances. How- ever, if for eachN_a,N_a^I,∅and all the instances are “identical”, they can be replaced by a single instance. Concretely, these instance have the same label set:

Definition 1 Given an interpretation I = (4^I,·^I), an atomic concept A is called a labelof an instance x if x∈A^I. The set of all the labels of x is named thelabel setof x inI, denoted by LS(x,I).

Such a replacement is called acondensation– it condenses all of the different instances into one instance, and thus it transformsIinto a model that satisfies the nominal axiom for N_a. If such a condensation can be done for all nominal placeholders, then we can create a model forO_o.

The strengthening axioms are designed to make these condensations possible. They have the form N_a v X and N_auX v ⊥ computed by Algorithm 4, and thus they force X to be a label of the nominal instanceN_a, or not to be one, respectively. By

“manipulating” labels of nominal instances through these strengthening axioms, we can force them all to be identical in certain models, so that the condensations can occur.

The models we construct for transformation are variants of the canonical model constructed forALCHontologies [17]. Our model construction is introduced in Ap- pendix A. Given F ∈ C^>,⊥, our approach ensures we can build a model ofO_s, which satisfies everyEu ¬F whereO_s 6|= E v F. And every such model can be condensed to a model ofO_o satisfyingEu ¬F. Stating the contrapositive: ifO_o |= E v F then O_s |=E vF. Thus classification ofO_s is complete. Soundness of our approach is en- sured by the verification stage, and completeness is proved in the following section 2.1.

2.1 Condensing Labels and Completeness

Due to space limitation, we only list the important definitions, lemmas, property and theorems in the paper, all their proofs are in Appendix B. Here we briefly introduce some notations used later. We writeO⁺for any intermediate (including final) version of strengthening axioms,O⁺_wfor the corresponding intermediate (including final) version of strengthened ontology whereO⁺_w = O_w∪ O⁺. We writeO_sfor the final version of strengthened ontology. GivenE,F ∈ C^>,⊥ such thatO⁺_w 6|= E vF, a canonical model I_[O+

w,≺F]ofO⁺_wis constructed by first computing asaturationSO⁺_wofO⁺_wand then defining a model based on a total order≺_F. The definition of construction of the canonical model is in Appendix A. The computation of saturation is as follows: GivenO⁺_w, the saturation SO⁺_wis initialized as

{init(A)|A∈C} ∪ {init(N_a)|N_a∈NP}

ThenSO⁺_w is expanded by iteratively applying the inference rules in Table 1 and adding the conclusions intoSO⁺_w until reaching a fixpoint. During this process, existing axioms inSO⁺_ware used as premises and axioms inO⁺_ware used as side conditions.SO⁺_wcontains axioms of the formsinit(H),HvMtAandHvMt ∃R.K. Note Table 1 is modified from Table 3 in [17] by usingR⁺_AandR_init to initialize contexts whenever necessary.

The conjunctionHthat occurs in the premises or conclusions of the inference rules is called the context of the inference.

Table 1.Complete Inference Rules for NormalizedALCHontologies

R⁺_A ^init(H)

HvA :A∈H R⁻_A HvNtA

HvN :¬A∈H Rinit

HvMt ∃R.K init(K)

Rⁿ_u {HvNitAi}ⁿ_i=1 HvFn

i=1NitM :dn

i=1AivM∈ O⁺_w R⁺_∃ HvNtA

HvNt ∃R.B :Av ∃R.B∈ O⁺_w R⁻_∃ HvMt ∃R.K KvNtA

HvMtBt ∃R.(Ku ¬A) : ∃S.AvB∈ O⁺_w

Rv^∗_OS R^⊥_∃ HvMt ∃R.K Kv ⊥ HvM

R∀

HvMt ∃R.K HvNtA

HvMtNt ∃R.(KuB) :Av ∀S.B∈ O⁺_w Rv^∗_OS

Definition 2 In an interpretationI = (4^I,·^I), an atomic concept L is called acon- densing labelif (1) L^I,∅and (2) for any x,y∈L^I, LS(x,I)=LS(y,I).

If a label applied to some instance is a condensing label then every instance to which it applies has the same label set. This means the label sets of all such instances are identical and can be condensed into one instance.

