The Logical Difference for EL: From Terminologies Towards TBoxes

Shasha Feng¹, Michel Ludwig², and Dirk Walther²

1 Jilin University, China fengss@jlu.edu.cn

2 Theoretical Computer Science TU Dresden, Germany

{michel,dirk}@tcs.inf.tu-dresden.de

Abstract. In this paper we are concerned with the logical difference problem between ontologies. The logical difference is the set of subsump- tion queries that follow from a first ontology but not from a second one.

We revisit our solution to logical difference problem forEL-terminologies based on finding simulations between hypergraph representations of the terminologies, and we investigate a possible extension of the method to generalEL-TBoxes.

1 Introduction

Ontologies are widely used to represent domain knowledge. They contain spec- ifications of objects, concepts and relationships that are often formalised using a logic-based language over a vocabulary that is particular to an application domain. Ontology languages based on description logics [2] have been widely adopted, e.g., description logics are underlying the Web Ontology Language (OWL) and its profiles.³ Numerous ontologies have already been developed, in particular, in knowledge intensive areas such as the biomedical domain, and they are made available in dedicated repositories such as the NCBO bioportal.⁴

Ontologies constantly evolve, they are regularly extended, corrected and re- fined. As the size of ontologies increases, their continued development and main- tenance becomes more challenging as well. In particular, the need to have auto- mated tool support for detecting and representing differences between versions of an ontology is growing in importance for ontology engineering.

The logical difference is taken to be the set of queries that produce differ- ent answers when evaluated over distinct versions of an ontology. The language and vocabulary of the queries can be adapted in such a way that exactly the differences of interest become visible, which can be independent of the syntac- tic representation of the ontologies. We consider ontologies formulated in the

? The second and third authors were supported by the German Research Foundation (DFG) within the Cluster of Excellence ‘Center for Advancing Electronics Dresden’.

3 http://www.w3.org/TR/owl2-overview/

4 http://bioportal.bioontology.org

lightweight description logic EL [1, 3] and queries that are EL-concept inclu- sions. The relevance of ELfor ontologies is emphasised by the fact that many ontologies are largely formulated inEL. For instance, the dataset of the ORE 2014 reasoner evaluation comprises 8 805 OWL-EL ontologies.⁵

The logical difference problem was introduced in [7] and investigated for EL-terminologies [6]. A hypergraph-based approach for EL-terminologies was presented in [4], which was subsequently extended toEL-terminologies with ad- ditional role inclusions, domain and range restrictions of roles in [8]. In this paper we investigate a possible extension of the method to generalEL-TBoxes.

Clearly, such an extension needs to account for the additional expressivity of general TBoxes w.r.t. terminologies. After normalisation, a terminology may contain at most one axiom of the form∃r.AvX orA1u. . .uAn vX for any concept nameX, whereas a general TBox does not impose such a restriction.

We first show that for every concept inclusion C v D that follows from a TBoxT, there exists a concept nameXinT that acts as aninterpolant between the conceptsCandD, i.e., we have thatT |=CvX andT |=X vD. Then we describe the set of all subsumeesC ofX in T using a concept ofELextended with disjunction and a least fixpoint operator, and the set of all subsumersDof XinT using a concept ofELextended with greatest fixpoint operators. Finally, we reduce the problem of deciding the logical difference between twoEL-TBoxes to fixpoint reasoning w.r.t. TBoxes in a hybridµ-calculus [10].

The paper is organised as follows. We start by recalling some notions re- garding the description logic EL and its extensions with disjunction and fix- point operators. In Section 3, we discuss how the logical difference problem for EL-terminologies could be extended to general EL-TBoxes, and we establish a witness theorem for general EL-TBoxes. In Section 4, we show how fixpoint reasoning can be used to decide whether two generalEL-TBoxes are logically different, and how witnesses to the logical difference can be computed. Finally we conclude the paper.

2 Preliminaries

We start by briefly reviewing the lightweight description logic EL and some notions related to the logical difference, together with some basic results.

LetN_C,N_R, andN_V be mutually disjoint sets of concept names, role names, and variable names, respectively. We assume these sets to be countably infinite.

We typically useA, B to denote concept names andrto denote role names.

