Forgetting in General EL Terminologies

Uniform Interpolation in General EL Terminologies

Nadeschda Nikitina

Karlsruhe Institute of Technology, Karlsruhe, Germany [email protected]

Abstract. Recently, different forgetting approaches for knowledge bases expressed in different logics were proposed. ForELterminologies containing each atomic concept at most once on the left-hand side, an approach has been proposed based on sufficient, but not necessary acyclicity conditions. In this paper, we propose an approach for computing a uniform interpolant for generalELterminologies.

We first show that a uniform interpolant of anyELterminology w.r.t. any sig- nature always exists inELenriched with least and greatest fixpoint constructors and show how it can be computed by reducing the problem to the computation of Most General Subconcepts and Most Specific Superconcepts for atomic concepts.

Then, we give the exact conditions for the existence of a uniform interpolant in ELand show how it can be obtained using our algorithms.

1 Introduction

The importance of non-standard reasoning services supporting knowledge engineers in modelling a particular domain or in understanding existing models by visualizing im- plicit dependencies between concepts and roles was pointed out by the research commu- nity [3], [5]. An example of such reasoning services supporting knowledge engineers in different activities is the uniform interpolation. In particular for the understanding and the development of complex knowledge bases, e.g., those consisting ofgeneral concept inclusions(GCIs), the appropriate tool support would be beneficial. However, the exist- ing approach [7] to uniform interpolation inELis restricted to terminologies containing each atomic concept at most once on the left-hand side of concept inclusions and ad- ditionally satisfying sufficient, but not necessary acyclicity conditions. Lutz et al.[10]

propose an approach to uniform interpolation in expressive description logics such as ALCw.r.t. general terminologies by encodingALCTBoxes as concepts, which, how- ever does not solve the problem of uniform interpolation inEL.

Clearly, the existence of the results for such reasoning problems is closely related to the notion of fixpoint semantics. For instance, Baader [2] shows that the structurally similar problems of computing Least Common Subsumer and Most Specific Concept can always be solved in cyclic classical TBoxes w.r.t. to greatest fixpoint semantics.

Similar results were obtained for generalEL TBoxes with descriptive semantics[9] , however extended with the greatest fixpoint constructor (EL_ν). In this paper, we extend the above results by showing that uniform interpolants preserving allELconsequences of generalELterminologies w.r.t. an arbitrary signature can always be expressed in an extension ofELwith least fixpoint and greatest fixpoint constructorsµ, νas well as the

disjunction used only on the left-hand side of concept inclusions. We extend the previ- ous approach [7] and propose the algorithms for computing such uniform interpolants for generalELterminologies based on the notion ofmost general subconceptsandmost specific superconcepts.

In the usual application scenarios it is rather useful to obtain uniform interpolants expressed in the DL of the original terminology instead of introducing additional lan- guage constructs. Therefore, in addition to the above algorithms, we derive the existence criteria for uniform interpolants inEL(i.e., expressed without the above extension) and show how such a uniform interpolant can be obtained using our algorithms. Full proofs are available in the extended version of this paper [11].

2 Preliminaries

LetNCandNRbe countably infinite and mutually disjoint sets of concept symbols and role symbols. AnELconceptCis defined as

C::=A|>|CuC|∃r.C

whereAandrrange overN_C andN_R, respectively. In the following, we use symbols A,Bto denote atomic concepts andC,Dto denote arbitrary concepts. Aterminology orTBoxconsists ofconcept inclusionaxiomsC vDandconcept equivalenceaxioms C ≡Dused as a shorthand forC vDandDvC. While knowledge bases in general can also include a specification of individuals with the corresponding concept and role assertions (ABox), in this paper we abstract from ABoxes and concentrate on TBoxes.

The signature of anELconceptCor an axiomα, denoted by sig(C) or sig(α), respec- tively, is the set of concepts and role symbols occurring in it. The signature of a TBoxT, in symbols sig(T), is analogouslyNC∪NR. In what follows, we denote the setNC∪ {>}

asN_C⁺.

