Most Specific Generalizations w.r.t. General EL-TBoxes

Technische Universität Dresden

Institute for Theoretical Computer Science Chair for Automata Theory

LTCS–Report

Most Specific Generalizations w.r.t. General EL-TBoxes

Benjamin Zarrieß Anni-Yasmin Turhan

LTCS-Report 13-06

Postal Address:

Lehrstuhl für Automatentheorie Institut für Theoretische Informatik TU Dresden

01062 Dresden

http://lat.inf.tu-dresden.de Visiting Address:

Nöthnitzer Str. 46 Dresden

Most Specific Generalizations w.r.t. General EL-TBoxes

Benjamin Zarrieß and Anni-Yasmin Turhan

Institute for Theoretical Computer Science Technische Universität Dresden, Germany {zarriess,turhan}@tcs.inf.tu-dresden.de

Abstract

In the area of Description Logics the least common subsumer (lcs) and the most specific concept (msc) are inferences that generalize a set of concepts or an individual, respectively, into a single concept. If computed w.r.t. a generalEL-TBox neither the lcs nor the msc need to exist. So far in this setting no exact conditions for the existence of lcs- or msc-concepts are known. This report provides necessary and suffcient conditions for the existence of these two kinds of concepts. For the lcs of a fixed number of concepts and the msc we show decidability of the existence in PTime and polynomial bounds on the maximal role- depth of the lcs- and msc-concepts. The latter allows to compute the lcs and the msc, respectively.

1 Introduction

Description Logics (DL) allow to model application domains in a structured and well-understood way. Due to their formal semantics, DLs can offer powerful reasoning services. In recent years the lightweight DL ELbecame popular as an ontology language for large-scale ontologies. EL provides the logical underpinning of the OWL 2 EL profile of the W3C web ontology language OWL [W3C09], which is used in important life science ontologies, as for instance, SNOMED CT [Spa00] and the thesaurus of the US national cancer institute (NCI) [SdH⁺07], which contain ten thousands of concepts. The reason for the success of ELis that it offers limited, but sufficient expressive power, while reasoning can still be done in polynomial time [BBL05].

In DLs basic categories from an application domain can be captured by concepts and binary relations by roles. Implications between concepts can be specified in the so-called TBox. A general TBox allows complex concepts on both sides of implications. Facts from the application domain can be captured byindividuals and their relations in theABox.

Classical inferences for DLs are subsumption, which computes the sub- and super-concept re- lationships of named concepts and instance checking, which determines for a given individual whether it belongs to a given concept. Reasoning support for the design and maintenance of large ontologies can be provided by thebottom-up approach, which allows to derive a new con- cept from a set of example individuals, see [BKM99]. For this kind of task the generalization inferences least common subsumer (lcs) and most specific concept (msc) are investigated for lightweight DLs likeEL. The lcs of a collection of concepts is a complex concept that captures

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all commonalities of these concepts. The msc generalizes an individual into a complex concept, that is the most specific one of which the individual is an instance of.

Unfortunately, neither the lcs nor the msc need to exist, if computed w.r.t. generalEL-TBoxes [Baa03] or cyclic ABoxes written inEL[KM02]. Let’s consider the TBox statements:

PenicillinvAntibioticu ∃kills.S-aureus, CarbapenemvAntibioticu ∃kills.E-coli,

S-aureusvBacteriumu ∃resistantMutant.Penicillin, E-colivBacteriumu ∃resistantMutant.Carbapenem

We want to compute the lcs of Penicillinand Carbapenem. Now, both concepts are defined by the type of bacterium they kill. These, in turn, are defined by the substance a mutant of theirs is resistant to. This leads to a cyclic definition and thus the common subsumer cannot be captured by a finiteEL-concept, since this would need to express the cycle. If computed w.r.t.

a TBox that in addition to the above ones also contains the axioms:

Antibioticv ∃kills.Bacterium,

Bacteriumv ∃resistantMutant.Antibiotic,

then the lcs exists. With the additional statements the lcs of Penicillin andCarbapenemis just Antibiotic. We can observe that the existence of the lcs does not merely depend on whether the TBox is cyclic. In fact, for cyclicEL-TBoxes exact conditions for the existence of the lcs have been devised [Baa04]. However, for the case of general EL-TBoxes such conditions are unknown.

There are several approaches to compute generalizations even in this setting. In [LPW10]

an extension of ELwith greatest fixpoints was introduced, where the generalization concepts always exist. Computation algorithms for approximative solutions for the lcs were devised in [BST07, PT11a] and for the msc in [KM02]. The last two methods simply compute the generalization concept up to a given k, a bound on the maximal nestings of quantifiers. If the lcs or msc exists and a large enoughkwas given, then these methods yield the exact solutions.

However, to obtain theleast common subsumer and themostspecific concept by these methods in practice, a decision procedure for the existence of the lcs or msc, resp., and a method for computing a sufficiently largekare still needed. This paper provides these methods for the lcs and the msc.

In this paper we first introduce basic notions for the DL ELand its canonical models, which serve as a basis for the characterization of the lcs introduced in the subsequent section. There we show that the characterization can be used to verify whether a given generalization is the most specific one and that the size of the lcs, if it exists, is polynomially bounded by the size of the input, which yields a decision procedure for the existence problem. In Section 4 we show the corresponding results for the msc. We end with some conclusions.