The sets ofEL-concepts C,ELU^µ-concepts D, andEL^ν-concepts Eare built according to the following grammar rules:

C::=> |A|CuC| ∃r.C

D::=> |A|DuD|DtD| ∃r.D|x|µx.D E::=> |A|EuE | ∃r.E|x|νx.E

5 http://dl.kr.org/ore2014/

whereA ∈NC, r, s∈NR, andx∈NV. For anELU^µ-conceptC, the set offree variables inC, denoted by FV(C) is defined inductively as follows: FV(>) =∅, FV(A) = ∅, FV(D1uD2) = FV(D1)∪FV(D2), FV(D1tD2) = FV(D1)∪ FV(D2), FV(∃r.D⁰) = FV(D⁰), FV(x) = {x}, FV(µx.D⁰) = FV(D⁰)\ {x}. The set FV(E) of free variables occurring in an EL^ν-conceptE can be defined analogously. An ELU^µ-concept C (an EL^ν-conceptD) is closed if C (D) does not contain free occurrences of variables, i.e. FV(C) = ∅ (FV(D) =∅). In the following we assume that every ELU^µ-concept C and every EL^ν-conceptD is closed.

AnEL-TBox T is a finite set of axioms, where an axiom can be a concept inclusion C vC⁰, or a concept equation C ≡C⁰, where C, C⁰ range over EL- concepts. AnEL-terminology T is an EL-TBox consisting of axioms α of the formAvC andA≡C, whereA is a concept name,C an EL-concept and no concept nameAoccurs more than once on the left-hand side of an axiom.

The semantics of EL, ELU^µ, and EL^ν is defined using interpretations I = (∆^I,·^I), where the domain∆^I is a non-empty set, and·^I is a function mapping each concept nameA to a subsetA^I of∆^I and every role namer to a binary relationr^I ⊆∆^I×∆^I. Interpretations are extended to concepts using a function

·^I,ξthat is parameterised by anassignment function that maps variablesx∈N_V to setsξ(x) ⊆∆^I. Given an assignment ξ, the extension of an EL, ELU^µ, or EL^ν-concept is defined inductively as follows: >^I,ξ := ∆^I, x^I,ξ := ξ(x) for x∈N_V, (C1uC2)^I,ξ :=C₁^I∩C₂^I, (∃r.C)^I,ξ :={x∈∆^I | ∃y ∈C^I,ξ : (x, y)∈ r^I}, (µx.C)^I,ξ =T

{W ⊆∆^I | C^I,ξ[x7→W] ⊆ W}, and (νx.C)^I,ξ =S {W ⊆

∆^I |W ⊆C^I,ξ[x7→W]}, whereξ[x7→W] denotes the assignmentξmodified by mappingxtoW.

An interpretation I satisfies a concept C, an axiom C vD or C ≡ D if, respectively, C^I,∅ 6= ∅, C^I,∅ ⊆ D^I, or C^I,∅ = D^I,∅. We write I |= α iff I satisfies the axiomα. An interpretation I satisfies a TBox T iff I satisfies all axioms inT; in this case, we say that I is a model of T. An axiom αfollows from a TBoxT, written T |=α, iff for all modelsI of T, we have thatI |=α.

Deciding whetherT |=C vC⁰, for twoEL-conceptsC andC⁰, can be done in polynomial time in the size ofT and C, C⁰ [1, 3]. For anELU^µ-conceptD and anEL^ν-conceptE, it is known thatT |=DvE can be decided in exponential time in the size ofT,D andE [10].

A signature Σ is a finite set of symbols from NC and NR. The signature sig(C), sig(α) or sig(T) of the concept C, axiom α or TBox T is the set of concept and role names occurring inC,αorT, respectively. AnEL^Σ-conceptC is anEL-concept such thatsig(C)⊆Σ.

AnEL-TBoxT isnormalised if it only containsEL-concept inclusions of the forms> vB,A1u. . .uAn vB,Av ∃r.B, or ∃r.AvB, whereA, Ai, B∈NC, r∈NR, andn≥1. EveryEL-TBox T can be normalised in polynomial time in the size ofT with a linear increase in the size of the normalised TBox w.r.t.T such that the resulting TBox is a conservative extension ofT [6]. Note that in a normalised terminologyT, we have that for every axiom of the form ∃r.Av B∈ T, there exists an axiom of the formBv ∃r.A∈ T; similarly for axioms of

the formA1u. . .uAn vB with n≥2. When convenient, we will abbreviate two axiomsAv ∃r.Band∃r.BvAby the single axiomA≡ ∃r.B; similarly for A≡B1u. . .uBn.