Before introducing the fixpoint operators, we recall the semantics of the above in- troduced DL constructs, which is defined by the means of interpretations. An interpre- tationIis given by the domain ∆^I and a function·^I assigning each conceptA ∈ NC

a subsetA^Iof∆^Iand each roler ∈ NR a subsetr^Iof∆^I×∆^I. The interpretation of

>is fixed to ∆^I. The interpretation of an arbitraryELconcept is defined inductively, i.e., (CuD)^I =C^I∩D^Iand (∃r.C)^I ={x| (x,y)∈r^I,y∈C^I}. An interpretationI satisfies an axiomC v DifC^I ⊆ D^I.Iis a model of a TBox, if it satisfies all of its axioms. We say that a TBoxT entails an axiomα, ifαis satisfied by all models ofT. In combination with fixpoint constructors, we will additionally useconcept disjunction CtD, the semantics of which is defined by (CtD)^I=C^I∪D^I.

We now introduce the logicsEL_µ(t),ν, a fragment of monadic second order log- ics that we use to compute uniform interpolants of generalELTBoxes.EL_µ(t),νis an extension ofELby the two fixpoint constructors,νX.Cν[9] andµX.Cµ[4].Xis an ele- ment of the countably infinite set of concept variablesNV andCν,Cµare constructed as follows:

Cν::=X|A|>|νX.Cν|CνuCν|∃r.Cν

C_µ::=X|A|>|µX.C_µ|C_µtC_µ|C_µuC_µ|∃r.C_µ

whereAranges over atomic concepts andXranges overNV. AllEL_νconcepts and all EL_µ(t)concepts are closedCνandCµexpression, i.e., all concept variables are bound by the corresponding fixpoint constructor. Note that we defineEL_ν concepts and all EL_µ(t)concepts in such a way, that no concept can contain both fixpoint constructors, i.e., we do not combine the two constructors within concepts. The semantics of the fixpoint constructors is defined using a mappingϑof concept variables to subsets of

∆^I. For an EL_µ(t),ν conceptC andW ⊆ ∆^I, we denote a replacement of X byW as C^I,ϑ[X→W]. The semantics ofEL_µ(t),νconcepts is defined by

(νX.C)^I,ϑ=[

{W ⊆∆^I|W⊆C^I,ϑ[X→W]} (µX.C)^I,ϑ=\

{W⊆∆^I|C^I,ϑ[X→W] ⊆W}.

In order to allow for more succinct concept expressions, we use an extended ver- sion of the fixpoint constructs allowing for mutual recursion [12], [9]. The extended constructors have the form ν_iX1...X_n.C_ν,1, ...,Cν,n andµ_iX1...X_n.C_µ,1, ...,Cµ,n with 1 ≤ i ≤ n. The semantics is defined as follows: (ν_iX1...X_n.C₁, ...,Cn)^I,ϑ = S{Wi} and (µ_iX₁...X_n.C₁, ...,C_n)^I,ϑ = T{W_i}such that there areW₁, ...,W_i−1,W_i+1, ...,W_nwith re- spectivelyW_j⊆C^I,ϑ[X_j ^{1→W1,...,Xn→Wn]}andC^I,ϑ[X_j ^{1→W1,...,Xn→Wn]} ⊆W_jfor 1≤ j≤n.

3 TBox Inseparability and Uniform Interpolation

Intuitively, two TBoxesT₁andT₂ are inseparable w.r.t. a signatureΣif they have the sameΣconsequences, i.e., consequences whose signature is a subset ofΣ. Depending on the particular application requirements, the expressivity of those Σ consequences can vary from subsumption queries and instance queries to conjunctive queries. In this paper, we investigate forgetting based on concept inseparability of generalELtermi- nologies defined analogously to previous work on inseparability, e.g., [8] or [7], as follows:

Definition 1. LetT₁andT₂be two generalELTBoxes andΣa signature.T₁andT₂ are concept-inseparable w.r.t.Σ, in symbolsT₁≡^c_ΣT₂, if for allELconcepts C,D with sig(C)∪sig(D)⊆ΣholdsT₁ |=CvD, iffT₂|=CvD.

