Ontology Materialization by Abstraction Refinement in Horn SHOIF

Ontology Materialization by Abstraction Refinement in Horn SHOIF

Birte Glimm, Yevgeny Kazakov, and Trung-Kien Tran University of Ulm, Germany,<first name>.<last name>@uni-ulm.de

Abstract. Abstraction refinement is a recently introduced technique which al- lows for reducing materialization of an ontology with a large ABox to material- ization of a smaller (compressed) ‘abstraction’ of this ontology. Although this ap- proach is sound for the very expressive languageSROIQ, so far it was proved to be complete only for HornALCHOI ontologies. In this paper we propose an extension of this method that is now complete for HornSHOIF ontolo- gies. Supporting functional roles, in particular, is nontrivial due to the sophis- ticated interaction with nominals and inverse roles. Additionally, we show how to compute not only the concept-materialization but also the role- and equality- materialization. To show completeness we use a tailored set of materialization rules that loosely decouple the ABox from the TBox, which is intrinsically inter- esting as reasoning with nominals in general requires both ABox and TBox.

1 Introduction

Description Logics (DLs) are popular languages for knowledge representation and rea- soning. They are the underlying formalism for the standardized Web Ontology Lan- guage (OWL) and its profiles, which are widely used in many application areas. Re- cent years have also seen an increasing interest in ontology-based data access (OBDA), where a TBox with background knowledge, often expressed in a DL language, is used to enrich datasets (ABoxes), which are then accessible via queries. Ontology materi- alization is a reasoning task that computes logical consequences of the dataset w.r.t.

the TBox and it is the most important task in some languages, e.g., OWL 2 RL [10].

In other languages, e.g., those that allow existential quantification, materialization is a stepping stone for query answering via rewriting [8].

To make ontology materialization useful in practice, especially for large datasets, scalable materialization is of great importance. Several approaches have been proposed to achieve this goal. The RDFox [11] and WebPIE [13] systems operate on the en- tire dataset and utilize parallel computing to perform a rule-based materialization for OWL 2 RL. Other approaches try to reduce the size of the dataset. Modules or so- called ‘individual islands’ [14] are used for reducing the set of ABox assertions to those that are sufficient for computing the entailed assertions for a given individual.

The SHER system [4, 2] improves consistency checking of a large ABox by computing a so-called ‘summary ABox’ in which several original individuals are merged into one.

If the resulting ABox is found consistent, then so is the original ABox. If not, then ex- planations [7] are used to pinpoint the contradictory axioms or relax the summary to avoid inconsistency. Similar to SHER, in the abstraction refinement method [5] several

individuals of the original ABox can be represented by one individual in a correspond- ing ‘abstract ABox’. In contrast to the summary ABox, the abstract ABox provides an under-approximation rather than the over-approximation of entailments (e.g., if the ab- straction is inconsistent then so is the original ABox but not necessarily vice versa).

To ensure completeness of the method, the so-called refinement step is used that re- computes the abstraction based on new (sound) entailments obtained from a previous abstraction. This has the added benefit that not only consistency but also the full mate- rialization of the ABox can be computed at once without (rather expensive) explanation computations or repeated consistency checks. This paper builds upon the abstraction refinement method and presents the following improvements:

1. We extend the approach to guarantee completeness in the presence of transitive and functional roles, thus fully supporting HornSHOIF ontologies. Reasoning with nominals, inverse roles, and functionality is known to be difficult due to the loss of the tree-model property.

2. We materialize not only concept assertions, but also role and equality assertions. In ALCHOI, role and equality assertions can be computed simply by expanding the role hierarchy and analyzing assertions of nominals. InSHOIF, this is no longer the case. Apart from transitivity, role assertions could result from the consequence that some roles are functional; and interactions of nominals, inverse roles, and func- tionality could lead to a new kind of nominals. In Section 4, we discuss this issue in more detail and provide a solution in Section 5.