3 Towards Logical Difference between General EL -TBoxes

The logical difference between two TBoxes witnessed by concept inclusions over a signatureΣis defined as follows.

Definition 1. The Σ-concept differencebetween twoEL-TBoxesT¹andT² for a signature Σ is the set cDiffΣ(T¹,T²) of allEL-concept inclusions αsuch that sig(α)⊆Σ,T¹|=α, andT²6|=α.

EL-TBoxes can be translated into directed hypergraphs by taking the sig- nature symbols as nodes and treating the axioms as hyperedges connecting the nodes. For normalised EL-TBoxes, the axiom > v B is translated into the hyperedge ({x_>},{xB}), the axiomA1u. . .uAn vB into the hyperedge ({xB1, . . . , xBn},{xA}), the axiomAv ∃r.Binto the hyperedge ({xA},{xr, xB}), and the axiom∃r.AvB into the hyperedge ({xr, xB},{xA}), where each node xY corresponds to the signature symbolY, respectively. A feature of the trans- lation of axioms into hyperedges is that all information about the axiom and the logical operators in it is preserved. In fact we can treat the ontology and its hypergraph representation interchangeably. The existence of certain simulations between hypergraphs forEL-terminologies characterises the fact that the corre- sponding terminologies are logically equivalent and, thus, no logical difference exists [4, 8].

As the setcDiffΣ(T¹,T²) is infinite in general, we make use of the following

“primitive witnesses” theorem from [6] that states that we only have to con- sider two specific types of concept differences. If there is an inclusionC vD∈ cDiffΣ(T¹,T²) for two terminologiesT¹andT², then we know that there is a con- cept nameA∈Σ such that A occurs either on the left-hand or the right-hand side of an inclusion in the set C v D ∈ cDiffΣ(T¹,T²). For checking whether cDiffΣ(T¹,T²) =∅, we only have to consider such simple inclusions. However, if T¹andT²are general EL-TBoxes, the situation is different.

Example 1. Let T¹ = {X ≡ A1 u A2, X v ∃r.>}, T² = ∅, and let Σ = {A1, A2, r}. Note that T¹ is not a terminology as the concept name X occurs twice on the left-hand side of an axiom. Then every differenceα∈cDiffΣ(T¹,T²) is equivalent to the inclusionA1uA2v ∃r.>. In particular, there does not exist a difference of the formψvθ, whereψor θis a concept name fromΣ.

As illustrated by the example, we need to account for a new kind of differences C v C⁰ ∈ cDiffΣ(T¹,T²) which are induced by a concept name X ∈ sig(T¹) such that X 6∈ Σ, T¹ |=C v X, and T¹ |= X v C⁰. We obtain the following witness theorem forEL-TBoxes as an extension of the witness theorem forEL- terminologies.

Theorem 1 (Witness Theorem). Let T¹,T² be two normalised EL-TBoxes and let Σ be a signature. Then, cDiffΣ(T¹,T²) 6= ∅ iff there exists an EL^Σ- inclusionα=ϕvψsuch that T¹|=αandT²6|=α, where

(i) ϕis an EL^Σ-concept andψ=A∈Σ, (ii) ϕ=A∈Σandψ is anEL^Σ-concept, or

(iii) there existsX ∈sig(T¹)\Σ such thatT¹|=ϕvX andT¹|=Xvψ.

The proof of the witness theorem for terminologies [6] is based on analysing the subsumptionT¹|=ϕvψsyntactically, using a sequent calculus [5]. A similar technique can be used for the proof of Theorem 2.

For deciding whethercDiffΣ(T¹,T²) =∅in the case of general TBoxes, we now have to additionally consider differences of Type (iii). Differences of types (i) or (ii) can be checked by using forward or backward simulations adapted to normalised EL-TBoxes, respectively, whereas Type (iii) differences require a combination of both techniques.

Before we illustrate how Type (iii) differences can be dealt with, we first introduce some auxiliary notions. We define cWtn^lhs_Σ(T¹,T²) as the set of all concept namesA from Σ such that there exists an EL^Σ-concept C with A v C ∈ cDiffΣ(T¹,T²). Similarly, cWtn^rhs_Σ (T¹,T²) is the set of all concept names A∈Σ such that there exists an EL^Σ-conceptC withC vA∈cDiffΣ(T¹,T²).