2 Preliminaries

2.1 The Description Logic EL

Let NC, NR and NI be disjoint sets of concept, role andindividual names. LetA ∈ NC and r∈NR. EL-concepts are built according to the syntax rule

C::=> |A|CuD| ∃r.C

An interpretation I = (∆^I,·^I) consists of a non-empty domain ∆^I and a function ·^I that assigns subsets of∆^I to concept names, binary relations on∆^I to role names and elements of

∆^I to individual names. The function is extended to complex concepts in the usual way. For a detailed description of the semantic of DLs see [BCM⁺03].

Let C, D denote EL-concepts. A general concept inclusions (GCIs) is an expression of the form C v D. A (general) TBox T is a finite set of GCIs. A GCIC v D is satisfied in an interpretationIifC^I ⊆D^I. An interpretationIis amodel of a TBoxT if it satisfies all GCIs in T.

Let a, b ∈ NI, r ∈ NR and C a concept, then C(a) is a concept assertion and r(a, b) a role assertion. An interpretationI satisfies an assertionC(a)ifa^I∈C^I andr(a, b)if(a^I, b^I)∈r^I holds. AnABox Ais a finite set of assertions. An interpretationI is a model of an ABox A if it satisfies all assertions in A. A knowledge base (KB) K consists of a TBox and an ABox (K= (T,A)). An interpretation is a model ofK= (T,A)if it is a model ofT andA.¹ Important reasoning tasks considered for DLs aresubsumptionandinstance checking. A concept C is subsumed by a concept D w.r.t. a TBox T (denoted C v_T D) if C^I ⊆ D^I holds in all models I ofT. A conceptC isequivalent to a conceptD w.r.t. a TBoxT (denoted C≡_T D) if C v_T D and D v_T C hold. A reasoning service dealing with a KB is instance checking.

An individual a is instance of the concept C w.r.t. K (denoted K |=C(a)) if a^I ∈C^I holds in all models I of K. These two reasoning problems can be decided for EL in polynomial time [BBL05].

Based on subsumption and instance checking our two inferences of interestleast common sub- sumer (lcs) and most specific concept (msc) are defined.

Definition 1. LetC, D be concepts andT a TBox. The concept E is the lcs ofC,D w.r.t.

T (lcsT(C, D)) if the properties 1. Cv_T EandDv_T E, and

2. Cv_T F andDv_T F impliesEv_T F.

are satisfied. If a conceptEsatisfies Property 1 it is acommon subsumer ofCandD w.r.t.T. Thus the lcs is unique up to equivalence, while common subsumers are not unique, thus we writeF ∈cs_T(C, D).

The role depth (rd(C)) of a concept C denotes the maximal nesting depth of ∃ in C. If, in Definition 1 the conceptsE andF are of role-depth up tok, then E is therole-depth bounded lcs (k-lcs_T(C, D)) ofC andD w.r.t.T.

NI,Ais the set of individual names used in an ABoxA.

Definition 2. Leta∈NI,A andK = (T,A)a KB. A concept C is the most specific concept of aw.r.t.K (mscK(a)) if it satisfies:

1. K |=C(a), and

2. K |=D(a)impliesCv_T D.

If in the last definition the concepts C and D have a role-depth limited to k, then C is the role depth bounded msc ofa w.r.t. K (k-mscK(a)). The msc and the k-msc are unique up to equivalence inEL.

1Since we only use the DLEL, we write ‘concept’ instead of ‘EL-concept’ and assume all TBoxes, ABoxes and KBs to be written inELin the following.

2.2 Canonical Models and Simulation Relations

The correctness proof of the computation algorithms for the lcs and msc depends on the char- acterization of subsumption and instance checking. In case of an empty TBox, homomorphisms between syntax trees of concepts [BKM99] were used. A characterization w.r.t. general TBoxes using canonical models and simulations was given in [LW10a], which we want to use in the following.

LetX be a concept, TBox, ABox or KB, thensub(X)denotes the set of subconcepts occurring in X.

Definition 3 (canonical model). Let C be a concept and T a TBox. The canonical model IC,T ofC andT is defined as follows:

• ∆^IC,T :={dC} ∪ {dC⁰ | ∃r.C⁰∈sub(C)∪sub(T)};

• A^IC,T :={dD|Dv_T A}, for allA∈NC;

• r^IC,T :={(dD, dD⁰)|Dv_T ∃r.D⁰ for∃r.D⁰∈sub(T)

or∃r.D⁰ is a conjunct inD}for allr∈N_R. The notion of a canonical model can be extended to a KB.

Definition 4 (canonical model of a knowledge base). Let K = (T,A)be a knowledge base.

The canonical modelIK ofKis defined as follows:

• ∆^IK:={d_a |a∈N_I,A} ∪ {d_C| ∃r.C ∈sub(K)};

• A^IK:={da | K |=A(a)} ∪ {dC|Cv_T A}, for allA∈NC;

• r^IK:={(dC, dD)|Cv_T ∃r.D,∃r.D∈sub(K)} ∪ {(da, d_b)|r(a, b)∈ A} ∪

{(da, dC)| K |=∃r.C(a),∃r.C ∈sub(K)}for allr∈NR;

• a^IK :=d_a, for alla∈N_I,A.

To identify some properties of canonical models we usesimulation relations between interpre- tations.

Definition 5(simulation). LetI1andI2be interpretations. S ⊆∆^I1×∆^I2is calledsimulation from I1to I2if all of the following conditions are satisfied:

(S1) For all concept namesA∈N_C and all(e₁, e₂)∈ S it holds: e₁∈A^I1 impliese₂∈A^I2. (S2) For all role namesr∈N_R and all(e₁, e₂)∈ S and all f₁∈∆^I1 with(e₁, f₁)∈r^I1 there

existsf₂∈∆^I2 such that(e₂, f₂)∈r^I2 and(f₁, f₂)∈ S.