3. We present a new set of materialization rules (Section 6), which loosely decouple the ABox from the TBox. Although we use them only for proving completeness of the method (Section 7), these rules can be of interest on its own, e.g., as a basis of an efficient implementation for ABox reasoning. This provides a fresh view of the approach as the completeness proofs are principally different from the previous proofs [5], and the method can potentially be extended to other languages having similar rules.

2 Preliminaries

The syntax and semantics of the Description Logic (DL)SHOIF is briefly presented in Table 1. ASHOIF ontologyOisHorn[9], if, for everyD(a)∈ OandCvD∈ O, the conceptsCandDsatisfy the following grammar definitions:

C_(i)::=> | ⊥ |A|o|C1uC2|C1tC2| ∃R.C (1) D_(i)::=> | ⊥ |A|o|D₁uD₂| ∃R.D| ∀R.D| ¬C (2) Given an ontologyO, a roleRisfunctional(inO) iffunc(R)∈ O andtransitive(in O) iftran(R) ∈ O. (Horn)ALCHOI is the fragment of (Horn)SHOIF in which functionality and transitivity are disallowed. For an ontologyO, letv^∗_Hbe the reflexive transitive closure of therole hierarchyH={R vS ∈ O}. IfR v^∗_H S, then we say thatRis asub-roleofSandSis asuper-roleofR; A roleRissimple(inO) if it has no transitive sub-roles; iffunc(R)∈ OthenRmust be simple.

Table 1.The syntax and semantics of the DLSHOIF

Syntax Semantics

Concepts:

atomic concept A A^I⊆∆^I

nominal o o^I ⊆∆^I,||o^I||= 1

top > ∆^I

bottom ⊥ ∅

negation ¬C ∆^I\C^I

conjunction CuD C^I∩D^I

disjunction CtD C^I∪D^I

existential restriction ∃R.C {d| ∃e∈C^I :hd, ei ∈R^I} universal restriction ∀R.C {d| hd, ei ∈R^I →e∈C^I}

Axioms:

concept inclusion CvD C^I⊆D^I

role inclusion RvS R^I ⊆S^I

role transitivity tran(R) R^I◦R^I ⊆R^I role functionality func(R) hd, ei,hd, e⁰i ∈R^I→e=e⁰

concept assertion C(a) a^I ∈C^I

role assertion R(a, b) ha^I, b^Ii ∈R^I equality assertion a≈b a^I =b^I

To simplify the presentation, we do not distinguish between axiomsR(a, b)and R⁻(b, a), a ≈ b andb ≈ a, R v S andR⁻ v S⁻, andtran(R)and tran(R⁻).

For example, ifR(a, b)∈ A, we assume thatR⁻(b, a)∈ A. We usecon(O),rol(O), ind(O),nom(O)for the sets of atomic concepts, atomic roles, individuals, and nominals occurring inO, respectively.

For an ontologyO, we say thatOisconcept-materialized ifO |= A(a)implies A(a)∈ Ofor eachA∈con(O)anda∈ind(O);Oisrole-materializedifO |=r(a, b) impliesr(a, b)∈ Ofor eachr∈rol(O)anda, b∈ind(O);Oisequality-materialized ifO |= a ≈bimpliesa ≈ b ∈ Ofor eacha, b ∈ ind(O);O is (fully)materialized if it is concept-, role-, and equality-materialized. Given an ontology O, the concept-, role-, equality-, and/or (full) materialization ofOis the smallest super-set ofOthat is concept-, role-, equality-, and/or fully materialized respectively.

3 Computing Materialization by Abstraction

The main idea of the abstraction refinement method [5] is to materialize an ontology O =A ∪ T with a potentially large ABoxAby constructing a smaller ABoxBsuch that the materialization ofB ∪ T can be computed by a general-purpose reasoner, and transfer the new entailments back toO. The ABoxBis usually called theabstractionof theoriginalABoxA(or just theabstract ABoxifAis clear from the context), and the individuals inBare calledrepresentativesof theoriginalindividuals inA. All results in this section apply to any DL with (classical) set-theoretic semantics, e.g.,SROIQ[6].