The concept names in cWtn^lhs_Σ(T¹,T²) are called left-hand side witnesses and the concept names incWtn^rhs_Σ (T¹,T²)right-hand side witnesses. Additionally, we define cWtn^mid_Σ (T¹,T²) as the set of all concept names X from sig(T¹) but not from Σ such that there exists C vC⁰ ∈cDiffΣ(T¹,T²) and T¹ |=C vX and T¹ |= X v C⁰. The concept names in cWtn^mid_Σ (T¹,T²) are called interpolating witnesses. To summarise, we have the following sets:

cWtn^lhs_Σ(T¹,T²) ={A∈Σ| ∃C∈ EL^Σ: AvC∈cDiffΣ(T¹,T²)} cWtn^rhs_Σ (T¹,T²) ={A∈Σ| ∃C∈ EL^Σ: CvA∈cDiffΣ(T¹,T²)} cWtn^mid_Σ (T¹,T²) ={X ∈sig(T¹)\Σ| ∃C, C⁰∈ EL^Σ:T¹|=CvX,

T¹|=X vC⁰,T²6|=CvC⁰} We illustrate the witness sets with the following example.

Example 2. Let T¹ ={X ≡A1uA2, X v ∃r.>, A3 v A2, A3 v ∃r.A2}, T² = {A2v ∃r.>}, and letΣ={A1, A2, r}. Then it holds thatcWtn^lhs_Σ(T¹,T²) ={A3} (e.g. {A3 v A2, A3 v ∃r.A2} ⊆ cDiffΣ(T¹,T²)), cWtn^rhs_Σ(T¹,T²) = {A2} (e.g.

A3vA2∈cDiffΣ(T¹,T²)), andcWtn^mid_Σ (T¹,T²) ={X} (e.g.T¹|=A1uA3vX, T¹|=X v ∃r.>andT²6|=A1uA3v ∃r.>).

We obtain as a corollary of Theorem 2 that, to decide the logical difference between two EL-TBoxes, it is sufficient to check the emptiness of the witness sets.

Corollary 1. LetT¹,T² be two normalisedEL-TBoxes and letΣbe a signature.

Then it holds that cDiffΣ(T¹,T²) = ∅ iff cWtn^lhs_Σ(T¹,T²) = cWtn^rhs_Σ(T¹,T²) = cWtn^mid_Σ (T¹,T²) =∅.

To characterise an interpolating witness of a Type-(iii) difference, we use the sets of its of subsumees and subsumers formulated using certain signature symbols only. A similar approach was used for the construction of uniform in- terpolants ofEL-TBoxes in [9].

Definition 2. Let T be an EL-TBox, let Σ be a signature and let C be an EL-concept. We define Premises^Σ_T(C) := {E ∈ EL^Σ | T |= E v C} and Conclusions^Σ_T(C) :={E∈ EL^Σ| T |=CvE}.

The set Premises^Σ_T(C) contains allEL-concepts formulated usingΣ-symbols only that entailCw.r.t.T; or are entailed byC in the case of Conclusions^Σ_T(C). The elements of Premises^Σ_T(C) are also called Σ-implicants or Σ-subsumees of C w.r.t.T, and the elements of Conclusions^Σ_T(C) are also named Σ-implicates or Σ-subsumers ofCw.r.t.T.

In [4], it was established that a concept name X is forward simulated by a concept nameY in an EL-terminologyT iff it holds that Conclusions^Σ_T(X) ⊆ Conclusions^Σ_T(Y); and similary,Xis backward simulated byY iff Premises^Σ_T(X)⊆ Premises^Σ_T(Y). We aim now at lifting this result to generalEL-TBoxes.