Given a signatureΣ and a TBoxT, the aim of uniform interpolation or forgetting is to determine a TBoxT⁰ with sig(T⁰) ⊆ Σ such that T ≡^c_Σ T⁰. T⁰ is also called a Uniform Interpolant (UI)ofT w.r.t.Σ. We callΣ=sig(T)\Σtheforgottensignature.

In practise, the uniform interpolants are required to be finite. Therefore, in this paper, we investigate the existence of such uniform interpolants inEL, i.e., uniform interpolants expressible by a finite set of finite axioms using only the language constructs ofEL. As demonstrated by the following example, in the presence of cyclic concept inclusions, a finite UI inELmight not exist for a particularT and a particularΣ.

Example 1. Forgetting the conceptAin the TBoxT ={A⁰vA,AvA⁰⁰,Av ∃r.A,∃s.Av A}results in an infinite chain of consequencesA⁰v ∃r.∃r.∃r....A⁰⁰and∃s.∃s.∃s....A⁰v A⁰⁰containing nested existential quantifiers of unbounded maximal depth.

Such infinite chain of consequences can be easily expressed in a finite way using the greatest fixpoint constructorνand the least fixpoint constructorµ, thereby resulting in the inclusion axiomsA⁰vνX.(A⁰⁰u ∃r.X) andµX.(A⁰t ∃s.X)vA⁰⁰. In the following, we show that for anyELTBoxT and any signatureΣ, a corresponding UI ofT w.r.t.

Σ inEL_µ(t),ν can always be computed. For this purpose, we reduce the problem of computing UI to the problem of computingmost general subconceptsMGS(Σ,T,A) and most specific superconceptsMSS(Σ,T,A) for each conceptA∈sig(T).

Definition 2. LetT be anELTBox andΣ a signature. Further, let A ∈ NC. A set of ELconcepts Cifor1≤i≤n isMSS(T, Σ,A), if:

– sig(C_i)⊆Σfor all i,

– T |=FCivA andT 6|=AvFCi,

– for allELconcepts D with sig(D)⊆Σholds:

T |=DvA, iffT |=Dv Ci.

A set ofELconcepts C_ifor1 ≤i ≤n isMGS(T, Σ,A)if the following conditions are fulfilled:

– sig(Ci)⊆Σfor all i, – T |=Av

CiandT 6|= CivA

– for allELconcepts D with sig(D)⊆Σholds:

T |=AvD, iffT |= CivD.

We denoteMSS(A) andMGS(A) expressed in logicLbyMSS^L(A) andMGS^L(A). IfMGS(T, Σ,A) consists of several incomparable disjunctsCi, it cannot be expressed by anELconcept.

However, in the following, it will come into notice that this is not further problematic for the computation of UI, since the disjunction appears only on the left-hand side and can therefore be expressed by the means of several inclusion axioms. If the TBoxT and the signatureΣdo not change, we omit them and simply writeMSS(A) andMGS(A).

For the remainder of this paper, we fixT to be a generalELTBox andΣa signature.

4 Normalization

In order to simplify the computation ofMGSandMSS, we apply the following normal- ization thereby restricting the syntactic form ofT. Analogously to the normalization employed in other approaches ([1], [6], [7]), we decompose complex axioms into syn- tactically simple ones. The decomposition is realized recursively by replacing expres- sionsB1u...uBn and∃r.Bwith fresh concept symbols until each axiom inT is one of{Av B,A ≡ B1u...uBn,A ≡ ∃r.B}, whereA,B,Bi ∈ NC∪ {>}andr ∈ NR. For this purpose, we introduce a minimal required set of fresh concept symbolsA⁰ ∈ ND

and the corresponding definition axioms (A⁰ ≡ C) for each of them. In what follows, we assume that knowledge bases are normalized and refer toNC ∪ND as NC. Since concept symbols inN_Dare fresh, they do not appear inΣ and are therefore elements of the forgotten signatureΣ. Further, we assume that equivalent concept symbols have