Definition 3 Given a modelIof anALCH OontologyO, a concept L inOand an individual name xL, we define acondensationfunction condense(L,xL,I)that trans- formsIinto an interpretationI⁰=(4^I0,·^I0)as follows:

1). Let n be a fresh instance which is not in4^I, and r be a replacement function r(x)=











n x∈L^I x otherwise 2).4^I0={r(x)|x∈ 4^I}

3). For each concept A, role R and individual o inO,

A^I0={r(x)|x∈A^I},R^I0 ={(r(x),r(y))|(x,y)∈R^I},o^I0=r(o^I),x^I_L⁰=n We say that each x∈L^Iiscondensedinto n. We also sayIis condensed toI⁰. Definition 4 H is apotentially supporting contextof A inO⁺_wif HvMtA∈S_O+

w. Definition 5 A concept X is called aPotentially Supporting concept (PS)of some A inO⁺_wif either X or¬X is a conjunct of a potentially supporting context H of A inO⁺_w. The set of allPSof A inO⁺_wis denoted byPS_[A,O+

w].

Example 2. (PS) Consider theALCHontologyO_w,exin Example 1, it can be seen as an ontologyO⁺_w,exwhereO⁺_ex =∅, andNP ={N_a}. The potentially supporting contexts ofN_aareN_a,EandEuD. SoPS[N_a,O_w,ex]={N_a,D,E}.

Property 6 We sayO⁺_w isdecisiveif for each N_a ∈ NP, each X ∈ PS_[Na,O+ w], either O⁺_w|=N_a vX orO⁺_w|=N_auXv ⊥holds.

Lemma 7 GivenO⁺_wand N_a ∈NP, if (1)O⁺_wis decisive, and (2)O⁺_w 6|= N_a v ⊥. Then for any F∈C^>,⊥, N_ais a condensing label inI_[O+

w,≺F].

Lemma 8 LetIbe a model of anALCH OontologyOsatisfying Eu ¬F,E,F∈C^>,⊥, where L is a condensing label inI. ThenI⁰=condense(L,x_L,I)is a model ofO∪{L= {x_L}}satisfying Eu ¬F.

Lemma 9 Given someO⁺_wand F ∈C^>,⊥, if inI_[O+w,≺_F]every N_a ∈NPis a condensing label, then for each E∈C^>,⊥s.t.O⁺_w6|=EvF, there is a model ofO_osatisfying Eu ¬F.

Theorem 10 If theO⁺_wwe compute satisfies: (1)O⁺_wis decisive and (2) all N_a∈NPare satisfiable inO⁺_w. Then the classification result ofO⁺_wis complete w.r.t.O_o.

Proof. LetE,F ∈C^>,⊥be concepts such thatO⁺_w 6|=EvF. Since conditions (1) and (2) of Lemma 7 are satisfied, allN_a∈NPare condensing labels in the canonical model I_[O+

w,≺F]. By Lemma 9, the model can be condensed to a modelI⁰ofO_ofor Eu ¬F, provingO_o6|=EvF. So the classification result ofO⁺_wis complete w.r.t.O_o.

In the next section, we demonstrate how we will compute anO⁺_wsatisfying condition (1) in Theorem 10 i.e., is decisive. Such anO⁺_w can also satisfy condition (2) in most cases as in our experiments, therefore completeness is achieved. If in a few cases that such anO⁺_wends up not satisfying condition (2), then we hand over the work toAR.The classification result is still complete.

3 Computing Strengthened Ontology

In this section we show how to create a decisiveO⁺_w. The property suggests that the strengthening axioms should be of the formN_a v X or N_a uX v ⊥. We first briefly introduce the strengthening axioms selection functionchooseStrAxiom. Based on that, we explain an initial idea to create a decisiveO⁺_w. Then, we introduce how to compute OPS[N_a,O⁺w] which is an overestimation ofPS[N_a,O⁺w] fromO⁺_wwithout doing saturation, and give the naive and improved algorithms to computeO⁺.

Definition 11 A strengthening axiom selectionchooseStrAxiomis a function that takes an N_a∈NP, X∈Co, andO_wand returns:

1. ∅only ifO_w|=N_avX orO_w|=N_auXv ⊥;

2. a choice between N_avX or N_auXv ⊥.