Example 3. LetT¹ ={A vX,∃r.X vX, X vB1, X vB2} and T² = {A v Y,∃r.Y v Y⁰,∃r.Y⁰ v Y,∃r.Y v Z1,∃r.Y v Z2, Y v B1, Y v B2, Z1 v B1, Z2 vB2} be twoEL-TBoxes. Let Σ={A, B1, B2, r} be a signature. Note thatX inT¹is cyclic and intuitively, the interpretation ofX in a modelI ofT¹ contains all finite r-chains “ending inA”. In T² the concept name Y is cyclic and its interpretation contains allr-chains ending in Athat are ofeven length, whereas the interpretations of the concept namesZ1andZ2 contain allr-chains ending inAthat are of odd length. Formally, we have that:

{A,∃r.A,∃r.∃r.A, . . .} ⊆Premises^Σ_T1(X) {A,∃r.∃r.A,∃r.∃r.∃r.∃r.A, . . .} ⊆Premises^Σ_T2(Y)

{∃r.A,∃r.∃r.∃r.A, . . .} ⊆Premises^Σ_T2(Zi) fori∈ {1,2} In particular, fori∈ {1,2}, we have

Premises^Σ_T1(X) = Premises^Σ_T2(Y)∪Premises^Σ_T2(Zi).

Intuitively, the set ofΣ-implicants ofX in T¹ are distributed over the concept namesY andZi inT². Moreover it holds that

Conclusions^Σ_T1(X) = Conclusions^Σ_T2(Y) = Conclusions^Σ_T2(Z1uZ2).

The concept nameX in T¹ could be forward simulated either byY orZ1uZ2

inT². Note thatZ1orZ2individually are not sufficient. Analogously,Xcould be backward simulated byY tZ1 orY tZ2. None of the concept namesX,Z1, or Z2are sufficient individually for the backward simulation. Combining backward and forward simulation,X could be simulated byY t(Z1uZ2).

In general, we hypothesise that non-Σ-concept names X in T¹ need to be

“simulated” by concepts of the formFn

i=1Ci,whereCi areEL-concepts.

4 Finding Logical Differences via Fixpoint Reasoning

We now show how fixpoint reasoning can be used to find difference witnesses between generalEL-TBoxes.

Given Theorem 2, we know that any differenceCvC⁰∈cDiffΣ(T¹,T²), for two EL^Σ-conceptsC andC⁰, is connected to some concept name X occurring inT¹ for which eitherT¹|=CvX, orT¹ |=X vC⁰ (or both) holds. To check whether X is indeed a difference witness, we construct concepts B^Σ_T1(X) and F^Σ_T1(X) formulated in ELU^Σµ and in EL^Σν, respectively, to describe the (poten- tially infinite) disjunction ofΣ-concepts that are subsumed byX w.r.t.T¹, and the conjunction of all the Σ-concepts that subsume X w.r.t. T¹, respectively.

Note that the use of fixpoint allows for a finite description of infinite disjunc- tions or conjunctions. TheELU^Σµ-concept B^Σ_T1(X) hence is a finite representation of the set Premises^Σ_T1(X), whereas the EL^Σν-concept F^Σ_T1(X) represents the set Premises^Σ_T1(X) in a finite way. Using the fixpoint descriptions of the premises and conclusions ofX w.r.t. T¹, we can verify whether X is a difference witness by checkingT²|= B^Σ_T1(X)vF^Σ_T1(X).

We first turn our attention to the set Premises^Σ_T1(X). Before we can give a formal definition for the concept B^Σ_T1(X), we have to introduce the following auxiliary notion to handle concept names X in the definition of B^Σ_T1(X) for which there exist axioms of the formZ1u. . .uZnvZ in a normalised TBoxT such that T |= Z v X. Intuitively, given a concept name X, we construct a set Conj_T(X) containing sets of concept names which has the property that for every EL-concept D withT |= D vX, there exists a set S ={Y1, . . . , Ym} ∈ Conj_T(X) such that T |=D vYi follows without involving any axioms of the formZ1u. . .uZn vZ. Nested implications between such axioms also have to be taken into account.

Definition 3. Let T be a normalisedEL-TBox and let X∈N_C. We define the set Conj_T(X)⊆2^sig(T)∩NC to be smallest set inductively defined as follows:

– {X} ∈Conj_T(X);

– if S ∈Conj_T(X), Y ∈S, andZ1u. . .uZn vZ ∈ T such thatn≥2 and T |=ZvY, thenS\ {Y} ∪ {Z1, . . . , Zn} ∈Conj_T(X).

Note that for every concept nameX the set Conj_T(X) is finite assig(T)∩N_C is finite.

