Completion-based computation of least common subsumers with limited role-depth for EL and Prob-EL 01

The least common subsumer (lcs) w.r.t general EL-TBoxes does not need to exists in general due to cyclic axioms. In this report we present an algorithm for computing role-depth bounded EL-lcs based on the completion algorithm for EL. We extend this computation algorithm to a recently introduced probabilistic variant of EL: Prob-EL.


Motivation
The least common subsumer (lcs) inference yields a concept description, that generalizes a collection of concept descriptions by extracting their commonalities.This inference was first introduced in [11].The lcs is used in many applications, most prominently, it is used in the bottom-up construction of knowledge bases [6], where a new concept is to be introduced that has a collection of selected individuals as instances.This new concept is then a candidate for a new concept definition in the TBox.This can be achieved by first generalizing each selected individual into a concept description (by computing its most specific concept (msc)) and in a second step apply the lcs to these concept descriptions.The concept description returned by the lcs can then be (edited and) added to the TBox.Further applications of the lcs include similarity-based Information Retrieval [18,1] or learning from examples [13,12].
The lightweight Description Logic EL and many of its extensions enjoy the nice property that concept subsumption and classification of TBoxes can be computed in polynomial time [4].Thus, despite of its limited expressiveness, EL is used in many practical applications -most prominently in the medical ontology Snomed [19] -and is the basis for EL profile of the OWL 2.0 standard 1 .However, some practical applications such as medical or context-aware applications have to represent information that holds only with a certain probability.For instance, context-aware applications may need to represent sensor data in their ontology, which is correct only with a certain probability.Whereas medical data may based on statistical analysis.This sort of information can be represented by the probabilistic DLs recently introduced [16].Here probabilistic information is assigned to concepts (and roles) directly and not, as in other probabilistic DLs, to concept axioms [15,14].In particular in [16] the DL Prob-EL and its sublanguage Prob-EL 01 was introduced.Latter allows to express limited probability values for EL-concepts, and it was shown in [16] that instance checking can be done in polynomial time.
In applications where different information sources supply varying information for the same entity, the generalization of this information, gives evidence for what the sources agree upon.For both, EL and Prob-EL, the computation of the lcs is a desirable task.Unfortunately, the lcs w.r.t.general or even cyclic EL-TBoxes does not need to exist (see [2]), due to cyclic definitions in the TBox.
In this report we present practical algorithms for computing the lcs up to a certain role-depth for general ELand Prob-EL 01 -TBoxes.The concept description obtained by the algorithm is still a generalization of the input concepts, but not necessarily the least one w.r.t.subsumption.We argue that this "good common subsumer" still is useful in practice.To the best of our knowledge this report gives the first computation algorithm for the lcs in a DL that captures (a limited form) probabilistic information.In [17] a probabilistic lcs for the DL ALN was investigated, with the idea to relax the notion of the lcs in order to avoid overfitting.This relaxed notion is helpful in cases where the concept obtained from the lcs is used as a search pattern.
Our algorithms to compute the lcs up to a certain role-depth for general ELand Prob-EL 01 -TBoxes are based upon the completion algorithms for computing classification in EL and Prob-EL 01 and thus can be easily implemented on top of completion-based reasoners for these two logics.

Description Logics
In Description logics (DLs), concept descriptions are inductively defined with the help of a set of concept constructors, starting with a set N C of concept names, a set N R of role names.From elements of these sets complex concept descriptions can be obtained by concept constructors.In this report we are interested to reasoning with concept descriptions.