Initial idea for computing a decisiveO⁺_w: To create such an ontology, a straightfor- ward idea is to start fromO_w, generate saturationSO_wofO_w, obtainPS_[Na,Ow]fromSO_w

for allN_a ∈ NP. For each pairhN_a,Xi,X ∈ PS_[Na,Ow], we add a strengthening axiom determined by chooseStrAxiom function into O¹⁺. Then, addO¹⁺ into O_w and ob- tain O¹_w⁺. Next, we generate saturation again based onO¹_w⁺. Since the impact ofO¹⁺, PS_[Na,O1+w] ⊇ PS[N_a,O_w]. Then we repeat the above process until a fixpoint at which PS_[N

a,Oⁱ⁺¹⁺_w ] = PS_[N

a,Oⁱ⁺_w] for all N_a ∈ NP, we call the final Oⁱ⁺¹⁺ as O⁺, final Oⁱ_w⁺¹⁺ asO_swhich is decisive. However to implement such a procedure generating the satu- ration forPS_[Na,O+

w]is costly. Thus, instead ofPS_[Na,O+

w], we compute the overestimated PS_[Na,O+

w]calledOPS_[Na,O+ w].

Definition 12 Given an ontologyO⁺_w,OPS[N_a,O⁺_w]is a set ofoverestimated potentially supporting conceptswe compute such thatOPS[N_a,O⁺_w]⊇PS[N_a,O⁺_w]for each N_a∈NP.

Algorithm 2:getOPS

Input: NormalizedALCHontologyO⁺_w, a conceptX, a set of nominal placeholdersNP, the set of atomic classesUin the original ontology

Output: A pairhOPS[X,O⁺_w],Pri[X,O⁺_w]i

1 OPS[X,O⁺_w]← ∅;Pri[X,O⁺_w]← ∅;ToProcess← {X};Exists← ∅;

2 whileToProcess,∅do

3 take out a labelWfromToProcess;

4 ifW<^Pri[X,O⁺_w]then

5 addWtoPri[X,O+w];

6 if W=>thenstop the procedure and useARto do the classification work;

7 foreachd

AivMF

W∈ O⁺_wdoselect oneAiand add it intoToProcess;

8 foreach∃S.YvW∈ O⁺_wand Rv^∗_OS and Bv ∃R.Z∈ O⁺_wdo

9 addBintoToProcess;

10 foreachW∈Pri[X,O⁺_w]do

11 ifW∈U or W∈NPthenaddWtoOPS[X,O+w];

12 foreachYv ∀S.W∈ O⁺_wand Rv^∗_OS and Bv ∃R.Z∈ O⁺_wdoadd∃R.ZtoExists;

13 foreachBv ∃R.W∈ O⁺_wdo add∃R.WtoExists;

14 foreach∃R.W∈Existsand Rv^∗_OSdo

15 addWtoOPS[X,O⁺_w];

16 foreachYv ∀S.Z∈ O⁺_wdo addZtoOPS[X,O⁺_w];

17 foreach∃S.ZvY∈ O⁺_wdo addZtoOPS[X,O⁺_w];

18 returnhOPS[X,O+w],Pri[X,O+w]i

We use Algorithm 2 to computeOPS[X,O⁺_w]. Based on Definition 4 and 5,PS[X,O⁺_w]

includes all the atoms ofHsuch thatHvMtX∈SO⁺_w. Without a real procedure gen- erating the saturation forPS[X,O⁺_w] as explained above, we actually computeOPS[X,O⁺_w]

based on analyzing the relationships among the premises, side conditions and conclu- sions of the inference rules in Table 1. The procedure of Algorithm 2 can be divided into three parts:

1. (line 2 to 9) We first conduct a search in the converse direction of all possible derivation paths(Definition 16 in Appendix B) of a conclusionHv MtXfor all possibleHs. In the search we maintain a setPri[X,O⁺_w]. Each conceptW ∈Pri[X,O⁺_w]

may be necessary to the derivation ofX, and appears prior toX in the derivation path. More precisely,Wcorresponds to some potential intermediate conclusionHv M⁰tW which is a necessary conclusion for derivingH v MtX. Since for any contextH,H vAis the only conclusion that can be derived frominit(H), at least one of suchA, which is a conjunct ofH, is inPri[X,O⁺_w]. We writeC^X_Hfor suchA.

Pri[X,O⁺_w]contains at least one conjunctC^X_Hfor any potentially supporting contextH ofX.