Example 4. Let T¹ = {A v X,∃r.X v X}. Then Conj_T1(X) = {{X}}. For T²={X1uX2vX, X3uX4vX1, Y1uY2vX}, we have that

Conj_T2(X) ={{X},{X1, X2},{X3, X4, X2},{Y1, Y2}}. We can now give a formal definition of the concept B^Σ_T(X).

Definition 4. Let T be a normalised EL-TBox, let Σ be a signature, and let X ∈ sig(T). For a mapping η: N_C → N_V, we define a closed ELU^µΣ-concept B^Σ_T(X, η)as follows. We set B^Σ_T(X, η) =>if T |=> vX; otherwise B^Σ_T(X, η) is defined recursively in the following way:

– If X∈dom(η), then

B^Σ_T(X, η) =η(X) – If X6∈dom(η), we set

B^Σ_T(X, η) =µx. G

S∈Conj_T(X) S={Y1,...,Ym}

(Y₁⁰u. . .uY_m⁰)

where x is a fresh variable, and Y_i⁰ (1 ≤ i ≤ n) is defined as follows for η⁰:=η∪ {X 7→x}:

Y_i⁰ = G

T |=BvYi

B∈Σ

Bt G

∃r.ZvY∈T r∈Σ T |=YvYi

∃r.B^Σ_T(Z, η⁰)

Finally, we set B^Σ_T(X) =B^Σ_T(X,∅).

Intuitively, the construction of B^Σ_T(X) starts fromX and recursively collects all the concept names contained inΣand all the left-hand sides of axioms inT that could be relevant for X to be entailed by a concept w.r.t. T. By taking into account all possible axioms that could lead to a logical entailment, it is guaranteed that we capture every Σ-concept from which X follows w.r.t. T. Reasoning involving axioms of the formZ1u. . .uZnvZ is handled by the set Conj_T(X). Infinite recursion over concepts of the form∃r.Cis avoided by keeping track of the concept names that been visited already using the mappingη.

We note that for a normalisedEL-terminologyT, the concept B^Σ_T(X) is of a simpler form than for normalisedEL-TBoxes. This is because the concept name X can occur on the right-hand side of at most one axiom of the form∃r.AvX orA1u. . .uAn vX withn≥2 inT, whereas in a TBox several such axioms may occur.

We illustrate the concept B^Σ_T(X) with the following examples.

Example 5. LetT¹ ={A1uA2 vX, A3 vA2,∃r.A2 vA1,∃r.A2 vX}, T² = T¹∪ {∃r.XvA2}, and letΣ={A1, A2, A3, r}. We obtain the followingELU^µΣ- concepts. We writeϕinstead ofµx.ϕifxdoes not occur freely inϕ.

B^Σ_T1(A1) =A1t ∃r.(A2tA3) B^Σ_T1(A2) =A2tA3

B^Σ_T1(X) = ((A1t ∃r.(A2tA3))u(A2tA3))t ∃r.(A2tA3) B^Σ_T2(X) =µx.(((A1t ∃r.(A2tA3t ∃r.x))u(A2tA3t ∃r.x))

t ∃r.(A2tA3t ∃r.x))

Example 6. LetT¹,T² be defined as in Example 3, and letΣ ={A, B1, B2, r}. We have that fori∈ {1,2}:

B^Σ_T1(X) =µx.(At ∃r.x) B^Σ_T1(Bi) =BitAt ∃r.µx.(At ∃r.x) B^Σ_T2(Y) =µy.(At ∃r.∃r.y) B^Σ_T2(Zi) =∃r.µy.(At ∃r.∃r.y) B^Σ_T2(Bi) =BitAt ∃r.µy1.(∃r.(At ∃r.y1))t ∃r.µy2.(At ∃r.∃r.y2)

By inspecting Definition 4 it is easy to see that|= B^Σ_T(X)≡ ⊥if there does not anEL^Σ-conceptCwithT |=CvX. Overall, one can establish the following correctness and completeness properties.

Lemma 1. Let T be a normalisedEL-TBox, letΣ be a signature, and letX∈ sig(T). Then the ELU^Σµ-concept B^Σ_T(X)satisfies the following properties:

(i) T |=B^Σ_T(X)vX, and (ii) for every D∈Premises^Σ_T(X),

T |=DvX iff |=DvB^Σ_T(X).

The following lemma states that theELU^µ-concept B^Σ_T(X) exactly captures the infinite set Premises^Σ_T(X). More formally, the concept B^Σ_T(X) is equivalent to the infinite disjunction over all the concepts contained in the set Premises^Σ_T(X).