Definition 1. LetAandBbe ABoxes. A mappingh : ind(B) → ind(A)is called a homomorphism (fromBtoA)if, for every assertionα∈ B, we haveh(α)∈ A, where

h(C(a)) :=C(h(a)),h(R(a, b)) :=R(h(a), h(b)), andh(a≈b) :=h(a)≈h(b). We say an individualb ∈ ind(B)is arepresentativeof an individuala ∈ ind(A)if there exists a homomorphismh:ind(B)→ind(A)such thath(b) =a.

Example 1. Consider the ABoxes A = {A(a), A(b)} and B = {A(u)}. Then the individualuofBis a representative for both individualsaandbsince the mappings h₁={u7→a}andh₂={u7→b}are homomorphisms fromBtoA.

The following property of homomorphisms allows transferring entailments from the abstraction to the original ABox.

Lemma 1. Leth: ind(B)→ind(A)be a homomorphism between the ABoxesBand A. Then, for every TBoxT and every axiomβ,B ∪ T |=βimpliesA ∪ T |=h(β).

Corollary 1. If an individualu ∈ ind(B) is a representative for an individual a ∈ ind(A), then, for every TBoxT and conceptC, ifB ∪ T |=C(u), thenA ∪ T |=C(a).

According to Corollary 1, in Example 1 one cantransferany newly entailed concept assertion foruto the corresponding assertions foraandb. In fact, in this particular case, all entailed concept assertions forAcan be computed this way because there is also a homomorphism fromAtoB.

Example 2 (Example 1 continued).Consider the homomorphismh:ind(A)→ind(B) defined byh={a7→u, b7→u}. Then by Lemma 1, for every TBoxT and conceptC ifA ∪ T |=C(a)orA ∪ T |=C(b)thenB ∪ T |=C(u).

In practice, computing a sufficiently small abstractionBofAsuch that there are homo- morphisms in both directions is rarely possible, so the set of concept assertions trans- ferred using Corollary 1 is usually incomplete. To ensure completeness, one can employ furtherrefinementsteps that recompute the abstraction based on the new information derived. This method was shown to be complete for concept materialization of Horn ALCHOI ontologies [5]. The aim of this paper is to extend this approach to Horn SHOIF. Before we go into further details explaining our extension, we first describe challenges that the new functionality and transitivity axioms pose for ontology materi- alization.

4 Full Materialization for Horn SHOIF

It is easy to show using model-theoretic arguments that anALCHOIontologyOwith- out equality assertions entails an equality between individualsa≈biff eithera=bor bothaandbare instances of some nominal conceptooccurring inO. To compute such entailed equality assertions, it is sufficient to compute instances of nominals, which can be accomplished by introducing an axiomo v A_o with a fresh conceptA_o for each nominal oand computing instances of A_o by concept materialization. IfO contains equality assertions, one needs to additionally perform the transitive symmetric closure of the resulting equality assertions. For role materialization, similarly, one can show that ifO |= R(a, b)then either(i)there exists an assertionR⁰(a⁰, b⁰) ∈ Osuch that O |=a≈a⁰,O |=b ≈b⁰, andR⁰ v^∗_H R, or(ii)ais an instance of∃R.oandbis an

instance ofofor some nominalo, or(iii)ais an instance ofoandbis an instance of

∃R⁻.ofor some nominalo. All these conditions can again be checked by introducing fresh concepts and computing the concept materialization.

That is, (full) materialization of HornALCHOIontologies can be reduced to con- cept materialization by simple syntactic transformations. The following examples illus- trate that for HornSHOIF ontologies such reductions do not work.

Example 3. Consider the ontologyO = A ∪ T withA = {A(a), A(b)}andT = {A v ∃F⁻.o,func(F)}. ThenO |= a ≈ b but neitheranorb are instances of the nominalo.

As Example 3 illustrates, equality testing in (Horn)SHOIF becomes less trivial;

the main reason is that using a combination of functional roles, inverse roles, and nom- inals one can expressentailed nominal conceptssuch asAin Example 3, which can be interpreted by at most one element.

In the following example, we demonstrate how functional roles can also result in some non-trivial entailments of role assertions, even if no equality or nominals are used.