Algorithm 1computingMGS_core(A) for anELTBoxT and a signatureΣ 1: M←SMGScond(D,A),D∈NCsuch thatT |=DvA,A,D

2: for allA≡

1≤i≤nBi∈ T do 3: M←M∪ {

C∈REDUCEMSS({C_i|1≤i≤n})C|(C1, ...,Cn)∈MGScond(B1,A)×...×MGScond(Bn,A)}

4: end for

5: for allA≡ ∃r.B∈ Twithr∈Σdo 6: M←M∪ {∃r.C|C∈MGScond(B,A)}

7: end for

8: return REDUCEMGS(M)

Algorithm 2computingMSS_core(A) for anELTBoxT and a signatureΣ 1: M←SMSScond(D,A),D∈N_C⁺such thatT |=AvD,A,D

2: for allA≡

1≤i≤nBi∈ T do 3: M←M∪ {C|C∈MSScond(B_i,A)}

4: end for

5: for allA≡ ∃r.B∈ Twithr∈Σdo 6: M←M∪ {∃r.

C∈MSScond(B,A)C}

7: end for

8: return REDUCEMSS(M)

been replaced by a single representative of the corresponding equivalence class.¹ The following lemma summarizes the properties of normalized TBoxes.

Lemma 1. AnyT can be extended into a normalized TBoxT⁰and each model ofT can be extended into a model ofT⁰.

Proof Sketch. All introduced concepts in ND are defined in terms of concepts with sig(C)⊆sig(T), therefore each model ofT can be extended into a model ofT⁰. ut

5 Computing MGS and MSS for Acyclic TBoxes

Given an acyclic, normalizedELTBoxT and a signatureΣ, Algorithms 1 and 2 compute for eachA∈ NCthe elements ofMGS(A) andMSS(A), respectively. The algo- rithms are derived from a Gentzen-style proof system and proceed along the definitions for Ain T as well as the inferred inclusions between atomic concepts involving A.

The computation is indirectly recursive and consists of the procedureMGS_core(MSS_core) containing the core computation and procedureMGS_cond (MSS_cond) realizing the termi- nation of the computation: if the first parameter ofMGS_cond(MSS_cond) – a concept B– is inΣ, it returnsBitself, which is the basecase of the computation; otherwise, it calls in turnMGS_core(B) (MSScore(B)). Thereby,MGS_cond(MSScond) ensures thatMGSandMSS only containΣ-concepts. To avoid confusion, we denoteMGS(A) andMSS(A) obtained using this simple definition ofMGS_cond(MSScond) byMGS_acyc(A) andMSS_acyc(A).

1The elimination of equivalent symbols does not affect the correctness or completeness of the uniform interpolation, since the removed symbols can easily be included into the resulting TBox.

Definition 3. LetT an acyclicEL TBox and A ∈ NC.MGS_acyc(A) = MGS_core(A)and MSS_acyc(A)=MSS_core(A).

While this separation of concerns betweenMGS_core(A) (MSS_core(A)) andMGS_cond(B,A) (MSS_cond(B,A)) appears not necessary in the simple case of acyclic TBoxes, in the next section we extend the computation to the general case of GCIs by adding further condi- tions toMGS_cond(B,A) (MSS_cond(B,A)) without changing the core computation presented in Algorithms 1 and 2. In particular, the role of second parameter ofMGS_condwill be- come clear.

The functionsREDUCE_MGSandREDUCE_MSSeliminate redundancy within the computed results, which is not just an optimization, but will also play an important role when deciding the existence of a uniform interpolant inEL. The two functions are defined as follows:

Definition 4. Let M={Ci|0≤i≤n}be a set ofELconcepts andT_e={}.

REDUCE_MSS(M)={Ci∈M|there is no Cj∈M such thatT_e|=CjvCi} REDUCE_MGS(M)={C_i∈M|the is no C_j∈M such thatT_e|=C_ivC_j}

Both,REDUCEandREDUCE_C, can be easily realized using standard reasoning procedures inEL_µ(t),ν, which is known to be decidable inE xpT ime[4]. It is easy to see that, in case of an acyclic TBoxT, both algorithms terminate, while, in case of cyclic terminologies, the algorithms do not need to terminate. In the next section, we extend the computation to be applicable to general TBoxes and ensure the termination also for this case.