To denote an interpretationI withd∈∆^I we write(I, d). It holds that(I, d)is simulated by (J, e) (written as (I, d).(J, e)) if there exists a simulation S ⊆ ∆^I×∆^J with (d, e)∈ S. The relation.is a preorder, i.e. it is reflexive and transitive. (I, d)issimulation-equivalent to (J, e)(written as(I, d)'(J, e)) if(I, d).(J, e)and(J, e).(I, d)holds.

Now we summarize some important properties of canonical models that were shown in [LW10a].

Lemma 6. Let C be a concept andT a TBox.

1. dE ∈E^IC,T for alldE ∈∆^IC,T. 2. I_C,T is a model ofT.

3. (IC,T, d_D)'(IC⁰,T, d_D), for all concepts C⁰ and alld_D∈∆^IC,T ∩∆^IC0,T. 4. For all models I of T and all d∈∆^I, the following conditions are equivalent:

(a) d∈C^I;

(b) (IC,T, dC).(I, d).

5. The following conditions are equivalent:

(a) Cv_T D;

(b) dC∈D^IC,T;

(c) (I_D,T, dD).(I_C,T, dC).

This lemma gives us a characterization of subsumption. A similar lemma was shown in [LW10b]

for the instance relationship.

Lemma 7. Let K be a knowledge base. I_K satisfies the following properties:

1. I_K is a model ofK.

2. The following conditions are equivalent:

(a) K |=C(a);

(b) da ∈C^IK.

Next we recall some known operations on interpretations.

Taking an element of the domain of an interpretation as the root, the interpretation can be unraveled into a possibly infinite tree. The nodes of the tree are words that correspond to paths starting ind. Now,π=dr1d1r2d2r3· · · is a path in an interpretationI if the domain elements di anddi+1 are connected viar^I_i+1 for alli.

Definition 8(tree unraveling of an interpretation). LetIbe an interpretation w.r.t. the names NC andNR withd∈∆^I. Thetree unraveling Id ofI indis defined as follows:

∆^Id:={dr1d1r2· · ·rndn|(di, di+1)∈r_i+1^I ∧0≤i < n∧d0=d};

A^Id:={σd⁰|σd⁰∈∆^Id∧d⁰∈A^I}, for allA∈N_C; r^Id:={(σ, σrd⁰)|(σ, σrd⁰)∈∆^Id×∆^Id}, for allr∈NR.

The length of an element σ∈ ∆^Id, denoted by |σ|, is the number of role names occurring in σ. If σis of the formdr₁d₁r₂· · ·r_md_m, then d_mis thetail ofσ denoted bytail(σ) =d_m. The interpretationI_d<sup>`</sup>denotes the finite subtree rooted indof the tree unravelingI<sub>d</sub> containing all elements up to depth `. Such a finite tree can be translated into a complex concept which is called characteristic concept.

Definition 9 (characteristic concept). Let (I, d) be an interpretation. The `-characteristic concept X<sup>`</sup>(I, d)is defined as follows: ²

X⁰(I, d) :=l

{A∈NC |d∈A^I}

X^`(I, d) :=X⁰(I, d)u l

r∈NR

l{∃r.X^`−1(I, d⁰)|(d, d⁰)∈r^I}

2For a setM of concepts we writed

M as shorthand ford

F∈MF. IfM is empty, thend

M is equal to>.

Later we will need the following basic property of characteristic concepts that was shown in [LPW10].

Lemma 10. Let (I, d) and (J, e) be interpretations. Then e ∈ (X^{`</sup>(I, d))<sup>J</sup> if and only if (I<sub>d</sub><sup>`}, d).(J, e).

Another operation that we will use later is the product of two interpretations that is defined as follows.

Definition 11 (product interpretation). LetI and J be interpretations. The product inter- pretation I × J is defined by

∆^I×J := ∆^I×∆^J;

A^I×J :={(d, e)|(d, e)∈∆^I×J ∧d∈A^I∧e∈A^J}, for allA∈N_C; r^I×J :={((d, e),(f, g))|((d, e),(f, g))∈∆^I×J ×∆^I×J

∧(d, f)∈r^I∧(e, g)∈r^J}, for allr∈NR.

3 Existence of the Least Common Subsumer

In this section we develop a decision procedure for the problem whether for two given concepts and a given TBox the least common subsumer of these two concepts exists w.r.t. the given TBox. If not stated otherwise, the two input concepts are denoted byC andD and the TBox byT.

Similar to the approach used in [Baa04] we proceed by the following steps:

1. Devise a method to identify lcs-candidates. The set of lcs-candidates is a possibly infinite set of common subsumers of C and D w.r.t. T, such that if the lcs exists then one of these lcs-candidates actually is the lcs.

2. Characterize the existence of the lcs. Find a condition such that the problem whether a given common subsumer of C and D w.r.t. T is least (w.r.t. v_T), can be decided by testing this condition.

3. Establish an upper bound on the role-depth of the lcs. We give a bound `such that if the lcs exists, then it has a role-depth less or equal`. By such an upper bound one needs to check only for finitely many of the lcs-candidates if they are least (w.r.t.vT).

The next subsection addresses the first two problems, afterwards we show that such a desired upper bound exists.

3.1 Characterizing the existence of the lcs

In this section canonical models and simulation relations are used to obtain in a first step a set of possible candidates for the lcs and then to characterize whether a common subsumer is least or not.