2. (lines 10 to 13) Check each conceptW ∈Pri_[X,O+

w]to see whether it can be a conjunct C^X_Hof some potential supporting contextHofX. If it can, we find the first conjunct

C¹_HofHfromC_H^X.C¹_H(seeH^∗in Lemma 13 in Appendix B), which is either inU orNPin line 11 or the filler of concepts inExistsfrom line 12 to 13, is the initial conceptHstarts from.

3. (line 14 to 17) Find all the other conjuncts ofHbased onC¹_Hby searching along the derivation paths.

Lemma 13 GivenO⁺_wand a concept A,OPS[A,O⁺_w] returned by Algorithm 2 preserves OPS[A,O⁺_w]⊇PS[A,O⁺_w].

Example 3. (OPS) Consider again the ontologyO_w,exin Example 1.

The Execution of Alg. 2:getOPS(O_w,ex,N_a,{N_a},{A,C,D,E,F,G}) Line 2 to 9 Pri_[Na,Ow,ex]={Na,E}

Line 11 OPS_[Na,Ow,ex]={N_a,E}

Line 13 Exists={∃R.E}

Line 16 OPS[N_a,O_w,ex]={N_a,E,D}

Line 18 ReturnOPS[N_a,O_w,ex]={Na,E,D},Pri[N_a,O_w,ex]={Na,E}

OPS[N_a,O_w,ex]=PS[N_a,O_w,ex]={Na,E,D} soOPS[N_a,O_w,ex] ⊇PS[N_a,O_w,ex]

For eachX∈OPS_[X,Oi+

w], we will use functionchooseStrAxiomto return a strength- ening axiom. Note in Definition 11 forchooseStrAxiom, how can we knowO_w|=N_av XorO_w |=N_auX v ⊥? Actually in Algorithm 1, line 4 (we usedgetNominalStrAx, if usinggetNominalStrAxNaive, it is the same) executesgetOPS(O_w,N_a,NP,U) inter- nally before line 3. Then for each N_a ∈ NP, each X ∈ OPS_[Na,Ow], we add a testing axiomX_avN_auXintoO_win the first round classification and obtainH_w, whereX_ais a fresh concept forO_w. In Algorithm 1 and Definition 11, we ignore this detail and only mention usingO_wjust for simplifying explanation. IfXa v ⊥is found inH_w, then we simply sayO_w|=N_auXv ⊥. IfN_avXorN_auXv ⊥is implied byH_w, we do not add any strengthening axiom forXintoOⁱ⁺¹⁺. We remove the extra testing results fromH_w in line 5 of Algorithm 1. When choosing betweenN_a vXandN_auXv ⊥, we use the heuristics that ifXcorresponds to a union concept before normalization, andN_avXis not implied, then we addN_auXv ⊥. For other cases, we addN_a vX.

Example 4. For the concepts inOPS[N_a,O_w,ex], since none of the 4 axioms N_a v E, Ea v ⊥,N_a vDorDa v ⊥is inH_w,ex, and assume EandDdo not associate union concepts before normalization, we addN_a v E andN_a v Das strengthening axioms O¹_ex⁺.

UsingOPS_[X,O+

w] instead ofPS_[X,O+

w], we design a naive Algorithm 3 based on the aboveInitial Ideato generateO⁺forO⁺_wso thatO⁺_wis decisive. The execution process is as follows:

1).O⁰⁺=∅. 2). ComputeOPS_[N

a,Oⁱ⁺_w]for∀N_a,N_a∈NPfor theOⁱ⁺. 3).Oⁱ⁺¹⁺={chooseStrAxiom(N_a,X)| ∀N_a,X,N_a∈NP,X∈OPS_[Na,Oi+w]}

In each loop, we add the new Oⁱ⁺ into O_w and generate Oⁱ_w⁺. When the process converges,Oⁱ⁺¹⁺ =Oⁱ⁺. SinceOⁱ⁺¹⁺is computed fromOPS_[N

a,Oⁱ⁺_w], andOPS_[N

a,Oⁱ⁺_w] ⊇ PS_[N

a,Oⁱ⁺_w]for allN_aand for anyi, thus Algorithm 3 guarantees thatO⁺_wis decisive.

Theorem 14 LetO⁺be strengthening axiom produced from Algorithm 3,O⁺_w=O_w∪O⁺ is decisive.