6 MGS and MSS for General TBoxes

As already mentioned, the computation based on the simple version ofMGS_cond(B,A) andMSS_cond(B,A) does not always terminate in case of general TBoxes such as the TBox in Example 1. In order to exactly specify the case, in which Algorithms 1 and 2 do not terminate, we use the following graph structures representing the possible flow of computation ofMGS_core andMSS_corefor a particular TBoxT (independent from a particular signature):

Definition 5. TheMSS- andMGS-graphsA_MSS(T)andA_MGS(T)are defined as

– A_MSS(T) =(Γ_MSS,Q,E_MSS)with the set of edge labelsΓ_MSS =N_R∪ {v}, the set of states Q=NCand the set of edges EMSS ={(A,r,B)|A≡ ∃r.B∈ T } ∪ {(A,v,B)|T |= AvB,A,B}, where A,B∈Q and r∈ΓMSS.

– A_MGS(T)=(ΓMGS,Q,EMGS)with the set of edge labelsΓMGS=NR∪ {w,u}, the set of states Q=NCand the set of edges EMGS ={(A,r,B)|A≡ ∃r.B∈ T } ∪ {(A,w,B)|T |= AwB,A,B}∪{(A,u,B)|A≡BuC∈ T for any conjunction C of elements from Q}, where A,B∈Q and r∈ΓMGS.

The two graphs can be constructed in linear time after the classification of the normalized TBox is finished. For X ∈ {MGS,MSS}, we denote the set of the paths in A_X(T, Σ) fromAtoBasL_X(A,B) and the set of the intersection-free paths from node

Fig. 1.MGS-graph (left) andMSS-graph (right) ofT.

Ato itself asL¹_X(A,A) , i.e., such paths not contain any node except for Amore than once. As illustrated by the example below, cyclic paths inA_MSS(T) andA_MGS(T) do not necessarily coincide.

Example 2. The correspondingMGS- andMSS-graphs of the TBoxT ={A1≡ ∃s.A2,A3≡

∃r.A₂,A3vA4,A5≡A3uA4,A5vA2,A5vA6}are shown in Fig. 1.

Given A_MSS(T) and A_MGS(T), we can easily determine for a particular signature Σ, whether the computation of theUIby the means of Algorithms 1 and 2 with the simple version ofMGS_cond(B,A) andMSS_cond(B,A) terminates: if neitherA_MSS(T) norA_MGS(T) contain cyclic paths formed only by concepts from Σ. Note that other cycles do not lead to a non-termination, since MGS_cond(B,A) = {B} for any B ∈ Σ andA ∈ NC, i.e., the computation terminates. We denote such cyclic intersection-free paths fromA containing only concepts fromΣbyL^1,Σ_X (A,A) and the sets of concepts involved in such cycles byΣC,MGS={A|A∈Σ,L^1,Σ_MGS(A,A),∅}andΣC,MSS ={A|A∈Σ,L^1,Σ_MSS(A,A),∅}.

In order to be able to computeMSSandMGSalso in case ofΣ_C,MSS∪Σ_C,MGS ,∅, we extendMGS_cond(A,B) andMSS_cond(A,B) by introducing a new condition for concepts A ∈ ΣC,MSS∪ΣC,MGS. Here, we require the second parameterBto determine when the quantification of the fixpoint expressions is necessary. IfMGS_cond orMSS_condis called from outside of the corresponding cycles (ΣC,MGSforMGS_condandΣC,MSSforMSS_cond), we return the complete fixpoint expression in its quantified form. Otherwise, we prefer to return the simplest possible value necessary, which can then be used as a part of a more complex, quantified concept expression. This second parameter is used by the caller – MGS_coreorMSS_core– to pass its own parameter to the calledMGS_condorMSS_condand let it then decide, whether to return a quantified fixpoint expression or a non-quantified one.