The DL EL
The DL EL allows for two concept constructors: conjunction and existential restrictions.The DL EL also contains the top-concept, denoted ⊤.
Definition 1 (Syntax of EL-concept descriptions) Let A denote a concept name, r denotes a role and C 1 , C 2 denote arbitrary EL-concepts.An EL-concept description C can be obtained by the following rule: The semantics of a concept description is defined in terms of an interpretation I = (∆, • I ).The domain ∆ of I is a non-empty set of individuals and the interpretation function • I maps each concept name A ∈ N C to a set P I ⊆ ∆ and each role name r ∈ N R to a binary relation r I ⊆ ∆×∆.Arbitrary EL-concept descriptions are interpreted as follows: Definition 2 (Semantics of EL-concept descriptions) The top-concept is interpreted as the domain (⊤ I = ∆).The extension of • I to arbitrary EL-concept descriptions is inductively defined, as follows: • (C ⊓ D) I = C I ∩ D I , and Concept descriptions can be assigned a name or sub-concept super-concept relationships between arbitrary concept descriptions established in the TBox.
is a general concept inclusion axiom (GCI).The semantics of GCIs is given by the interpretation function.
If the concept axioms in the TBox contain only EL-concept descriptions, we call it an EL-TBox.
It is easy to see that concept equivalence between two concept descriptions (written C 1 ≡ C 2 ) can be stated by two GCIs: C 1 ≡ C 2 and C 1 ⊑ C 1 .A special form of GCI are primitive concept definitions.Primitive concept definitions allow only for concept names on the left-hand side of the concept axiom.
Based on the semantics of concept descriptions and TBoxes a number of inference problems for DLs have been defined.The most relevant ones can be described as follows: Since the DL EL cannot express contradictions, every EL-concept is trivially satisfiable.In the remainder of this report, we will thus concentrate on subsumption as the basic reasoning task.

Prob-EL
We first introduce Prob-EL, a probabilistic variant of EL that allows reasoning with probabilistic concepts.We later present the variant Prob-EL 01 in which probabilistic expressions can only express that a concept is necessarily true, i. e. occurs with probability equal to 1, or that it is not impossible, i. e. occurs with probability greater than 0. These probabilistic logics were first introduced in [16].
The probabilistic DL Prob-EL extends EL with the constructor P * .That is, Prob-EL concepts are constructed as where A is a concept name, r a role name and n a rational number from the interval [0, 1].Intuitively, P * C expresses that the concept C is true with probability at least n.
The semantics of Prob-EL generalize the interpretation-based semantics of EL.
A probabilistic interpretation is of the form where ∆ I is the (non-empty) domain, W is a set of worlds, µ is a discrete probability distribution on W , and for each world w ∈ W, I w is a classical EL interpretation with domain ∆ I .
The probability that a given element of the domain d ∈ ∆ I belongs to the concept name A is given by The interpretation function I w and p I d are extended to complex concepts in the usual way for the classical constructors, while the extension to the new constructor P * is defined as The variant probabilistic DL Prob-EL 01 extends EL with the probability constructors P >0 and P =1 , with the straightforward semantics.Intuitively, this logic allows us to express the possibility or necessity of a concept, but does not allow a fine-grained description of the probability with which the concept is satisfied.This simple probabilistic variant retains the good complexity properties held by EL.Moreover, as it will be shown in the following section, reasonig in Prob-EL 01 can be performed through a variant of the completion algorithm for EL.

Completion-based subsumption algorithms
In this section we recapitulate the completion algorithms for testing subsumption in EL from [8,4] and for testing subsumption in prob-EL from [16]. 2 Completionbased methods compute the subsumption relations between all concept names from the TBox simultaneously, i.e., they classify the Tbox.