In [PT11a] so called role-depth bounded least common subsumers were introduced as approxi- mations of the lcs, denoted by k-lcs_T(C, D). For a fixed natural numberkthek-lcs_T(C, D)is a common subsumer that is the least one of all common subsumers with a role-depth≤k. To obtain thek-lcsT(C, D)we build the product of the canonical models(IC,T, dC)and(ID,T, dD) and then take thek-characteristic concept of this product model. This product construction is

adopted from [Baa03, LPW10], where a similar construction was used to define the lcs in EL with gfp-semantics and in the DLEL^ν respectively.

In order to prove that the k-lcs can be computed as described above, we first show some properties of product models and their characteristic concepts.

Lemma 12. Let I_C,T× I_D,T andI_E,T× I_F,T be products of canonical models with(d_G, d_H)∈

∆^IC,T×ID,T ∩∆^IE,T×IF,T.

1. For anyk∈Nit holds thatX^k(I_C,T × I_D,T,(dG, dH)) =X^k(I_E,T × I_F,T,(dG, dH)) 2. LetN be a concept. (dG, dH)∈N^IC,T×ID,T iffGv_T N andHv_T N.

Proof. 1. By Claim 3 of Lemma 6 it is implied that for anyk X^k(IC,T, d_G) =X^k(IE,T, d_G) andX^k(ID,T, d_H) =X^k(IF,T, d_H), respectively. Obviously, this implies the claim.

2. This claim follows directly from the definition of products of interpretations and Claim 5 of Lemma 6.

Now we show that thek-characteristic concept of(IC,T×ID,T,(dC, dD))yields thek-lcsT(C, D).

Lemma 13. Letk be a natural number.

1. X^k(I_C,T × I_D,T,(d_C, d_D))∈cs_T(C, D).

2. LetE be a concept withrd(E)≤kandCv_T E andDv_T E.

It holds thatX^k(I_C,T × I_D,T,(dC, dD))v_T E.

Proof. 1. We show the claim by induction onk.

k= 0 : By Definition 9 it holds that

X⁰(IC,T × ID,T,(d_C, d_D)) =l

{A∈N_C|(d_C, d_D)∈A^IC,T×ID,T}. (1) For any concept name A in this conjunction it holds that (dC, dD) ∈ A^IC,T×ID,T and thereforedC∈A^IC,T anddD∈A^ID,T. From point 5 of Lemma 6 it follows thatCvT A and D vT A and therefore C vT X⁰(IC,T × ID,T,(dC, dD)) and D vT X⁰(IC,T × ID,T,(d_C, d_D)).

k >0 : By applying the definition ofX^k we get

X^k(I_C,T × I_D,T,(dC, dD)) =X⁰(I_C,T × I_D,T,(dC, dD))u l

r∈NR

l{∃r.X^k−1(I_C,T × I_D,T,(d_E, d_F))

|((d_C, d_D),(d_E, d_F))∈r^IC,T×ID,T}.

(2)

From Lemma 12.1 it follows thatX^k−1(I_C,T×I_D,T,(dE, dF)) =X^k−1(I_E,T×I_F,T,(dE, dF)).

Now the induction hypothesis can be applied as follows:

EvT X^k−1(IE,T × IF,T,(dE, dF)) F vT X^k−1(IE,T × IF,T,(d_E, d_F)).

By Lemma 12.1 it is implied that

E vT X^k−1(IC,T × ID,T,(dE, dF)) F vT X^k−1(IC,T × ID,T,(d_E, d_F))

and by Lemma 6.5

d_E∈(X^k−1(I_C,T × I_D,T,(d_E, d_F)))^IE,T dF ∈(X^k−1(I_C,T × I_D,T,(dE, dF)))^IF,T.

From Lemma 6.3 it follows(I_E,T, dE)'(I_C,T, dE) and (I_F,T, dF)'(I_D,T, dF) conse- quently

dE∈(X^k−1(I_C,T × I_D,T,(dE, dF)))^IC,T dF ∈(X^k−1(IC,T × ID,T,(dE, dF)))^ID,T.

and by definition of the product of interpretation it holds that

(d_E, d_F)∈(X^k−1(IC,T × ID,T,(d_E, d_F)))^IC,T×ID,T.

Since(d_E, d_F)is anr-successor of (d_C, d_D)in I_C,T × I_D,T it is implied that (dC, dD)∈(∃r.X^k−1(I_C,T × I_D,T,(dE, dF)))^IC,T×ID,T

and with Lemma 12.2 we obtain

Cv_T ∃r.X^k−1(I_C,T × I_D,T,(dE, dF)) DvT ∃r.X^k−1(IC,T × ID,T,(dE, dF)).

As shown in the base caseX⁰(IC,T × ID,T,(dC, dD))is also a common subsumer of C andD w.r.t.T. It is now implied thatX^k(IC,T × ID,T,(dC, dD))is a common subsumer ofCandD w.r.t.T.

2. The claim is proven by induction on the role-depth of an arbitrary common subsumerE of C andD w.r.t.T withrd(E)≤k.

rd(E) = 0 : Eis a conjunction of concept names of the formd

iA_i. We show that the concept namesA_ioccur in the conjunctionX⁰(I_C,T×I_D,T,(d_C, d_D)). SinceCv_T EandDv_T E holds, it follows by Lemma 6.5 that dC ∈ E^IC,T and dD ∈ E^ID,T. So we have that dC ∈A^I_i^C,T anddD∈A^I_i^D,T for alli and(dC, dD)∈A^I_i^C,T×ID,T for all i. By definition ofX^k(IC,T × ID,T,(dC, dD))and (1) it is implied thatX⁰(IC,T × ID,T,(dC, dD))vT E.