Example 4. Consider the ontologyO = A ∪ T withA ={A(a), R(a, b)}andT = {Av ∃S.>, RvF, SvF,func(F)}. ThenO |=S(a, b)butO 6|=RvS.

As can be seen from Examples 3 and 4, the computation of equality- and role- materialization becomes a non-trivial problem for Horn SHOIF ontologies. Fortu- nately, using the following corollary of Lemma 1, one can extend the main idea be- hind concept materialization described in the previous section to equality- and role- materialization.

Corollary 2. Leth : ind(B) → ind(A)be a homomorphism between the ABoxesB andA,u, v ∈ ind(B),a=h(u), b=h(v)∈ ind(A), andT aSROIQTBox. Then B ∪T |=u≈vimpliesA∪T |=a≈b, andB ∪T |=R(u, v)impliesA∪T |=R(a, b) for every roleR.

Unfortunately, the abstract ABoxes that are sufficient to guarantee completeness of concept-materialization are not sufficient to guarantee completeness of equality- and role-materialization as demonstrated in the following example.

Example 5. Consider the ABox Aand its abstraction B in Example 1. As stated in Example 2, for any TBoxT all entailed concept assertions ofT ∪ Acan be obtained using Corollary 1 for the abstractionB. However, the abstractionBmay be insufficient for computing all entailed role or equality assertions using Corollary 2. Indeed, consider T = {A v o}. Then A ∪ T |= a ≈ b, but, clearly, there is no homomorphism h:ind(B)→ind(A)such thath(u) =aandh(u) =brequired to derive this assertion using Corollary 2. Similarly, it is possible thatA ∪ T |=R(a, b)for some roleR, but we are not able to derive this assertion using Corollary 2, for example, forT ={Av

∃T.o, Av ∃T⁻.o,tran(T), T vR}.

5 Abstraction Refinement for Horn SHOIF

The general algorithm for ontology reasoning using the abstraction refinement method can be summarized as follows:

1. Build asuitableabstraction of the original ontology;

2. Compute the entailments from the abstraction using a reasoner andtransferthem to the original ontology using homomorphisms (Lemma 1);

3. Compute the deductive closureof the original ontology using some (light-weight) rules;

4. Repeat from Step 1 until no new entailments can be added to the original ontology.

The efficiency and theoretical properties of this method depend on the choices of how the abstraction is computed in Step 1, which entailments are transferred in Step 2, and which rules are used to compute the deductive closure in Step 3. In the following we detail these choices for HornSHOIF.

To compute the abstraction of the original ABox (Step 1), we definetypesof indi- viduals based on their assertions.

Definition 2. LetAbe an ABox and aan individual. Theconcept type ofais a set of concepts τC(a) = {C|C(a)∈ A}. The role typeofais a set of rolesτR(a) = {R| ∃b:R(a, b)∈ A}. The (combined) type ofa is a pairτ(a) = hτC(a), τR(a)i whereτC(a)is the concept type ofaandτR(a)is the role type ofa.

Note that we assume w.l.o.g. that, for every individuala,>(a)is included in the on- tology and, therefore,>occurs in every (concept) type. To simplify the presentation, however, we usually omit>in the remainder.

Example 6. Let A = {A(a), A(b), R(a, b)}. Then τ_C(a) = τ_C(b) = τ₁ = {A}, τ_R(a) = {R},τ_R(b) = {R⁻},τ(a) = τ₂ = h{A},{R}i, andτ(b) = τ₃ = h{A}, {R⁻}i.

The abstract ABox is then constructed by introducing one representative for each type with the respective assertions.

Definition 3. Theabstractionof an ABoxAis an ABoxB=S

a∈ind(A)(Bτ_C(a)∪Bτ(a)), where:

– for each concept typeτC,BτC ={C(uτC)|C∈τC},

– for each combined typeτ =hτC, τRi,Bτ ={C(vτ) |C ∈ τC} ∪ {R(vτ, w_τ^R) | R∈τR},

whereu_τC,v_τ, andw^R_τ are distinguishedabstract individualsfor each concept typeτ_C and each combined typeτ.