Completion-based subsumption algorithm for EL
Given a TBox T , we use BC T to denote the set of basic concept descriptions for T , i.e., the smallest set of concept descriptions which contains • all concept names used in T ; Based on this, a normal form for TBoxes can be defined as follows.
Definition 4 (Normal Form for EL-TBoxes) An EL-TBox T is in normal form if all concept inclusions have one of the following forms, where C 1 , C 2 ∈ BC T 2 Actually, we prune the completion algorithm given for instance checking in prob-EL down to a method for testing subsumption in this DL.
Any TBox T can be turned into a normalized TBox T ′ by introducing new concept and role names.EL-TBoxes can be transformed into normal form by applying the normalization rules displayed in Figure 1 exhaustively.These rules replace the GCI on the left-hand side of the rules with the set of GCIs on the right-hand side of the rule.
The signature of a concept description C (denoted sig(C)) is the set of concept names and role names that appear in C. The signature of a TBox T (denoted sig(T )) is the set of concept names and role names that appear in T .Clearly, the signature of T may be extended during normalization.However, this extension does not affect subsumption tests for EL-concept descriptions formulated w.r.t. the signature of T .To capture the relation between T and its normalized variant, we introduce the notion of a conservative extension.
• T 1 and T 2 are sig(T 1 )-inseparable w.r.t.concept inclusion in EL , iff for all EL-concept descriptions C, D with sig(C) ∪ sig(D) ⊆ sig(T 1 ), we have and The normalized TBox T ' is a conservative extension of T as already stated in [5], i.e., every model of T ′ is also a model of T , and every model of T can be extended to a model of T ′ by appropriately adding the interpretations of the additional concept names introduced by normalization.
Let T be the TBox to be classified and let R T denote the set of all role names appearing in T .The completion algorithm works on two kinds on completion sets: S(C) and S(C, r), which contain concept names from BC T .The intuition is that the completion rules make implicit subsumption relationships explicit in the following sense: By S T we denote the set of completion sets of T .In the algorithm, the completion sets are initialized as follows: Then the sets S(C) and S(C, r) are extended by applying the completion rules shown in Table 2 until no more rule applies.After the completion has terminated, the subsumption relation between two named concepts A and B from T can be tested by checking whether B ∈ S(A).Soundness and completeness of the ELcompletion algorithm has been shown in [9] as well as that it runs in polynomial time.This algorithm has recently been extended in [16] to a probabilistic variant of EL, both of which we introduce in the next section.

Completion-based Subsumption Algorithm for Prob-EL
In Prob-EL 01 , basic concepts also include the probabilistic constructors; that is, the set BC T of Prob-EL 01 basic concepts for T is the smallest set that contains (1) ⊤, (2) all concept names used in T , and (3) all concepts of the form P * A, where A is a concept name in T .
Definition 6 (Normal Form for Prob-EL 01 -TBoxes) A Prob-EL 01 -TBox T is in normal form if all its axioms are of one of the following forms The normalization rules in Figure 1 can also be used to transform a Prob-EL 01 -TBox into this extended notion of normal form.In the following, we denote as P T 0 and P T 1 the set of all concepts of the form P >0 A and P =1 A, respectively, occuring in a normalized TBox T .
The completion algorithm for Prob-EL 01 follows the same idea as the algorithm for EL, but uses several completion sets to deal with the information of what needs to be satisfied in the different worlds of a model.We define the set of worlds V := {0, ε, 1} ∪ P T 0 , where the probability distribution µ assigns a probability of 0 to the world 0, and the uniform probability 1/(|V | − 1) to all other worlds.For each concept name A, role name r and world v, we store the completion sets S 0 (A, v), S ε (A, v), S 0 (A, r, v), and S ε (A, r, v).
The algorithm initializes the sets as follows for every A ∈ N C , r ∈ R T : These sets are then extended by exhaustively applying the rules shown in Figure 3, where * ∈ {0, ε} and γ : V → {0, ε} is defined by γ(0) = 0, and γ(v) = ε for all v ∈ V \ {0}.The first four rules are simple adaptations of the completion rules for EL, while the last four rules deal with probabilistic concepts.This algorithm terminates in polynomial time.After termination it holds that, for every pair of concept names A, B, B ∈ S 0 (A, 0) if and only if A ⊑ T B [16].