rd(E) =n >0 : Let

E=A1u · · · uA`u ∃r1.E₁⁰ u · · · u ∃rm.E_m⁰

It can be shown like in the base case that the conjunctionA₁u...uA`subsumesX^k(IC,T× ID,T,(d_C, d_D)). Let ∃rj.E_j⁰ with 1 ≤ j ≤ m be an existential restriction in E. Since it holds that C v_T ∃r_j.E_j⁰ and D v_T ∃r_j.E⁰_j, we get d_C ∈ (∃r_j.E⁰_j)^IC,T and d_D ∈ (∃r_j.E_j⁰)^ID,T by Lemma 6.5. There are r_j-successors d_G and d_H of d_C and d_D in I_C,T andI_D,T, respectively, withd_G∈(E_j⁰)^IC,T andd_H ∈(E_j⁰)^ID,T. It holds that

dG ∈(E⁰_j)^IC,T

⇒(I_E⁰

j,T, d_E⁰

j).(I_C,T, d_G)'(I_G,T, d_G)(by Lemma 6.4 and 6.3)

⇒Gv_T E_j⁰ (by Lemma 6.5).

The same argument holds fordH. By induction hypothesis andrd(E_j⁰) =n−1 we now have thatXⁿ⁻¹(IG,T × IH,T,(d_G, d_H))vT E_j⁰. From Lemma 12.1 it follows that

Xⁿ⁻¹(IG,T × IH,T,(dG, dH)) =Xⁿ⁻¹(IC,T × ID,T,(dG, dH))

and thereforeXⁿ⁻¹(IC,T × ID,T,(dG, dH))vT E_j⁰ and

∃rj.Xⁿ⁻¹(I_C,T × I_D,T,(dG, dH))v_T ∃rj.E_j⁰.

Since∃rj.Xⁿ⁻¹(IC,T × ID,T,(dG, dH))is a conjunct inXⁿ(IC,T × ID,T,(dC, dD)), it is implied thatXⁿ(IC,T × ID,T,(dC, dD))vT ∃rj.E_j⁰.

In the following we take X^k(I_C,T × I_D,T,(dC, dD)) as representation of k-lcs_T(C, D). It is implied by Lemma 13 that the set of k-characteristic concepts of the product model (I_C,T × I_D,T,(dC, dD))for allkis the set of possible candidates for the lcs_T(C, D). This can be stated as follows.

Corollary 14. The lcsT(C, D)exists if and only if there exists ak∈Nsuch that for all`∈N: k-lcsT(C, D)vT `-lcsT(C, D).

Obviously, this doesn’t yield a decision procedure for the problem whether thek-lcs_T(C, D)is the lcs, since subsumption cannot be checked for infinitely many`in finite time.

Next, we address step 2 and show a condition on the common subsumers that decides whether a common subsumer is the least one or not. The main idea is that the product model captures all commonalities of the input concepts by means of canonical models. Thus we compare the canonical models of the common subsumers and the product model using . and simulation equivalence'.

First it can be stated that the canonical model of thek-lcs simulates the tree unraveling of the product model limited to depthk.

Lemma 15. Let J_(dC,dD) be the tree unraveling of(I_C,T × I_D,T,(d_C, d_D))in(d_C, d_D)andK the k-lcsT(C, D)w.r.t.T. It holds thatJ_(d^k

C,dD).(IK,T, dK).

Proof. The concept X^k(I_C,T × I_D,T,(dC, dD)) is by Lemma 13 a common subsumer ofC, D w.r.t. T. Since X^k(I_C,T × I_D,T,(dC, dD)) has role-depth ≤ k, it is implied that K v_T X^k(I_C,T × I_D,T,(dC, dD)) and therefore dK ∈ (X^k(I_C,T × I_D,T,(dC, dD)))^IK,T by Lemma 6.5. From Lemma 10 it now followsJ_(d^k

C,dD).(I_K,T, dK).

The following lemma recalls a simple property about products of interpretations.

Lemma 16 ( [LPW10]). Let (J, e),(I1, d₁)and(I2, d₂)be interpretations. If(J, e).(I1, d₁) and(J, e).(I2, d₂), then(J, e).(I1× I2,(d₁, d₂)).

Now we show that a common subsumer is the lcs if and only if its canonical model is simulation- equivalent to the product of the canonical models of the input concepts.

Lemma 17. LetE be a concept.

E is the lcs of C andD w.r.t. T iff(I_C,T × I_D,T,(dC, dD))'(I_E,T, dE).

P,C

S,E

∈A^I ∈B^I kills^I resistantMutant^I P,C

S,E S,B

P,A B,E

A,C

B,B

A,A

A

B Au ∃kills.(Bu ∃resistantMutant.A)

Bu ∃resistantMutant.A

A

IP,T1× IC,T1 IAu∃kills.(Bu∃resistantMutant.A),T1 IP,T2× IC,T2 I_A,T2

Figure 1: Product of canonical models ofT₁ andT₂ The proof idea of this claim can be outlined as follows:

Assume(IE,T, d_E)is simulation-equivalent to the product model. We need to show thatE≡T

lcs_T(C, D).

For any F ∈cs_T(C, D) it holds by Lemma 6.5 that(I_F,T, dF)is simulated by (I_C,T, dC) and by(I_D,T, dD)and therefore also by(I_C,T × I_D,T,(dC, dD)). By transitivity of.it is implied that (I_F,T, dF).(I_E,T, dE)andEv_T F by Lemma 6. ThereforeE≡_T lcs_T(C, D).