Algorithm 3:getNominalStrAxNaive(Naive algorithm for computing strength- ening axioms)

Input: NormalizedALCHontologyO_w, a set of nominal placeholdersNP, the set of atomic classesUin the original ontology

Output: Strengthening axiomsO⁺

1 O⁺← ∅;

2 repeat

3 newAxioms=∅;

4 foreachN_a∈NPdo

5 O⁺_w← O_w∪ O⁺;

6 hOPS[X,O⁺_w],Pri[X,O⁺_w]i ← getOPS(O⁺_w,N_a,NP,U) ; /* from Algorithm 2 */

7 foreachX∈_OPS[X,O+w]do

8 newAxioms← newAxioms∪chooseStrAxiom(N_a,X);

9 O⁺← O⁺∪newAxioms;

10 untilnewAxioms=∅;

11 returnO⁺

Theorem 15 LetO⁺be strengthening axioms computed from Algorithm 4,O⁺_w=O_w∪ O⁺is decisive.(Proof see Appendix B, the following is just an intuitive explanation) Algorithm 4 improves Algorithm 3 by avoiding repetitive search process. In each loop of Algorithm 3, Algorithm 2getOPSbases on the axioms of the inputOⁱ_w⁺¹⁺to compute OPS_[N

a,Oⁱ⁺¹⁺_w ]forN_a, where only search onOⁱ_w⁺¹⁺\ Oⁱ_w⁺is new, the majority work – search onO_wis repeated in each iteration, while Algorithm 4 improves this and only executes getOPSbased onO_wfor once in line 3. In Algorithm 2 and 3, a strengthening axiom αcan take effect only in the following situation: Ifα={Nb vX} ∈ Oⁱ_w⁺,X ∈Pri_[N

a,Oⁱ⁺_w], thenNb ∈ OPS_[Na,Oi+

w]. IfchooseStrAxiom(N_a,Nb) returnN_a v NbinOⁱ_w⁺¹, thenN_a ∈ OPS_[Nb,Oi+1+

w ]. That means because ofα, in the end,OPS[N_a,O⁺_w]=OPS[Nb,O⁺_w].

In Algorithm 4, without really computing each ofOⁱ_w⁺and repeatedly search on them, we achieve similar results based on the merge criteria in line 6 and the merge operation from lines 7 to line 7. AssumeNb∈gi.nominalsandN_a∈gj.nominals,X∈gi.ops∩gj.pri,

∅in line 6. Then,X ∈ gi.opsmeans Nb vX will possibly be a strengthening axiom, X∈ gj.primeansXis possibly inN_a’sPri, thengiandgjare merged into one group, and finallyOPS[N_a,O⁺_w] =OPS[N_b,O⁺_w], too, similar with Algorithm 3.

Example 5. In Algorithm 3,O¹_ex⁺=O²_ex⁺. Thus, the loop from line 2 to 9 repeats twice.

In Algorithm 4, sinceNP ={N_a}, the merge process from line 6 to 7 does not happen.

For both algorithms, that meansO¹_ex⁺does not have impact in later saturation.

Note the naive Algorithm 3 is only used for demonstrating our initial idea. In our implementation, we used the improved Algorithm 4. All three Algorithms 2, 3 and 4 have polynominal complexity. In Algorithm 2, the number of iterations in all levels of loops are bounded by the number of axioms or concepts inO_w, and thus have polyno- mial complexity. In Algorithm 3, the number of iterations is bounded by the size ofO⁺, which is also polynomial. In Algorithm 4, the merge loop can only continue for at most

||NP||times. So all the algorithms are polynomial of the size ofO_wand terminate.

Algorithm 4:getNominalStrAx(Calculate strengthening axioms for nominals) Input: NormalizedALCHontologyO_w, a set of nominal placeholdersNP, the set of

atomic classesUin the original ontology Output: Strengthening axiomsO⁺

1 groups← ∅;

2 foreachN_a∈NPdo

3 hOPS[N_a,O_w],Pri[N_a,O_w]i ← getOPS(O_w,N_a,NP,U); /* from Algorithm 2 */

4 create a groupgwithg.nominals={N_a},g.ops=^OPS[N_a,O_w],g.pri=^Pri[N_a,O_w];

5 addgintogroups;

6 whilethere exists gi,gj∈groupssuch that gi.ops∩gj.pri,∅do

7 mergegi,gjinto one groupg, whose properties are unions of corresponding properties ofgiandgj; removegi,gjfromgroupsand addg;