Example 7. The abstraction forAin Example 6 is the ABoxB =Bτ_C(a)∪ Bτ_C(b)∪ B_τ(a)∪ B_τ(b), whereB_τC(a) =B_τC(b)=B_τ1 ={A(u_τ1)},B_τ(a) =B_τ2 ={A(v_τ2), R(vτ₂, w^R_τ2)},B_τ(b)=Bτ₃ ={A(vτ₃), R⁻(vτ₃, w^R_τ3⁻)}.

Intuitively, the abstraction is a disjoint union of ABoxes simulating concept and com- bined types. Note that each mappingh:ind(B)→ind(A)such that:

h(u_τC)∈ {a∈ind(A)|τ_C(a) =τ_C}, (3)

h(vτ)∈ {a∈ind(A)|τ(a) =τ}, (4)

h(w_τ^R)∈ {b∈ind(A)|R(h(v_τ), b)∈ A}, (5) is a homomorphism fromBtoA. This allows us to transfer entailments from the ab- straction back to the original ABox using Corollaries 1 and 2. Note that each original individualainAhas at least two representatives inB:uτ_C(a)that has exactly the same concept assertions asaandv_τ(a)which additionally has assertions with the same roles.

Two representatives are necessary to solve the problem mentioned in Example 5 for transferring role and equality assertions.

Example 8. Consider the ABoxA = {A(a), A(b)} and TBoxT = {A v o}men- tioned in Example 5. We haveτ_C(a) =τ_C(b) = τ₁ ={A}, andτ(a) =τ(b) =τ₂= h{A},∅i. The abstraction ofAis defined asB = Bτ₁ ∪ Bτ₂ withBτ₁ = {A(uτ₁)}, Bτ₂ = {A(vτ₂)}. SinceB ∪ T |= uτ₁ ≈ vτ₂, andh = {uτ₁ 7→ a, vτ₂ 7→ b} is a homomorphism fromBtoA, using Corollary 2 we obtainA ∪ T |=a≈b.

Next, we detail how entailed assertions are transferred fromBtoAin Step 2 of the algorithm. To achieve completeness it is not necessary to transfer all of them and we detail which assertions are to be transferred after the definition.

Definition 4. LetBbe the abstraction of an ABoxA(according to Definition 3), and

∆Ba set of assertions. Theupdate ofA(using∆B)is the smallest set of assertions

∆Asuch that:

C(v_τ(a))∈∆B ⇒ C(a)∈∆A, (6) R(a, b)∈ A, C(w^R_τ(a))∈∆B, ⇒ C(b)∈∆A, (7) R(a, b)∈ A, S(v_τ(a), w^R_τ(a))∈∆B ⇒ S(a, b)∈∆A, (8) S(v_τ(a), v_τ(a))∈∆B ⇒ S(a, a)∈∆A (9) u_τC(a)≈v_τ(b)∈∆B ⇒ a≈b∈∆A, (10) R(u_τC(a), v_τ(b))∈∆B ⇒ R(a, b)∈∆A. (11) Lemma 2. LetBbe the abstraction ofA,∆Aan update for∆B, andT a TBox. Then B ∪ T |=∆BimpliesA ∪ T |=∆A.

Proof. It is easy to see that for eachα ∈ ∆Aandβ ∈ ∆Bin cases (6)–(11), there is a homomorphismhfromBtoAsatisfying conditions (3)–(5) such thath(β) = α.

Hence, by Lemma 1,B ∪ T |=βimpliesA ∪ T |=h(β) =α. ut LetB⁰be the materialization ofB ∪ T, then we transfer assertions from∆B=B⁰\ B according to Definition 4 in Step 2. Next, we compute the (deductive) closure of the ABoxAunder equality, transitivity, and functionality in Step 3.