For the other direction assumeE≡T lcsT(C, D). It has to be shown that(IE,T, dE)simulates the product model. LetJ(d_C,d_D)be the tree unraveling of the product model. SinceEis more specific than the k-characteristic concepts of the product model for all k (by Corollary 14), (IE,T, d_E) simulates the subtree J_(d^k

C,d_D) of J(d_C,d_D) limited to elements up to depth k, for allk. For eachk we consider the maximal simulation from J_(d^k

C,d_D)to (I_E,T, dE). Note that ((d_C, d_D), d_E)is contained in any of these simulations. Letσbe an element of∆^{J(dC ,dD)} at an arbitrary depth `. We show how to determine the elements of ∆^IE,T, that simulate this fixed element σ. Let Sn(σ)be the maximal set of elements from ∆^IE,T that simulateσ in each of the trees J_(dⁿ

C,d_D) withn≥`. We can observe that the infinite sequence(S`+i(σ))i=0,1,2,... is decreasing (w.r.t. ⊇). Therefore at a certain depth we reach a fixpoint set. This fixpoint set exists for anyσ. It can be shown that the union of all these fixpoint sets yields a simulation from the product model to(I_E,T, dE).

Proof of Lemma 17. “⇒":

Assume that E is the lcs of C, D w.r.t. T, thus C v_T E and D v_T E and by Lemma 6.5 (I_E,T, d_E). (I_C,T, d_C)and (I_E,T, d_E) .(I_D,T, d_D) holds. It is now implied by Lemma 16 that

(I_E,T, dE).(I_C,T × I_D,T,(dC, dD)). (3) We now show(I_C,T × I_D,T,(dC, dD)).(I_E,T, dE)by constructing a simulation from the tree unraveling J_(dC,dD) of (I_C,T × I_D,T,(dC, dD)) to (I_E,T, dE). We first write J_(dC,dD) as an infinite union of the subtreesJ_(d^k

C,d_D).

∆^{J(dC ,dD)} = [

k=0

∆^{J(kdC ,dD)}, (4)

A^{J(dC ,dD)} = [

k=0

A^{J(kdC ,dD)}, for allA∈NC (5) r^{J(dC ,dD)} = [

k=0

r^{J(kdC ,dD)}, for allr∈NR (6)

LetK be thek-lcs_T(C, D)for an arbitraryk. By Lemma 15 we have:

J_(d^k_C,dD).(IK,T, dK). (7) Since E is the lcs,E is subsumed by K w.r.t. T and therefore it holds (by Lemma 6.5) that (IK,T, d_K).(IE,T, d_E). With (7) and transitivity of.we have

J_(dⁿ_C,dD).(IE,T, d_E)

for all n∈N. If J_(dC,dD) is finite, then there exists anm∈N such thatJ_(d^m

C,d_D) =J_(dC,dD). In this case we are done. It remains to be shown that J(d_C,d_D) . (IE,T, d_E) also holds if J_(dC,dD) is an infinite tree. Consequently, there exists for eachn a maximal simulation S_n ⊆

∆^{J(ndC ,dD)}×∆^IE,T with((dC, dD), dE)∈ Sn. For the infinite sequence of subtrees J_(d⁰

C,d_D),J_(d¹

C,d_D),J_(d²

C,d_D), ...

of J_(dC,dD) there exists an infinite sequence S0,S1,S2, ... of maximal simulations. Using this sequence we construct now a simulation that showsJ_(dC,dD).(I_E,T, dE). To do this we select an`∈Nand an arbitrary elementσ∈∆<sup>J(dC ,dD)</sup> with|σ|=`.

The elementσoccurs in all subtreesJ_(d^m

C,dD)withm≥`. So there are pairs in the corresponding maximal simulations Smthat consist ofσand an elementd∈∆<sup>IE,T</sup>. For thisσand allm≥` we now collect exactly those pairs that occur in the maximal simulation Smand denote it by:

Sm(σ) := ({σ} ×∆^IE,T)∩ Sm.

For allmthe corresponding setsSm(σ)⊆ Smare non-empty.

We can also observe, that if an elementσis simulated by dinSi+2 (i.e. (σ, d)∈ Si+2(σ)) it is also simulated by the same din Si+1 since these simulations are maximal. Therefore the sets Sm(σ)don’t increase with increasing m. This is shown in the following claim.

Claim. Let σ∈∆^{J(dC ,dD)} with `=|σ|. It holds that:

S_{`</sub>(σ)⊇ S<sub>`+1}(σ)⊇ S_`+2(σ)...

Proof of the claim. We show by induction onn≥` that

Sn(σ)⊆ Sn−1(σ)⊆...⊆ S`+1(σ)⊆ S`(σ).

This obviously holds for the base casen=`.

Letn > `and(σ, d)∈ Sn(σ). It has to be shown that(σ, d)∈ Sn−1(σ)and thereforeSn(σ)⊆ S_n−1(σ). LetSn⊆∆^{J(ndC ,dD)}×∆^IE,T be the maximal simulation fromJ_(dⁿ

C,dD)to(I_E,T, dE).

LetSnn−1 defined as

S_nn−1 :=S_n∩(∆^J

n−1

(dC ,dD)×∆^IE,T)

be the restriction ofSnto pairs, whose first components are elements of the tree unraveling with depth less or equaln−1. SinceSnn−1is a simulation fromJ_(dⁿ⁻¹

C,d_D)to(I_E,T, dE), it holds that Snn−1 is contained in the maximal simulationSn−1. We have now (σ, d)∈ Sn(σ)⊆ Snn−1 ⊆ S_n−1, because|σ|< n. Then it is implied that(σ, d)∈ S_n−1(σ)and thereforeS_n(σ)⊆ S_n−1(σ).