8 foreachg∈groupsdo

9 foreachN_a∈g.nominalsdo

10 foreachX∈g.opsdo

11 O⁺← O⁺∪chooseStrAxiom(N_a,X);

12 returnO⁺

Example 6. : The Overall Execution Results from Alg. 1:

(1) Weakened OntologyO_w:Na≡ {a}removed Line 2 of Alg. 1 (2) Class HierarchyH_w:AvC EvNa DvG Line 3 of Alg. 1 (3) Strengthening axiomsO⁺: two axioms added:NavE NavD (4) For (3): Line 4 of Alg. 1 return from Alg. 4 For (5): Line 8 of Alg. 1 (5) Class HierarchyH_s:AvC EvNa DvG AvF NavE Na vD (6) Verify the following 6 pairs (indirect subsumptions included): Line 16 to 17 (7)AvF EvD EvG NavD NavE NavG, 1 pair validated:AvF (8) Class HierarchyH_ws=H_o:AvC AvF EvN_a DvG. Line 18 of Alg. 1

4 Related Work

Optimization techniques for ontology classification have been extensively studied in the literature [4,16,7,12]. For tableau-based reasoners, Enhanced Traversal (ET) [4]

and KP [16,7] are the most widely used techniques. Optimizations for consequence- based classification ofELOontologies were also studied [12], and the most effective technique is overestimation. Firstly, the algorithm saturates the ontology using infer- ence rules forEL and obtains sound subsumptions. Next, potential subsumptions are obtained by continuing saturation with a new overestimation rule added. Finally, the potential subsumptions are checked using a sound and complete but slower procedure forELO. Comparing with this procedure, we support a more expressive DLALCH O.

In the area of hybrid reasoning, Romero et al. [1,2] proposed classification based on modules given to aSROIQreasonerRand an efficientL-reasonerRLsupporting a fragmentLofSROIQ. GivenO_o, they find a set of classesΣ^Lwhose superclasses inO_ocan be computed by classifying a subsetM^LofO_oin DLL. The superclasses of remaining classes are computed usingR. However, because of the restriction of locality- based modular approach used for computingΣ^L, nominal axiomsN_a ≡ {a}cannot be

moved out fromM^L[6]. Therefore, in order to guarantee completeness, eitherRLsup- ports nominals or all the work is assigned toR. In current implementation of MORe, RLdoes not support non-safe [12] use of nominals in the Galen ontologies, soRhas to do all the work. In sum, module extraction technique MORe uses for classification is a more general technique that is primarily intended for ontologies in DLs whose expres- sivity is beyond ALCHO. In contrast, our approach currently only supports language ALCH O, combines the two reasoners differently, handles nominals, and improves on its full reasoner more often for complex and highly cyclic ontologies.

Our approach generally belongs to theory approximation [15]. Yuanet. al. [14]

encodedSROIQontologies intoEL⁺⁺ with additional data structures, and classified by a tractable, sound but incomplete algorithm [14]. A strengthened approximation of SROIQTBoxes with the OWL 2 RL profile [21] is used for query answering.

5 Empirical Results and Conclusion

We have implemented our prototype hybrid classifier WSClassifier in Java. The classi- fier uses ConDOR as the mainALCHreasoner and HermiT as the assistant reasoner for DLALCH O. WSClassifier adopts a well-known preprocessing step to eliminate transitive roles [10], hence supports DLSH O. We set the Java heap space to 12GB and the time limit to 9 days for all reasoners. We compared the classification time of WSClassifier with main-stream tableau-based reasoners and another hybrid reasoner MORe on all large and highly cyclic ontologies available to us, on the ORE dataset and on some proposed variants. For classifying FMA-constitutionalPartForNS(FMA- C) which is the only real-world large and highly cyclic ontologies with nominals we have access to, WSClassifier used 21.2 seconds, while HermiT used 140,882 seconds with configuration of simple core blocking and individual reuse. Other reasoners did not get a result because they ran out either of time or memory. We put the evaluation result Table 2 of comparison of classification performance in Appendix C. The results of Table 2 show, excluding ORE dataset, that WSClassifier is significantly faster than the tableau-based reasoners on 7 out of 10 ontologies. For the other 3 of 10 ontologies, WSClassifier detected that strengthening axioms made some concepts unsatisfiable in O_s, and so failed over to HermiT. We see a major speedup for WSClassifier on ORE’s FMA-lite which is highly cyclic and is our targeted ontology. For the remaining 112 ORE ontologies which are not highly cyclic and thus not our target, WSClassifier failed over to HermiT on one of them. Our average reasoning time on these 112 ontologies is longer than that of the other reasoners mainly due to our overheads: computing normal- ized axioms and transmitting the ontology to and from ConDOR.