Computing Least Common Subsumers using Completion
The least common subsumer was first mentioned in [11] and has since been investigated for several DLs.However, most lcs computation algorithms were devised for concept descriptions only or for unfoldable TBoxes (see e.g.[6]) and are not capable for handling general TBoxes.In case of EL the lcs has been investigated for more expressive TBoxes.It turned out in [3] that the lcs w.r.t.cyclic TBoxes the lcs does not need to exist w.r.t.descriptive semantics -the usual semantics for DLs.One approach to compute the lcs even in the presence of GCIs is to use different semantics for the underlying DL.This approach was pursued in [2,10], where greatest fixed-point semantics have been employed.A different approach was followed in [7], where the lcs was investigated for unfoldable TBoxes written in a "small" DL using concept names defined in a more expressive and general background TBox.
All approaches for proving the (non-)existence of the lcs or devising computation algorithms for the lcs are built on a characterization of subsumption for the respective DL and for the underlying TBox formalism.For instance, the lcs algorithm for EL-concept descriptions (and unfoldable TBoxes) [6] uses homomorphisms between so-called EL-description trees, which are basically syntax trees of EL-concept descriptions.The work on the lcs w.r.t.cyclic EL-TBoxes [2, 3] uses (synchronized) simulations between EL-description graphs to characterize subsumption.In this paper we use the completion algorithm from [4] as the underlying characterization of subsumption to obtain a role-depth bounded lcs in EL.
Formally the least common subsumer inference is defined as follows.Note, that the lcs is defined w.r.t. to a certain L. In cases the lcs is computed for concept descriptions alone, we can using an empty TBox.
Due to the associativity of the lcs operator, the lcs can be defined as a n-ary operation.We stick to its binary version for simplicity of the presentation.

Role-depth bounded lcs in EL
As mentioned, the lcs does not need to exists due to cycles in the TBox.To avoid such infinite nestings, we limit the role-depth of the concept description to be computed.Intuitively, the role-depth is the maximal number of nested quantifiers of a concept description.Please note that in case the exact lcs has a role-depth less than k the role-depth bounded lcs is the exact lcs.

Computing the role-depth bounded ELlcs
The computation algorithm for the role-depth bounded lcs w.r.t.general EL-TBoxes, constructs the concept description from the set of completion sets.More precisely, it combines and intersects the completion sets in the same fashion as in the cross-product computation in the lcs algorithm for EL-concept descriptions from [6].
The idea is that the normalization of the TBox introduces a new named concepts for each top-level conjunct and for each qualification of a top-level existential restriction that appears in the TBox.We call these new names normalization names in the following.During completion the normalization names and the concept names from the initial TBox are then collected in the completion sets of those concepts they subsume.Thus each normalization name in a completion set of, say C, provides a "pointer" to a complex concept description that is a subsumer of C. We compute the lcs by introducing new names for the input concepts of the lcs, do the normalization and completion and then collect the lcs concept from the completion sets of the input concepts.However, the returned lcs-concept description should only contain concept names that appear in the initial TBox, thus we need to "de-normalize" the concept descriptions obtained from the completion sets.

De-normalizing EL-concept descriptions
The signature of a concept description C (denoted sig(C)) is the set of concept names and role names that appear in C. The signature of a TBox T (denoted sig(T )) is the set of concept names and role names that appear in T .However, the extension of the signature does not affect subsumption tests for EL-concept descriptions formulated w.r.t. the signature of T .
Lemma 10 Let T be an EL-TBox and T ′ the TBox obtained from T by applying the EL normalization rules, C, D be EL-concept descriptions with sig(C) ∪ sig(D) ⊆ sig(T ′ ) and C ′ (D ′ ) be the concept description obtained by removing all names A ∈ sig(T ′ ) \ sig(T Proof.Since T ' is a conservative extension of T w.r.t.concept inclusion in EL, it is implied that T and T ′ are sig(T )-inseparable w.r.t.concept inclusion in EL.
Thus the claim follows directly.❏ Lemma 10 guarantees that subsumption relations w.r.t. the normalized TBox T ′ between arbitrary EL-concept descriptions C and D, that contain only names from sig(T ′ ), also hold w.r.t. the original TBox T for C and D with the names from sig(T ′ ) \ sig(T ) removed.