Definition 5. We say that an ABoxAisequality-closedif:

a∈ind(A)impliesa≈a∈ A (12) {a≈b, b≈c} ⊆ Aimpliesa≈c∈ A, (13) {a≈b, A(a)} ⊆ AimpliesA(b)∈ A, (14) {a≈b, R(a, c)} ⊆ AimpliesR(b, c)∈ A, (15) Aisclosed under the axiomtran(T)if:

{T(a, b), T(b, c)} ⊆ AimpliesT(a, c)∈ A. (16) Aisclosed under the axiomfunc(F)if:

{F(a, b), F(a, c)} ⊆ Aimpliesb≈c∈ A. (17) Theclosure ofA(w.r.t. a TBoxT) under equality, transitivity, and/or functionalityis the smallest super-set ofAthat is closed under equality, for eachtran(T)∈ T and/or eachfunc(F)∈ T, respectively.

Computing the closure of an ABox under equality, functionality, and transitivity is a relatively lightweight operation that does not require using a reasoner. Note that all these assertions must be derived in order to compute the full materialization. In the previous method [5], the abstraction of an ABoxAis defined asB=S

a∈ind(A)B_τ(a), and there is no computation of the closure as in Step 3. One can easily check that the previous method is incomplete not only for role and equality materializations but also for concept materialization of HornSHOIFontologies, even with the computation of the closure in Step 3.

Since Steps 2 and 3 can only extend the original ABox with entailed atomic as- sertions, the procedure issound, and since the number of such assertions is bounded, the procedure eventually terminates. We next show that the procedure iscompletefor HornSHOIF ontologies, that is, the resulting ontology is (fully) materialized when the procedure terminates.

6 A Criterion for Ontology Materialization

To prove completeness of the abstraction refinement procedure in the case of Horn SHOIF, we characterize when such ontologies are fully materialized by means of closure of the ABox assertions under certain rules. The rules are similar to the rules for reasoning in HornSHIQand HornSHOIQ[12, 3] in the sense that they derive logical consequences of axioms. Since we are only interested in ABox consequences and not going to use these rules for computing the materialization (but merely for prov- ing completeness of our abstraction refinement algorithm), however, we will not derive TBox axioms explicitly but use their entailments in side conditions of the rules.

Recall from the discussion after Example 3, that inSHOIFone can express some non-trivialnominal concepts. To be able to reason about such concepts, we need to extend the language with new axiom types.

Definition 6. A concept cardinality restrictionis an axiom of the form|C| ≥ n or

|C|=nwithCa concept andn∈N. An interpretationIsatisfies|C| ≥n(|C|=n), writtenI |=|C| ≥n(I |=|C|=n), iff|C^I| ≥n(|C^I|=n).

We also userole conjunctionsRuS, interpreted by(RuS)^I =R^I∩S^I. The new constructors and axioms are used only in the conditions of rules and not in the ontology.

In the following, we denote byN andM conjunctions of atomic concepts and/or nominals and byH conjunctions of roles (possibly with indexes and superscripts). If we writeC ∈N,N ⊆N⁰,N (N⁰, we treatN andN⁰ as the corresponding sets of conjuncts, whereN =>denotes the empty conjunction. Similarly for conjunctions of roles. For a role conjunctionH we denote byH⁻the conjunction of inverses of roles inH. We writeN(a)∈ AifC(a)∈ Afor everyC ∈N. Note that ifN(a)∈ Athen A |=|N| ≥1.

Definition 7. LetAbe an ABox. ThenN(A) = {|N| ≥1 | N(a)∈ A}is theset of cardinality axioms induced byA.

R¹≈

a≈a :a∈ind(A) R²≈

a≈b b≈c a≈c R³≈

a≈b R(a, c) R(b, c) R⁴≈

a≈b A(a) A(b) R¹t

T(a, b) T(b, c)

T(a, c) :tran(T)∈ T R²t

N(a)

T(a, a) :T⁰|=Nv ∃(TuT⁻).>,tran(T)∈ T R³t

N(a) M(b)

T(a, b) :T⁰|={Nv ∃T.N⁰, M v ∃T⁻.N⁰,|N⁰|= 1},tran(T)∈ T R||

N(a) N(b)

a≈b :T⁰|=|N|= 1 R∀

N(a) R(a, b)

B(b) :T⁰|=Nv ∀R.B R∃

N(a) M(b)

R(a, b) :T⁰|={Nv ∃R.M,|M|= 1} R¹v

R(a, b)

S(a, b) :RvS∈ T R¹≤

F(a, b) F(a, c)

b≈c :func(F)∈ T R²v

N(a)

B(a) :T⁰|=NvB R²≤

M(a) F(a, b)

H(a, b) N(b) :T⁰|=Mv ∃(FuH).N,func(F)∈ T Fig. 1.Materialization rules withT⁰=T ∪N(A)

The materialization rules are presented in Figure 1. These rules are complete for ontology materialization provided the input ontology is in normal form.