Definition 6. Let n,m be the cardinality ofΣC,MSSandΣC,MGS, respectively. Further, let Ai∈ΣC,MSSwith0≤i≤n and Aj ∈ΣC,MGSwith0≤ j≤m. Let{X(Ai)|Ai∈ΣC,MSS}and {Y(Aj)|Aj ∈ΣC,MGS}be two disjoint sets of concept variables. Then, we define for each Ai∈Σ_C,MSSand each Aj∈Σ_C,MGS:

N(A_i)=ν_iX(A₁), ...,X(A_n).u_{C∈MSScore(A1)}C, ...,u_{C∈MSScore(An)}C M(A_j)=µ_jY(A₁), ...,Y(A_m).t_{C∈MGScore(A1)}C, ...,t_{C∈MGScore(Am)}C.

MSS_cond(A,B)andMGS_cond(A,B)for any A,B∈N_Cis then given by:

MSS_cond(A,B)=































{A} if A∈Σ {X(A)} if A∈Σ_C,MSS,

B∈ΣC,MSS

{N(A)} if A∈ΣC,MSS, B<Σ_C,MSS MSS_core(A) otherwise

MGS_cond(A,B)=































{A} if A∈Σ {Y(A)} if A∈Σ_C,MGS,

B∈ΣC,MGS

{M(A)} if A∈ΣC,MGS, B<Σ_C,MGS MGS_core(A) otherwise , andMGS(A)=MGS_cond(A,>)andMSS(A)=MSS_cond(A,>).

Note that, in case of an acyclic TBox,MGS(A) coincides withMGS_acyc(A) described in Section 5, and the same holds forMSS(A).

Theorem 1 (Termination).Let A ∈ NC. The computation ofMSS(A)andMGS(A)al- ways terminates in at most exponential time.

Proof Sketch. We first show by induction in caseΣC,MSS = ∅that the computation of MSS(A) for eachA∈NCterminates. Then, we consider the caseΣC,MSS,∅.MSS_conden- capsulates all concepts inΣC,MSSinto a single computational unit with direct or indirect incoming dependencies from concepts referencing concepts inΣC,MSSand direct or in- direct outgoing dependencies to concepts referenced from any concept inΣC,MSS. These two sets of referencing and referenced concepts are disjoint by definition of cyclicity.

In the computation ofN(A), either concept variables or results of acyclic computations ofMSS(B) forBnot referencingΣC,MSSare used, therefore each computation terminates.

OnceN(A) is computed, references toA∈Σ_C,MSSdo not require further computation and the remaining computation terminates as shown in caseΣ_C,MSS ,∅. Since the structure ofMGS_condandMSS_condis analogous andMGS_corealso only iterates through the finite in- put directly, the argumentation for the termination ofMGSis identical. The complexity is due to the conjunction constructs introduced in line 3 of Algorithm 1. ut Theorem 2 (Correctness MSS and MGS). Let A ∈ NC. The computed MSS(A) and MGS(A)satisfy the conditions stated in Definition 2.

The proof of Theorem 2 is based on a Gentzen-style proof system for normalized TBoxes.

7 Computing Uniform Interpolants

Given MGS(A) and MSS(A) for each A ∈ NC, we can compute the UI in EL_µ(t),ν as follows:

Definition 7. UI(T, Σ)=UI_Σ,MSS(T, Σ)∪UI_Σ,MGS(T, Σ)∪UI_Σ(T, Σ)with – UI_Σ,MSS(T, Σ)={AvD|A∈NC∩Σ,D∈MSS(A)},

– UI_Σ,MGS(T, Σ)={CvA|A∈NC∩Σ,C∈MGS(A)},

– UI_Σ(T, Σ)={CvD|there is A∈NC∩Σ, such that C∈MGS(A)and D∈MSS(A)}.

Now, we have to prove thatUI(T, Σ)≡^c_Σ T, i.e., the TBoxUI(T, Σ) is in fact a uniform interpolant ofT w.r.t.Σ.