A computation algorithm for k-lcs
We assume that the role-depth of each input concept of the lcs has a role-depth less or equal to k.This assumption is motivated by the applications of the lcs and due to the simplicity of presentation than a technical necessity.The algorithm for computing the role-depth bounded lcs of two EL-concept descriptions is depicted in Algorithm 1.It consists of the procedure k-lcs, which calls the recursive procedure k-lcs-r.∃r.k-lcs-r (E, F, S, k − 1)

7: end if
The procedure k-lcs first adds concept definitions for the input concept descriptions to (a copy of) the TBox and transforms this TBox into a TBox in normalization form T '.Next, it calls the procedure apply-completion-rules, which applies the EL-completion rules exhaustively to the TBox T ', and stores the obtained set of completion sets in in S. Then it calls the function k-lcs-r with the concept names A and B for the input concepts, the set of completions sets S, and the role-depth k.The result is then de-normalized and returned (lines 4 to 6).More precisely, in case a complex concept description is returned from k-lcs-r, the procedure remove-normalization-names removes concept names that were added during the normalization of the TBox.
The function k-lcs-r gets a pair of concept names, a set of completion sets and a natural number as inputs.First, it tests whether one of the input concepts subsumes the other w.r.t.T ′ .In that case the name of the subsuming concept is returned.Otherwise the set of concept names that appear in the completion sets of both input concepts is intersected (line 4) and stored in common-names. 3In case k = 0, i.e. the role-depth bound is reached, the conjunction of the elements in common-names is returned.Otherwise, the elements in common-names are conjoined with a conjunction over all roles r ∈ R T , where for each r and each element of the cross-product over the r-successors of the current A and B a recursive call to k-lcs-r is made with the role-depth bound reduced by 1 (line 6).This conjunction is then returned to k-lcs.
For L = k-lcs(C, D, T , k) it holds by construction that rd(L) ≤ k. 4 We now show that the result of the function k-lcs is a common subsumer of the input concept descriptions.Proof.We show the claim by induction on k.By T ′ we denote the TBox obtained from T by applying the normalization rules. Case: • one of the input concept descriptions C, D subsumes the other one and thus appears in the subsumer set of the (name of) the other concept.Then its name is returned from k-lcs-r (lines 1 and 2) and then the initial concept description is returned by k-lcs (lines 4 and 5) and thus the claim holds.
• or, L is obtained from: L =remove-normalization-names(S(A) ∩ S(B)) and thus Case: k = n, with n ≥ 0. Assume that the result holds for some k ≥ 0, and we show it for k+1.Since k ≥ 0, L = remove-normalization-names( A i ∈S(A)∩S(B) A i ⊓ r∈R T (E,F ) ∈ S(A,r)×S(B,r) ∃r.k-lcs-r (E, F, S, k − 1)).It follows for the conjunction of concept names in L A i ∈S(A)∩S(B) A i that F ⊑ T remove-normalization-names(S(A) ∩ S(B)) holds for all F ∈ {C, D} by the same argument as in the base case.It remains to show that the conjunction of existential restrictions is a subsumer of C and D. Due to the assumption that result holds for k − 1, we know that for all r ∈ R T and for all E ∈ S(A, r) it holds that A ⊑ T ′ ∃r.E, and hence A ⊑ T ′ (E,F )∈S(A,r)×S(B,r) ∃r.klcs(E, F, T ′ , k).The argument for D ⊑ T L is analogous.❏ Next, we show that the result of the function k-lcs obtained for EL-concept descriptions C and D is the least (w.r.t.subsumption) concept description of roledepth up to k that subsumes the input concepts.Proof.We prove the claim by induction on rd(E).
The concept E is the concept name A or B or a conjunction of concept names.In the first case we obtain the result L = C from k-lcs and thus E ≡ T A ≡ T C ≡ T L.
In the second case we obtain L = D from k-lcs-r and thus E ≡ T B ≡ T D ≡ T L.
In the third case E is a conjunction of concept names: Thus, if the lcs exists and has a role-depth of less than k, the algorithm k-lcs yields the exact lcs.
The complexity of the overall method is polynomial for the binary lcs and exponential for the n-ary lcs due to the cross-product computation-just as in the case for the lcs for EL-concept descriptions, see [6].However, in contrast to the lcs algorithm for EL-concept descriptions, the algorithm k-lcs does not need to perform unfolding, i.e. copying of concepts, but proceeds by structure sharing if a concept from the TBox needs to be examined.Thus it is even advantageous for unfoldable EL-TBoxes such as Snomed [19].