By applying the induction hypothesis to S_n−1(σ)we get Sn(σ)⊆ Sn−1(σ)

I.H.

⊆ ...

I.H.

⊆ S`+1(σ)

I.H.

⊆ S`(σ)

which finishes the proof of the claim.

From this claim it follows that there exists anf ∈Nsuch that Sf(σ) =

∞

\

`≥|σ|

S`(σ). (8)

We construct a relationS ⊆∆^{J(dC ,dD)}×∆^IE,T as follows:

S := [

σ∈∆^{J(dC ,dD)}

^∞

\

`≥|σ|

S`(σ)

To showJ_(dC,dD).(I_E,T, dE)it has to be shown thatSis a simulation with((dC, dD), dE)∈ S. For alln∈Nwe have((dC, dD), dE)∈ Sn((dC, dD))and therefore((dC, dD), dE)∈ S. Next we show that S fulfills the conditions(S1) and(S2) of Definition 5.

(S1) : Let (σ, d) ∈ S with σ ∈ A^{J(dC ,dD)} for a concept name A. It has to be shown that d∈A^IE,T.

There exists an x ∈ N with (σ, d) ∈ Sx. From σ ∈ A^{J(dC ,dD)} and (5) it follows that σ ∈ A^{J(xdC ,dD)}. S_x is a simulation from J_(d^x

C,d_D) to (I_E,T, d_E) and satisfies (S1). It follows thatd∈A^IE,T.

(S2) : Let (σ, d) ∈ S and (σ, σre) ∈ r^{J(dC ,dD)}. It has to be shown that there is a g with (d, g)∈r^IE,T and(σre, g)∈ S.

By (8) there are numbersn, mwithSn(σ) =T∞

i≥|σ|Si(σ)andSm(σre) =T∞

j≥|σre|Sj(σre).

Letm > n w.l.o.g. It is implied that S_m(σ) =S_n(σ). Since (σ, d) ∈ S_m and (σ, σre)∈ r^{J(mdC ,dD)} (by (6)), there is ag with(d, g)∈r^IE,T and(σre, g)∈ Sm(σre)⊆ Sm, because Smis a simulation and satisfies(S2). The numbermwas chosen such thatSm(σre)⊆ S holds and therefore it is implied that(σre, g)∈ S.

It is implied thatJ_(dC,dD).(I_E,T, dE)and therefore also(I_C,T×I_D,T,(dC, dD)).(I_E,T, dE).

Together with (3) we have(I_C,T × I_D,T,(dC, dD))'(I_E,T, dE).

“⇐":

Assume E is a common subsumer of C and D and (I_C,T × I_D,T,(dC, dD))'(I_E,T, dE). It has to be shown that E is the least common subsumer. LetF be an arbitrary concept with CvT F andDvT F. From Lemma 6.5 it follows that

(I_F,T, dF).(I_C,T, dC) (I_F,T, d_F).(I_D,T, d_D) From Lemma 16 it follows that

(I_F,T, d_F).(I_C,T × I_D,T,(d_C, d_D)) and by assumption

(I_F,T, dF).(I_C,T × I_D,T,(dC, dD)).(I_E,T, dE).

We now have(I_F,T, dF).(I_E,T, dE)andE v_T F by Lemma 6.5. SoE is the least common subsumer ofC andDw.r.t.T.

By the use of this Lemma it can be verified whether a given common subsumer is the least one or not, which we illustrate by an example.

Example 18. Consider again the TBox from the introduction (now displayed with abbreviated concept names)

T1={PvAu ∃kills.S, SvBu ∃resistantMutant.P, CvAu ∃kills.E, EvBu ∃resistantMutant.C} and the following extended TBox

T2=T1∪ {Av ∃kills.B, Bv ∃resistantMutant.A}.

In Figure 1 we can see that

Au ∃kills.(Bu ∃resistantMutant.A)∈csT1(P,C),

but it is not the lcs, because its canonical model cannot simulate the product model(I_P,T1× I_C,T1,(dP, dC)). The concept A, however, is the lcs of P and C w.r.t. T2. We have (I_P,T2 × I_C,T2,(dP, dC)).(I_A,T2, dA)since any element from∆^IP,T2×IC,T2 inA^IP,T2×IC,T2 orB^IP,T2×IC,T2 is simulated by A or B, respectively.

The characterization of the existence of the lcs given in Corollary 14 can be reformulated using Lemma 17.

Corollary 19. The lcsT(C, D) exists iff there exists a k such that the canonical model of X^k(IC,T × ID,T,(d_C, d_D))w.r.t. T simulates (IC,T × ID,T,(d_C, d_D)).

This corollary still doesn’t yield a decision procedure for the existence problem of the lcs, since the depthkis still unrestricted. Such a restriction will be developed in the next section.

3.2 A Polynomial Upper Bound on the Role-depth of the LCS

In this section we show that, if the lcs exists, its role-depth is bounded by the size of the product model. First, consider again the TBox T2 from Example 18, where A v_T2 ∃kills.(Bu

∃resistantMutant.A) holds, which results in a loop in the product model through the elements A,A and ^B,B. Furthermore, the cycles in the product model involving the roles kills and resistantMutantare captured by the canonical modelI_A,T2. ThereforeA≡_T2lcs_T2(P,C). On this observation we build our general method.

We call elements(d_F, d_F⁰)∈∆^IC,T×ID,T synchronous ifF =F⁰ andasynchronous otherwise.

The structure of(IC,T× ID,T,(d_C, d_D))can now be simplified by considering only synchronous successors of synchronous elements.