Theorem 3 (UI).UI(T, Σ)always exists and it holds thatUI(T, Σ)≡^c_Σ T.

The proof of Theorem 3 is also based on a Gentzen-style proof system for normalized TBoxes.

Deciding the Existence of UI inEL

Clearly, ifT does not contain pureΣcycles, theUI(T, Σ) only containsELconstructs and, therefore, aUIinELexists. This would be a sufficient, but not necessary criterion for the existence of a UI. From Definition 7, we can deduce a very general form of criterion requiring the deductive closure of any UI² to contain an (arbitrary) finiteEL justification for the set of all non-ELaxioms in theUI(T, Σ). Interestingly, given theEL TBoxUI^EL(T, Σ) obtained by extracting theELpart of each fixpoint concept within the non-mutual representation ofUI(T, Σ), this criterion can be easily checked, since it is equivalent to a very simple criterion, which is an immediate consequence of the following theorem:

Theorem 4 (Existence).LetUI^EL(T, Σ)be the ELTBox obtained by extracting the ELpart of each fixpoint concept within the non-mutual representation ofUI(T, Σ)and letT⁰be anELTBox with sig(T⁰)⊆Σsuch thatT⁰≡^c_ΣT. ThenUI^EL(T, Σ)≡ T⁰. The theorem claims that, if a finiteELjustification for the set of all non-ELaxioms in UI(T, Σ) exists, it is already a contained in the non-mutual representation ofUI(T, Σ).

Subsequently, a UI ofT w.r.t.ΣinELexists, iffUI^EL(T, Σ)|=UI(T, Σ). The proof of this theorem is based on the ideas stated in Lemmas 2 and 3, which show that there is a close relation between the existence of a UI inELand redundancy inUI(T, Σ).

Lemma 2. LetT⁰be anELTBox with sig(T⁰) ⊆Σsuch thatT⁰ ≡^c_Σ T. Further, let A∈Σ_C,MGSwith C1 ∈MGS(A)and C2 ∈MSS(A). Then there is anELconcept C⁰such that

– T 6|=C⁰≡C1andT 6|=C⁰≡C2

– {} |=C1vC⁰ – UI(T, Σ)|=C⁰vC₂.

Let A ∈ ΣC,MSS with C1 ∈ MGS(A)and C2 ∈ MSS(A). Then there is anELconcept C⁰ such that

– T 6|=C⁰≡C1andT 6|=C⁰≡C2

– {} |=C⁰vC2

– UI(T, Σ)|=C1vC⁰.

2The deductive closure is the same for anyUIby definition.

Proof Sketch. ConsiderC1 = µX.(At ∃r.X), which is the simplest possible non-EL concept inMGS.C1 is semantically equivalent to an infinite disjunction of more and more specificELconcepts. The language constructs ofELdo not allow us to specify a concept, which captures exactly the subset of the interpretation domainC^I₁ in all models. LetC₂ be an arbitrary concept withUI(T, Σ) |= C₁ v C₂. IfC₁ v C₂ is a consequence ofT⁰, then there must be anELconceptC⁰₁, which subsumesC^I₁ in all models. SinceT⁰is a finiteELTBox, it must hold thatT⁰|=C1 vC⁰₁, i.e., the latter inclusion axiom must be derived from the finiteEL TBox itself (e.g.,C₁⁰ = Bwith {∃r.BvB,AvB} ∈ T⁰). MoreoverC⁰₁vC2must have a justification inT⁰consisting of finitely manyELaxioms. The same argumentation applies toC2 as a concept with

greatest fixpoint constructs. ut

The above proof is the first step towards a connection between the redundancy in UI(T, Σ) and the existence of a UI in EL. Since {C1 v C⁰,C⁰ v C2} |= C1 v C2