Computing the Role-depth Bounded Prob-EL 01 -lcs
The computation of the role-depth bounded Prob-EL 01 -lcs follows the same steps as in Section 4.1.1.First, it adds concept definitions for the input concepts to the TBox and normalizes it.It then applies the completion rules exhaustively to produce the set of completion sets S. It then calls a variation of the function k-lcsr that can deal with probabilistic concepts.The new function k-lcs-r is identical to the one presented in Algorithm 1, except that in line 6 it now returns: P =1 (∃r.k-lcs-r(E,F, S, k − 1)) , where common-names := S 0 (A, 0)∩S 0 (B, 0) and S >0 0 (A, r) := v∈V \{0} S 0 (A, r, v).This new computation generalizes the idea for obtaining the lcs for EL, taking also into consideration the probabilistic concepts.Basically, if there is a concept name E ∈ S 0 (A, r, v) for some world v ∈ V \ {0}, then it holds that A ⊑ P >0 (∃r.E).Thus, the last two conjunctions express the existential restrictions that subsume both concepts A and B with probability greater than 0 and 1, respectively.
The result is then de-normalized by removing all concept names that were introduced during the normalization phase.The correctness of this procedure can be shown in a similar way as it was done for EL before.Proof.We show the result by induction on k.First, if k = 0, then rd(L) = 0.If one of the concepts C, D subsumes the other, then the algorithm returns it.Otherwise, we have that L = E∈S 0 (A,0)∩S 0 (B,0) E and thus, since C ≡ T A, C ⊑ T E∈S 0 (A,0) E ⊑ T L. An analogous argument shows that D ⊑ T L. Assume now that the result holds for some k ≥ 0, and we show it for k + 1.First, using the same argument for the base case, it is easy to see that C ⊑ T E∈S 0 (A,0)∩S 0 (B,0) E. Let now r ∈ R T .Then, for all E ∈ S 0 (A, r, 0) it holds that A ⊑ T ∃r.E, and hence A ⊑ T (E,F )∈S 0 (A,r,0)×S 0 (B,r,0) ∃r.k-lcs(E, F, T , k).Additionally, for all E ∈ S 0 (A, r, v) with v ∈ V \ {0} it holds that A ⊑ T P >0 (∃r.E), and for all E ∈ S 0 (A, r, 1) it holds A ⊑ T P =1 (∃r.E).Hence, A ⊑ T (E,F )∈S 0 (A,r,0)×S 0 (B,r,0) P >0 (∃r.k-lcs(E,F, T , k)) Let now rd(E) > 0 and let ∃r.E ′ be a (top level) conjunct of E, where E ′ is not of the form P * F .Then, as C ⊑ T E and D ⊑ T E, there must exist A ′ ∈ S 0 (A, r, 0) and B ′ ∈ S 0 (B, r, 0) such that A ′ ⊑ T E ′ and B ′ ⊑ T E ′ .By induction hypothesis, we then have that ∃r.k-lcs(A ′ , B ′ , T , k − 1) ⊑ T E ′ .Let now P >0 (∃r.E ′ ) be a conjunct in E. Then there must exist A ′ ∈ S >0 0 (A, r) and B ′ ∈ S >0 0 (B, r) such that A ′ ⊑ T E ′ and B ′ ⊑ T E ′ .But then, we have that P >0 (∃r.k-lcs(A ′ , B ′ , T , k − 1)) ⊑ T P >0 (∃r.E ′ ).Analogously, we can prove that for every conjunct P =1 (∃r.E ′ ) in E, there exist A ′ ∈ S 0 (A, r, 1) and B ′ ∈ S 0 (B, r, 1) such that A ′ ⊑ T E ′ and B ′ ⊑ T E ′ , and hence P =1 (∃r.k-lcs(A ′ , B ′ , T , k − 1)) ⊑ T P =1 (∃r.E ′ ).All this together implies that L ⊑ T E. ❏ The following theorem is then a trivial consequence of Lemmas 14 and 15.Similar to the completion algorithm, the extension of the algorithm to compute the role-bounded lcs in EL to the "light-weight" probabilistic DL Prob-EL 01 is fairly straight-forward.