Definition 8. A HornSHOIF ontologyOis innormal formif(1)for everyD(a)∈ O,Dis an atomic concept;(2)for everyC vD ∈ O, the conceptsCandDsatisfy the following grammar:

C_(i)::=> | ⊥ |A|o|C₁uC₂|C₁tC₂ (18) D(i)::=> | ⊥ |A|o| ∃R.A| ∀R.A| ¬C (19)

and(3)for everyCv ∀R.A∈ Oand every transitive sub-roleT ofR, there exists an atomic conceptB that occurs only in{Cv ∀T.B, Bv ∀T.B, B vA} ⊆ Oand not inCorA.

Theorem 1. LetO=A ∪ T be a normalized HornSHOIF ontology andT⁰=T ∪ N(A). ThenAis closed under the rules in Figure 1 w.r.t.T iffOis fully materialized.

Proof (sketch).Theif direction is trivial since all rules derive logical consequences of the axioms in premises and side conditions. For theonly if direction, if T⁰ is incon- sistent then we can derive all assertions using rulesR²_v,R_∃, andR_||. Ifind(O) = ∅, then Theorem 1 trivially holds. We assumeind(O)6=∅andT⁰is consistent. We then construct a modelJ ofO such thatJ |=αimpliesα ∈ A, for every atomic asser- tionα. Then,O |= αimpliesJ |=α, which impliesα ∈ A, i.e.,Ois materialized.

Such modelJ is obtained in two steps: 1) construct an interpretationI satisfying all but transitivity axioms inOsuch thatI |=αimpliesα∈ Afor every atomic assertion α; 2) obtainJ fromI by extending the interpretation of non-simple roles to satisfy transitivity axioms.

Intuitively,Ihas a forest-like structure consisting of a graph part and tree parts. The graph part contains domain elements for individuals and concepts that have exactly one instance. The tree parts grow from the graph part to satisfy entailed existential axioms.

To constructI, we use achase-liketechnique [1] in which new domain elements are introduced to satisfy axioms of the formM v ∃H.N entailed byOsuch thatH and N are maximal (w.r.tM), i.e., there is noM v ∃H⁰.N⁰ entailed byOandH∪N ( H⁰∪N⁰. This guarantees that new domain elements (and their respective assertions) also satisfy universal restrictions and role inclusions. Additionally, a new domain element is introduced only when an axiom cannot be satisfied by the existing domain elements.

This ensures all the existential axioms are satisfied while the functionality axioms are not violated.

To obtain the modelJ fromI that also satisfies the transitivity axioms inO, we extend the interpretation of non-simple roles to include the transitive closure, i.e., for every transitive roleT,(d0, dn)is added toT^J and to the interpretations of its super- roles if(di−1,di)∈T^I,1≤i≤n. This extension makesJ satisfy transitivity axioms and the rulesR¹_t–R³_tensure thatJ |=R(a, b)impliesR(a, b)∈ Afor every roleRand individualsa, b. Moreover, since only the interpretation of non-simple roles has been extended, it is also possible to showJ entails the other axioms inO, i.e.J |=O, and J |=αimpliesα∈ Afor every atomic assertionα, i.e., we obtainJ as required. ut

7 Completeness

Once the abstraction refinement procedure terminates, we claim that the input ontology is fully materialized by showing that it is closed under the rules in Figure 1.