and any minimal justification of {C1 v C⁰,C⁰ v C2} in any T⁰ does not contain C1 vC2, it also holds thatUI(T, Σ)∪UI^EL(T, Σ)\ {C1 v C2} |=C1 v C2. There- fore, if T⁰ exists, each non-EL axiom is redundant, i.e., it could be removed from UI(T, Σ)∪UI^EL(T, Σ) without losing any consequences. To avoid confusion, we de- note the non-mutual representation ofMSS(A) andMGS(A) with the correspondingEL parts explicitly appearing outside of all fixpoint quantifiers byMSS(A)∪EL(MSS(A)) andMGS(A)∪EL(MGS(A)). The functionsREDUCE_MSSandREDUCE_MGShave to be applied also when computingMSS(A)∪EL(MSS(A)) andMGS(A)∪EL(MGS(A)). Therefore, the re- dundancy can only appear during the construction ofUI(T, Σ)∪UI^EL(T, Σ). From the definition ofMGSandMSSfollows that the setsUI_Σ,MSS(T, Σ) andUI_Σ,MGS(T, Σ) in Defi- nition 7 cannot be redundant if the setsMSS(A)∪EL(MSS(A)) andMGS(A)∪EL(MGS(A)) contain only incomparable elements. Therefore, it remains to consider the redundancy introduced during the construction ofUI_Σ(T, Σ). We denote byP_Σ ={(C1,C2)|there is A∈ Σ s.t.C1 ∈MGS(A)∪EL(MGS(A)),C2 ∈ MSS(A)∪EL(MSS(A))}the set of all con- cept pairs relevant for the construction ofUI_Σ(T, Σ) and the subset ofP_Σ containing the “redundant” concept pairs byR={(C1,C2)∈P_Σ|(UI(T, Σ)∪UI^EL(T, Σ))\ {C1v C2} |= C1 v C2}. I.e., Ris the set of concept pairs that are potentially nonessential for the construction of a UIdue to entailment of the corresponding inclusion axiom by the remainder of a UI if the axiom itself is omitted. Due to possible dependen- cies between the elements of R, there may be several different maximal subsets M of Rsuch that (UI(T, Σ)∪UI^EL(T, Σ))\ {C₁ v C₂|(C₁,C₂) ∈ M} |= UI(T, Σ). We denote the set of all such maximal subsets ofR as R_MAX = {M|M ⊆ R,(UI(T, Σ)∪ UI^EL(T, Σ))\ {C1 v C2|(C1,C2) ∈ M} |= UI(T, Σ),for all (C⁰₁,C⁰₂) ∈ P_Σ \M holds (UI(T, Σ)∪UI^EL(T, Σ))\({C⁰₁vC⁰₂} ∪ {C1vC2|(C1,C2)∈M})6|=UI(T, Σ)}. The next lemma states that if a concept pair with at least one non-ELconcept is contained in one setM∈ R_MAX, it is contained in allM∈ R_MAX.

Lemma 3. LetT⁰be anELTBox with sig(T⁰) ⊆Σsuch thatT⁰ ≡^c_Σ T. Further, let A∈ Σ_C,MSS∪Σ_C,MGS with C₁ ∈ MGS(A)and C₂ ∈ MSS(A). Let let M⁰ ∈ R_MAX such that (C₁,C₂)∈M⁰. Then for each M∈ R_MAXholds(C₁,C₂)∈M.

Note that all concept pairs with at least one non-ELconcept are contained in the inter- section ofR_MAX, iffUI^EL(T, Σ)≡ T⁰. As a consequence of the above two lemmas and

the fact that for any (C1,C2) ∈ Rthere exists at least one M ∈ R_MAX, it is sufficient to check whether all concept pairs with at least one non-ELconcept are contained inRto determine whether theT⁰in Theorem 4 exists.

8 Summary

In this paper, we provide anE xpT imealgorithm for computing a uniform interpolant of generalELterminologies preserving allELconcept inclusions for a particular signa- ture based on the notion ofmost general subconceptsandmost specific superconcepts.

The result of the computation is expressed in logicEL_µ(t),ν—an extension ofELwith least fixpoint and greatest fixpoint constructorsµ, νas well as the disjunction used only on the left-hand side of concept inclusions. We also state the exact existence criteria for anELinterpolant and show how it can be obtained from the corresponding interpolant expressed inEL_µ(t),ν.

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