Summary and Outlook
In this report we presented a practical approach for computing least common subsumers of limited role-depth in the DL EL and the moderately probabilistic DL Prob-EL 01 .It turns out that the complexity of computing the role-depth limited lcs is the same for concept descriptions.
Althought simply limiting the role-depth of the result concept description is not elegant from the theoretical perspective, we argue that it will turn out to be sufficient for many practical applications.
The computation algorithm k-lcs can be easily be implemented on top of the completion algorithm.In case a k-lcs turns out to be too general and a bigger role depth of the k-lcs is desired, the completion of the TBox does not have to be redone for a second computation.The completion sets can simply be "traversed" further.It remains an interesting question whether the completion sets obtained from the classification of the initial TBox can simply be extended, when C ≡ A and D ≡ B are added.
In our future work we will try to extend the computation algorithm for the roledepth limited lcs to full Prob-EL.

Definition 7 (
Least common subsumer) Let T be a TBox and C, D concept descriptions in the DL L, then L is the least common subsumer (lcs) of C, D w.r.t.T (written lcs T (C, D)) iff 1. C ⊑ T L and D ⊑ T L, and 2. for all L-concept descriptions E it holds that, C ⊑ T E and C ⊑ T E implies L ⊑ T E.
Consider the TBox T = {A ⊑ ∃r.A ⊓ C, B ⊑ ∃r.B ⊓ C}.The lcs of A and B is then ∃r.(C ⊓∃r.(C ⊓∃r.(C ⊓• • • and cannot be expressed by a finite concept description.

Definition 8 (
role-depth) Let C, D be EL-concept descriptions.The roledepth of a concept description C (denoted rd(C)) is: • 0 for concept names and ⊤ • max(rd(C), rd(D)) for a conjunction C ⊓ D, and • 1 + rd(C) for existential restrictions of the form ∃r.C.Now we can define the lcs with limited role-depth for EL.Definition 9 (role-depth bounded EL-lcs) Let T be an EL-TBox and C, D EL-concept descriptions and k ∈ IN.Then the EL-concept description L is the role-depth bounded EL-least common subsumer of C, D w.r.t.T and role-depth k (written k-lcs T (C, D)) iff 1. rd(L) ≤ k, 2. C ⊑ T L and D ⊑ T L, and 3. for all EL-concept descriptions E with rd(E) ≤ k it holds that, C ⊑ T E and C ⊑ T E implies L ⊑ T E.

Algorithm 1
Computation of a role-depth bounded EL-lcs.Procedure k-lcs (C, D, T , k) Input: C, D: EL-concept descriptions; T : EL-TBox; k: natural number Output: k-lcs(C, D): role-depth bounded EL-lcs of C and D w.r.t T and k. 1: T ′ := normalize(T ∪ {A ≡ C, B ≡ D}) 2: S T ′ := apply-completion-rules(T ′ ) 3: L := k-lcs-r (A, B, S T ′ , k) 4: if L = A then return C 5: else if L = B then return D 6: else return remove-normalization-names(L) 7: end if Procedure k-lcs-r (A, B, S, k) Input: A, B: concept names; S: set of completion sets; k: natural number Output: k-lcs(A, B): role-depth bounded EL-lcs of A and B w.r.t T and k. 1: if B ∈ S(A) then return B 2: else if A ∈ S(B) then return A 3: end if 4: common-names := S(A) ∩ S(B) 5: if k = 0 then return P ∈common−names P 6: else return P ∈common−names P ⊓ r∈R T (E,F ) ∈ S(A,r)×S(B,r)