We have presented a hybrid reasoning technique for soundly and completely clas- sifying anALCH Oontology based on a weakening and strengthening approach. The input ontology is approximated by twoALCHontologies, one weakenedO_wand one strengthenedO_s, which are classified by a fast consequence-based reasoner. The sub- sumptions ofO_wandO_sare a subset and a superset of the subsumptions of the original ontology, respectively. Subsumptions implied byO_sbut not byO_ware further checked by a (slower)ALCH O reasoner. This approach is possibly applied to different lan- guage classes, each requiring different strengthening axioms. The implementation can be improved with heuristics for a tighterOPS[N_a,O⁺_w]and better strengthening axioms.

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A Canonical Model Construction

GivenF ∈ C^>,⊥ andO⁺_w, our target is to construct a modelI_[O+

w,≺F] forO⁺_w satisfying Eu ¬F for anyE ∈ C^>,⊥ such thatO⁺_w 6|= E v F. We writeIforI_[O+

w,≺F] when the

parameters are not important. The construction is done by first computing asaturation S_O+

w ofO⁺_w and then defining a model based on it.S_O+

w contains axioms of the forms init(H),HvMtAandHvMt ∃R.Kderived using the inference rules.

(1) Computation of saturation

Given anALCHontologyO⁺_w, the saturationS_O+

wis initialized as {init(A)|A∈C^>} ∪ {init(N_a)|N_a∈NP}

ThenSO⁺_w is expanded by iteratively applying the inference rules in Table 1 and adding the conclusions intoSO⁺_w until reaching a fixpoint. An inference rule is ap- plied by using existing axioms inSO⁺_was premises and axioms inO⁺_was side condi- tions. We writeO⁺_w `αfor everyα∈SO<sup>+</sup><sub>w</sub> derived fromO<sup>+</sup><sub>w</sub>. The inference process is obviously sound, i.e. ifO<sup>+</sup><sub>w</sub>`αthenO⁺_w|=α.

(2) Model construction

We define a total order≺_F over all the concepts inO⁺_w such that F has the least order. IfFis⊥or>, the order can be any arbitrary order. We define the domain4^I ofI_[O+

w,≺F]as

4^I := {x_H|init(H)∈SO⁺_wandHv ⊥<SO⁺_w}

wherex_His an instance introduced forH.4^Iis nonempty because ifO⁺_wis consis- tent,> v ⊥<SO⁺_w. Sinceinit(>)∈SO⁺_w,x>exists.

To define the interpretation for atomic concepts, we first construct the label set LS(xH,I) for each instancexH. In this section, we writeI_HforLS(xH,I). LetAi

be the concept with theith order from the smallest to the largest according to≺_F. For convenience we writeM≺_F Aiif for each disjunctAinM,A≺_F Ai. LetIⁱ_Hbe a sequence whereI⁰_H:=∅, andIⁱ_His defined as

Iⁱ_H :=















Iⁱ_H∪ {Ai}if there existsM ≺_F Aisuch that O⁺_w`HvMtAiandM∩ Iⁱ⁻¹_H =∅ Iⁱ⁻¹_H otherwise

The last element of the sequence is defined asLS(xH,I). With theLS(xH,I) de- fined, the interpretation of an atomic conceptAis defined as

A^I := {xH|A∈LS(xH,I)}

The roles are interpreted to satisfy the axiomsHvMt ∃R.K. For each roleRand eachHsuch thatxH∈ 4^I, define

I^R_H:={K| ∃M:O⁺_w`HvMt ∃R.K,M∩ I_H=∅}

A conjunctionKis said to be maximal inLS^R(xH,I) if there is noK⁰∈ I^R_Hwith a superset of conjuncts ofK. SinceHv ⊥<SO⁺_w, byR^⊥_∃ rule we haveKv ⊥<SO⁺_w. And byRinit rule we haveinit(K)∈SO⁺_w. SoxKis well-defined. The interpretation of roles is defined as