Lemma 2. Let T be a normalisedEL-TBox, letΣ be a signature, and letX∈ sig(T). Then for every interpretationI it holds that

(B^Σ_T(X))^I,∅=[

{C^I,∅|C∈Premises^Σ_T(X)}.

Analogously to the concept B^Σ_T(X), it is possible to construct an EL^Σν- concept F^Σ_T(X) which exactly captures the set Conclusions^Σ_T(X) for a concept nameX and anEL-TBoxT. Due to lack of space, we cannot give a full definition of the concept F^Σ_T(X). Instead, we state its existence and its essential property in the following lemma.

Lemma 3. Let T be a normalised EL-TBox, let Σ be a signature, and let X∈sig(T). Then there exists anEL^ν-concept F^Σ_T(X)such that for every inter- pretationI it holds that

(F^Σ_T(X))^I,∅=[

{C^I,∅|C∈Conclusions^Σ_T(X)}.

We can now state the following theorem, which establishes how the concepts B^Σ_T(X) and F^Σ_T(X) can be used to search for difference witnesses.

Theorem 2. Let T¹,T² be two normalisedEL-TBoxes. Then it holds that:

(i) A6∈cWtn^lhs_Σ(T¹,T²)iffT²|=AvF^Σ_T1(A), for everyA∈Σ;

(ii) A6∈cWtn^rhs_Σ(T¹,T²) iffT²|=B^Σ_T1(A)vA, for everyA∈Σ; and

(iii) X 6∈cWtn^mid_Σ (T¹,T²)iffT²|=B^Σ_T1(X)vF^Σ_T1(X), for everyX ∈sig(T¹)\Σ.

Theorem 2 together with Corollary 1 give rise to an algorithm for deciding the logical difference betweenEL-TBoxes. Procedure 1 is such an algorithm based on reasoning in the hybridµ-calculus, which allows for fixpoint reasoning w.r.t.

TBoxes [10].

Theorem 3. Procedure 1 runs in ExpTime.

Procedure 1Deciding existence of logical difference Input: NormalisedEL-TBoxesT¹,T²and signatureΣ Output: trueorfalse

1: for every concept nameX∈sig(T¹)∪Σdo 2: B:= B^Σ_T1(X)

3: F:= F^Σ_T1(X) 4: if X ∈Σthen

5: if T²6|=Xv F or T²6|=B vX then

6: return true

7: end if

8: else if T²6|=B v F then 9: return true

10: end if 11: end for 12: return false

We note that the lower bound for the running time of Procedure 1 may also be exponential as the underlying problem of deciding the logical difference of two EL-TBoxes is ExpTime-complete [6, 7].

Example 7. Continue Example 3, where sig(T¹)∪Σ = {A, X, B1, B2, r}. For A, B = B^Σ_T1(A) = A, and F = F^Σ_T1(A) = B1 uB2. As T² |= A v F and T² |= B v A, the loop continues. Then for X, B = B^Σ_T1(X) = µx.(At ∃r.x) andF = F^Σ_T1(X) =B1uB2 (cf. Example 6). Since it holds that T² |=B v F, the loop continues. For B1, B = B1tAt ∃r.µx.(At ∃r.x) and F = B1. As T²|=B1v F andT²|=B vB1, the loop continues. The case ofB2 is similar to that ofB1. Finally, the algorithm returns false.

Procedure 1 can be modified to obtain witnesses to difference subsumption.

Example 8. We run Procedure 1 onT¹,T³=T²\{Z1vB1}andΣ, whereT¹,T² andΣ are the same as in Example 3. Then, forX, we have thatB= B^Σ_T1(X) = µx.(At ∃r.x) andF = F^Σ_T1(X) =B1uB2. However,T³6|=B v F, which means X∈cWtn^mid_Σ (T¹,T³). Similarly, it can readily be seen that B1∈cWtn^rhs_Σ(T¹,T³).

5 Conclusion

We have revisited our solution to logical difference problem forEL-terminologies which was based on finding simulations between hypergraph representations of the terminologies [4]. We have shown that there is a new type of witness in the logical difference between twoEL-TBoxes. We have shown that deciding the logical difference between twoEL-TBoxes can be reduced to fixpoint reasoning w.r.t. TBoxes.

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