Lemma 11
Let C and D be EL-concept descriptions, T an EL-TBox, k ∈ IN and L = k-lcs(C, D, T , k).Then C ⊑ T L and D ⊑ T L.
and by application of Lemma 10 we obtain C ⊑ T L. Argument for D ⊑ T L is analogous.

Lemma 12
Let C and D be EL-concept descriptions, T an EL-TBox, k ∈ IN and L = k-lcs(C, D, T , k) and E an EL-concept description with rd(E) ≤ k.If C ⊑ T E and D ⊑ T E, then L ⊑ T E.
and D ⊑ T E holds, we have C ⊑ T A i and D ⊑ T A i for all A i with 1 ≤ i ≤ m.From this follows A i ∈ S(A) ∩ S(B) for all i, which implies thatL ⊑ T C ′ ∈S(A)∩S(B) C ′ ⊑ T E.Case: rd(E) = n, with n ≥ 0. E can contain two kinds of conjuncts: (possibly zero) concept names and at least one existential restriction.The concept names in E must appear in L as well by an argument analogous to the base case.Let ∃r.E ′ be a (top-level) conjunct of E. Since C ⊑ T ′ E and D ⊑ T ′ E, there must exist A ′ ∈ S(A, r) and B ′ ∈ S(B, r) such that A ′ ⊑ T E ′ and B ′ ⊑ T E ′ .By induction hypothesis, we then have that ∃r.k-lcs(A ′ , B ′ , T , k − 1) ⊑ T E ′ .By L ⊑ T ∃r.k-lcs(A ′ , B ′ , T , k − 1), it is implied that L ⊑ T E. ❏We obtain together with Lemma 11 and Lemma 12 that all conditions of Definition 9 are fulfilled for k-lcs(C, D, T , k).Theorem 13 Let C and D be EL-concept descriptions, T an EL-TBox, k ∈ IN, then k-lcs(C, D, T , k) ≡ T k-lcs T (C, D).

Lemma 14
Let C and D be Prob-EL 01 -concept descriptions, T a Prob-EL 01 -TBox, k ∈ IN and L = k-lcs(C, D, T , k).Then C ⊑ T L and D ⊑ T L.
F )∈S 0 (A,r,0)×S 0 (B,r,0) P =1 (∃r.k-lcs(E,F, T , k)).And thus, C ⊑ T L. An analogous argument yields the result for D ⊑ T L. ❏ Lemma 15 Let C and D be Prob-EL 01 -concept descriptions, T an Prob-EL 01 -TBox, k ∈ IN and L = k-lcs(C, D, T , k) and E an Prob-EL 01 -concept description with rd(E) ≤ k.If C ⊑ T E and D ⊑ T E, then L ⊑ T E.Proof.By induction on rd(E).First, if rd(E) = 0, thenE = A 1 ⊓ . . .⊓ A m for A i ∈ N C .Since C ⊑ T E and D ⊑ T E, then C ⊑ A i and D ⊑ A i for all i, 1 ≤ i ≤ m.But then A i ∈ S 0 (A, 0) ∩ S 0 (B, 0) for all i.This implies that L ⊑ T C ′ ∈S 0 (A,0)∩S 0 (B,0) C ′ ⊑ T E.

Theorem 16
Let C and D be Prob-EL 01 -concept descriptions, T a Prob-EL 01 -TBox, and k ∈ IN; then k-lcs(C, D, T , k) ≡ T k-lcs(C, D).