Lemma 20. Let(d_E, d_E)∈∆^IC,T×ID,T. (I_C,T × I_D,T,(d_E, d_E))'(I_E,T, d_E).

Proof. We define relations S ⊆ ∆^IC,T×ID,T ×∆^IE,T and Z ⊆ ∆^IE,T ×∆^IC,T×ID,T with ((dE, dE), dE)∈ S and(dE,(dE, dE))∈ Z as follows.

S:={((dF, dG), dF)|(dF, dG)∈∆^IC,T×ID,T, dF ∈∆^IE,T} Z:={(dF,(d_F, d_F))|d_F ∈∆^IE,T,(d_F, d_F)∈∆^IC,T×ID,T}

Obviously S and Z satisfy(S1)and (S2)of Definition 5. Since ((d_E, d_E), d_E)∈ S andS is a simulation,(I_C,T × I_D,T,(d_E, d_E)).(I_E,T, d_E). And analogous we have(d_E,(d_E, d_E))∈ Z, Z is a simulation and therefore(I_E,T, dE).(I_C,T× I_D,T,(dE, dE)). The compositionS ◦ Z ⊆

∆^IC,T×ID,T ×∆^IC,T×ID,T is also a simulation with ((dE, dE),(dE, dE))∈ S ◦ Z. The second component of the pairs inS ◦Zare synchronous by definition ofZ. Therefore any asynchronous successor of(dE, dE)is simulated by its synchronous counterparts inS ◦ Z.

In order to find a numberk, such that the product model is simulated by the canonical model of K=X^k(IC,T× ID,T,(dC, dD)), we first represent the model(IK,T, dK)as a subtree of the tree unraveling of the product modelJ(d_C,d_D)with root(dC, dD). We construct this representation by extending the subtree J_(d^k

C,d_D) by new tree models at depthk. We need to ensure that the resulting interpretation, denoted byJb_(d^k

C,dD), is a model ofT, that is simulation-equivalent to (IK,T, dK). The elements σ∈∆^{J(kdC ,dD)} with |σ| =k we extend and the corresponding trees we append to them are selected as follows:

First we consider elements that have a tail that is a synchronous element. Iftail(σ) = (d_F, d_F), thenF is calledtail conceptofσ. To select the elements with a synchronous tail, that we extend by the canonical model of their tail concept, we use embeddings of J_(d^k

C,d_D) into (I_K,T, dK).

We show that such an embedding exists.

Lemma 21. Let I_K,T be the canonical model ofX^k(I_C,T × I_D,T,(dC, dD))w.r.t.T. For any k there exists a simulationZ ⊆∆^{J(kdC ,dD)}×∆^IK,T that is functional andZ((dC, dD)) =dK. Proof. It holds by Definition 9 and by definition of the tree unraveling that:

X^k((I_C,T × I_D,T,(d_C, d_D))) =X^k(J_(dC,dD),(d_C, d_D)) =X^k(J_(d^k

C,d_D),(d_C, d_D)).

By Definition 3 (I_K,∅, dK)is a subinterpretation of(I_K,T, dK), which means∆^IK,∅ ⊆∆^IK,T, A^IK,∅⊆A^IK,T for all concept namesAandr^IK,∅⊆r^IK,T for all role namesr. From Definition 3 and 9 it follows that there even exists a bijective total functionZbetween∆^IK,∅and∆^{J(kdC ,dD)} such thatσ∈A^{J(kdC ,dD)}iffZ(σ)∈A^IK,∅for allAand(σ, σ⁰)∈r^{J(kdC ,dD)}iff(Z(σ), Z(σ⁰))∈r^IK,∅

for allr. Z is a functional simulation from(J_(d^k

C,d_D),(d_C, d_D))to(IK,T, d_K).

Let H ={Z1, ..., Zn} be the set of all functional simulations Zi from J_(d^k

C,dD) to (I_K,T, dK) withZ_i((d_C, d_D)) =d_K. We say thatσwith tail conceptF ismatched byZ_iifZ_i(σ)∈F^IK,T. The set of elements σ∈∆^{J(kdC ,dD)} with |σ| =k, that are matched by a functional simulation Z_i is called matching set denoted byM(Zi).

The elements from∆^{J(kdC ,dD)}, we extend, are calledstubs.

Definition 22. Letσ∈∆^{J(kdC ,dD)} with|σ|=k. σis contained in theset of stubs ofJ_(d^k

C,dD), denoted bystubs(J_(d^k

C,d_D)), if it satisfies one of the following properties:

1. LetM be a conjunction of concept names and∃r.F ∈sub(T). It holds thatσ∈M^{J(kdC ,dD)} andM v_T ∃r.F

2. LetM(H) :={M(Z)|Z ∈ H}be the set of all matching sets. It holds thatσis contained in all maximal sets inM(H).

Now we define the set of treesΥ(σ)that are appended to a stubσ. Considerσ∈stubs(J_(d^k

C,d_D)) that satisfies the first condition for ∃r.F. Let (I∃r.F,T, d∃r.F) be the canonical model. By definition ofJ(d_C,d_D)it holds thatσr(dF, dF)∈∆^{J(dC ,dD)}and the subtreeJσr(d_F,d_F)ofJ(d_C,d_D)

is simulation-equivalent to(I∃r.F,T, d_∃r.F)(by Lemma 20). ThusΥ(σ)containsJσr(d_F,d_F). Assume σ ∈stubs(J_(d^k

C,dD)) satisfies the second property for the tail concept F. In this case the subtreeJσ ofJ_(dC,dD)is simulation-equivalent to(I_F,T, dF)as shown in Lemma 20. Thus Υ(σ)containsJσ.