R^I:= [

R⁰v_O+

wR

{(xH,xK)|Kis maximal inI^R_H⁰}

The inference rules in Table 1 is modified from Table 3 in [17] by usingR⁺_AandRinit

to initialize contexts only when necessary. The change affects only the validity ofxKin the construction forR^Iwhich has been explained above, and the proof thatIsatisfies each type of axiom can be kept unchanged from Simanciket. al.[17]. SoIis a model of theALCHontologyO⁺_w. Moreover, for anyE∈C^>,⊥, ifO⁺_w6|=EvF,O⁺_w0EvF.

Since F has the least order in≺_F, by the definition of LS(x_E,I) we know x_E < F^I. Thusx_E ∈(Eu ¬F)^I, andI_[O+w,≺F]satisfiesEu ¬F.

B Proofs of Lemmas and Theorems

Lemma 7 GivenO⁺_wand N_a ∈NP, if (1)O⁺_wis decisive, and (2)O⁺_w 6|= N_a v ⊥. Then for any F∈C^>,⊥, N_ais a condensing label inI_[O+w,≺F].

Proof. For simplicity we writeIforI_[O+w,≺F] in this proof. Sinceinit(N_a) ∈ SO⁺_w and O⁺_w6|=N_a v ⊥, by the construction ofI,xN_aexists andxN_a∈N_a^I. Hence it is equivalent to show that for eachxH∈N_a^I,LS(xH,I)=LS(xN_a,I).

We first show that for eachHsuch thatxH∈N_a^I,xN_a ∈H^I. SincexH∈N_a^I,His a potentially supporting context ofN_a. LetH=dn

i=1Cⁱ_H, whereC_Hⁱ isAor¬A. SinceO⁺_w is decisive, we have the following two cases:

– O⁺_w |=N_a vCⁱ_Hholds for allCⁱ_H,1≤i≤n, thenO⁺_w |=N_a vH. Since xN_a ∈ N_a^I, xN_a ∈H^I.

– There exists someisuch thatO⁺_w |= N_auCⁱ_H v ⊥, thenO⁺_w |= N_auH v ⊥. By Lemma 3 of paper [17],xH∈H^I, which contradicts with our assumptionxH∈N_a^I. Next we proveLS(xN_a,I) ⊆ LS(xH,I) by contradiction. AssumeLS(xN_a,I) * LS(xH,I), letXbe the concept inLS(xN_a,I)\LS(xH,I) with the smallest order. Since X∈LS(xN_a,I), there existsN≺_F Xsuch thatO⁺_w`N_avNtXandN∩LS(xN_a,I)=∅.

∵O⁺_w`N_avNtX ∴O⁺_w|=N_avNtX

∵xH∈N_a^I∧xH<X^I ∴xH∈N^I ∴LS(xH,I)∩N,∅

In the above proof, ifN =⊥, a contradiction arises withxH ∈N^I. Otherwise, letY ∈ LS(xH,I)∩N, there must existN⁰≺_F Ys.t.O⁺_w`HvN⁰tYandLS(xH,I)∩N⁰=∅.

∵O⁺_w`HvN⁰tYandxN_a∈H^I ∴ xN_a∈(N⁰tY)^I

∵N⁰≺_FY andY ∈NandN≺_F X∴N⁰≺_F X

SinceXis the smallest inLS(xN_a,I)\LS(xH,I),N⁰ ≺_F X andLS(xH,I)∩N⁰ =∅, we have LS(xN_a,I)∩N⁰ = ∅ (it is trivially true if N⁰ = ⊥), and xN_a < N^0I. Given xN_a∈(N⁰tY)^I, we haveY ∈LS(xN_a,I), this contradicts withN∩LS(xN_a,I)=∅. So we conclude thatLS(x_Na,I)\LS(x_H,I)=∅andLS(x_Na,I)⊆LS(x_H,I).

Finally we need to proveLS(x_H,I)⊆LS(x_Na,I). For eachX ∈ LS(x_H,I), there existsN≺_F Xsuch thatO⁺_w`HvNtXandN∩LS(x_H,I)=∅.

∵LS(x_Na,I)⊆LS(x_H,I) ∴N∩LS(x_Na,I)=∅ ∴x_Na<N^I