Lemma 3. LetA ∪ T be an ontology s.t.Ais equality-, transitivity-, and functionality- closed,Bthe abstraction ofA,B⁰an ABox s.t.B ⊆ B⁰,N(B⁰) =N(A)and∆A ⊆ A with∆Athe update ofAusingB⁰\ B. Then,B⁰is closed under the rules in Figure 1 w.r.t.T implies thatAis also closed under the rules w.r.t.T.

Proof. SinceN(B⁰) = N(A), the side condition of each rule holds forB⁰ ∪ T iff it holds forA ∪ T. We examine each rule in Figure 1 and show thatAis closed under that rule. Clearly,Ais closed underR¹_≈–R⁴_≈,R¹_≤,R¹_t. For the other rules, the intuition is: if the premises of a ruleR hold for some assertionsγ inA, then the premises of Ralso hold for the corresponding abstract assertionsγ⁰ inB, and inB⁰consequently.

SinceB⁰is closed underR, the conclusionκ⁰ ofγ⁰w.r.t.Ris already inB⁰. Then, the condition∆A ⊆ Aguarantees that the conclusionκofγw.r.t.Ris already inA, which implies thatAis closed under R. We present one interesting case:R³_t; the remaining cases are analogous. Let {N(a), M(b)} ⊆ A, and the corresponding side condition holds. We have{N(v_τ(a)), M(v_τ(b))} ⊆ B ⊆ B⁰. SinceB⁰is closed underR³_t, we have T(vτ(a), vτ(b))∈ B⁰. Ifvτ(a) =vτ(b) =vτ, i.e.aandbhave the same typeτ, we are not able to obtainT(a, b)to add toA. That explains why concept types are required.

In this case, since{N(uτC), M(vτ)} ⊆ B ⊆ B⁰ andB⁰ is closed underR³_t, we have T(u_τC, v_τ) ∈ B⁰, and consequentlyT(u_τC, v_τ) ∈ B⁰ \ B. By Definition 4, we have T(a, b)∈∆A ⊆ A. Therefore,Ais closed underR³_t. ut Using Lemma 3, we show that the procedure is complete.

Theorem 2. LetO=A ∪ T be the ontology obtained from the abstraction refinement procedure in Section 5. Then,Ois fully materialized.

Proof. LetB be the abstraction ofA,B⁰ ∪ T the materialization of B ∪ T,∆B = B⁰ \ B, and∆Athe update ofAusing∆B. For everyA(a) ∈ A, we have A(vτ(a))

∈ B ⊆ B⁰, which impliesN(A) ⊆ N(B⁰)(?). There always exists a homomorphism hfromBtoBs.t.h(u_τC(a)) = v_τ(a). Therefore, for everyA(u_τC(a)) ∈ B⁰, we have A(v_τ(a))∈ B⁰. Since the procedure has terminated, i.e.∆A ⊆ A, for everyA(v_τ(a))∈ B⁰ or A(w^R_τ(a)) ∈ B⁰ we haveA(a) ∈ A for somea ∈ ind(A). Hence, we obtain N(B⁰) ⊆ N(A)(†). From (?)and(†)we have N(A) = N(B⁰), which allows us to apply Lemma 3.

By Theorem 1,B⁰is closed under the rules in Figure 1. Therefore, by Lemma 3,A is closed under those rules. Consequently, by Theorem 1,Ois materialized. ut

8 Discussion and Future Work

Our next step is to evaluate the procedure in practice. We believe that it is particularly efficient for ontologies with large ABoxes. We have demonstrated in our previous work [5] that for such ontologies, the abstractions are often significantly smaller than the original ones. Given the (intentionally) close similarity with the previous procedure, it is reasonable to assume that the proposed extension would yield similar reductions.

We also plan to extend the approach to allow more constructors in the Horn fragments.

A possibility to handle qualified number restrictions is to extend the combined type of individuals by considering also concept assertions of their predecessors/successors.

Materialization of concept assertions in the presence of role chains could be done us- ing well-known encoding techniques but computing role assertions, especially for non- simple roles, might be more involved. Handing non-Horn ontologies requires major changes in the current approach as the abstraction can communicate only deterministic entailments; this will be the subject of future work.

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