Computing Minimal Projection Modules for Conjunctive Queries
?Jieying Chen1, Michel Ludwig2, Yue Ma1, and Dirk Walther3
1 Laboratoire de Recherche en Informatique, Universit´e Paris-Sud, France {jieying.chen,yue.ma}@lri.fr
2Luxembourg, [email protected]
3 Fraunhofer IVI, Dresden, Germany [email protected]
Abstract. We consider the problem of extracting modules of an ontol- ogy that contains the knowledge as represented by a second ontology.
The knowledge to be preserved is specified using entailment of conjunc- tive queries over a given vocabulary. We propose a novel module no- tion calledprojection module that preserves the answers to conjunctive queries as they follow from a reference ontology. We present an algo- rithm for computing minimal projection modules for conjunctive queries.
As target and reference ontology we take ELHr-terminologies. The al- gorithm is based on simulation notions developed for detecting logical differences betweenELHr-terminologies.
1 Introduction
Ontology comparison can help understanding the overlap and differences among ontologies which is often desired while a user manipulates multiple knowledge sources. In this paper, we propose the notion ofprojection modulewhich allows to compare the entailment capacities of two ontologies about a given vocabulary.
A projection module characterizes the relative knowledge of one ontology by taking another one as a reference. This can thus lead to a fine-grained ontology comparison measurement between two ontologies.
Various approaches to comparing ontologies have been suggested, including ontology mapping or alignment [12], and logical difference [19–21, 23]. Ontol- ogy matching is the process of determining correspondences, in particular, the subsumption, equivalence, or disjointness relations between two concept or role names from different ontologies. A good concept similarity [1, 22] is often helpful for ontology matching. In contrast, logical difference focuses on the compari- son of entailed logical consequences from each ontology and returns difference witnesses if differences are present.
When an ontology has no logical difference compared to another one, our approach further extracts sub-ontologies of the first ontology that contain the
?This work is partially funded by the ANR project GoAsQ (ANR-15-CE23-0022).
knowledge as represented by the second ontology. For example, suppose T1 = {A1vA2, A2 vA3},T2 ={A1vA3uB1, B1v ∃r.A3}, andΣ={A1, A3, r}.
ThenT2entails all queries aboutΣthat follow fromT1. However, the projection module of T2 with respect to T1 and Σ consists of A1 vA3uB1. This means that a strict sub-ontology ofT2is sufficient to capture all the information ofT1
about Σ. Moreover, T2 entails the consequence A1 v ∃r.A3, which is not the case forT1. Intuitively,T2 is richer in information aboutΣ thanT1.
Ontology modularity [9, 16, 19, 21, 24, 25] is about the extraction of sub- ontologies that preserve all logical consequences over a given signature. In con- trast, the proposed projection module is different from modules of a single on- tology. For the example above, the basic minimal module [9],⊥>?-module, and MEX-module [18] of T2 w.r.t. Σ are allT2 itself. The extra axiom in T2 com- pared to its projection module with respect toT1 confirms thatT1conveys less information about Σ thanT2, which gives a way to compare these two ontolo- gies. For latest results on logical inseparability (in particular, inseparability w.r.t.
conjunctive queries), see [7,8,13], and for a survey on query inseparability, see [6].
To compute projection modules, in this paper, we generalize the notion of justification to the notion ofsubsumption justificationas a minimal set of axioms that maintains a consequence. Our algorithm employs the classical notion of justification to compute subsumption justification. Currently, the approaches for computing all the justifications of an ontology w.r.t. a consequence can be classified into two categories: “glass-box” [2,5,14,15] and “black-box” [10,14,26].
We proceed as follows. Section 2 gives a brief review of Description Logic EL and its extensions as well as logical difference. In Section 3, we introduce the notion of project module. In Section 4, the notion and the computation of role subsumption justifications are presented, which are employed in Section 5 to compute project modules. Finally, Section 6 concludes the paper.
2 Preliminaries
We start by briefly reviewing the description logic ELand several of its exten- sion with range restrictions and conjunction of roles and the universal role as well as concept subsumptions based on these extensions. For a more detailed introduction to description logics, we refer to [3, 4].
LetNCandNRbe sets of concept names and role names. We assume these sets to be mutually disjoint and countably infinite. The sets ofEL-concepts C,ELran- concepts D, ELu-concepts E, and ELu,u-concepts F, and the sets of ELHr- inclusionsαandELran,u,u-inclusionsβare built according to the grammar rules:
C ::= A|CuC| ∃r.C|dom(r)
D ::= A|DuD| ∃r.D|dom(r)|ran(r) E ::= A|EuE| ∃R.E
F ::= A|FuF | ∃R.F | ∃u.F
α ::= CvC|ran(r)vC|ran(r)uCvC|C≡C|rvs β ::= DvF |rvs
where A ∈ NC, r, s ∈ NR, u is a fresh logical symbol (the universal role) and R =r1u. . .urn withr1, ..., rn ∈NR, for n≥1. We refer to inclusions also as axioms. A Γ-TBox is a finite set ofΓ-inclusions, whereΓ ranges over the sets ofELHr- andELran,u,u-inclusions.
The semantics is defined as usual in terms of interpretations, which interpret concept and role names and are inductively extended to complex concepts. The notions of satisfaction of a concept, axiom and TBox as well as the notions of a model and the logical consequence relation are defined as usual. We skip a detailed introduction here.
A signature is a finite set of symbols from NC and NR. We write sigNC(ξ) and sig(ξ) for the set of concept names and the set of concept and role names occurring in a syntactic objectξ. The symbol Σis used as a subscript to a set of concepts or inclusions to denote that the elements only use symbols fromΣ.
AnELHr-terminologyT is anELHr-TBox consisting of axiomsαof the form AvC, A≡C,rvs,ran(r)vC or dom(r)vC, whereAis a concept name, C anEL-concept and no concept name occurs more than once on the left-hand side of an axiom.1 To simplify the presentation we assume that terminologies do not contain axioms of the form A ≡B or A ≡ > (after removing multiple
>-conjuncts) for concept names A and B. For a terminology T, let ≺T be a binary relation over NC such that A ≺T B iff there is an axiom of the form AvC orA≡C in T such that B ∈sig(C). A terminologyT isacyclic if the transitive closure≺+T of≺T is irreflexive; otherwiseT iscyclic. A concept name Ais said to beconjunctive inT iff there exist concept namesB1, . . . , Bn,n >0, such thatA≡B1u. . .uBn∈ T; otherwiseAis said to benon-conjunctive inT. An ELHr-terminology T is normalised iff it only contains axioms of the formsϕvB1u. . .uBn,Av ∃r.B,Avdom(r),rvs, andA≡B1u. . .uBm, A ≡ ∃r.B, where ϕ∈ {A,dom(s),ran(s)}, n≥1,m≥2,A, B, Bi ∈ NC, r, s∈ NR, and each conjunctBi is non-conjunctive in T. EveryELHr-terminologyT can be normalised in polynomial time such that the resulting terminology is a conservative extension ofT [17]. A subsetM ⊆ T is called ajustification for an ELH-concept inclusion α from T iff M |=α and M0 6|=αfor every M0 ( M. We denote the set of all justifications for an ELH-concept inclusionαfrom an ELH-terminologyT with JustT(α). The latter may contain exponentially many justifications in the number of axioms inT.
We now recall the notion of logical difference for concept subsumption queries, instance queries and conjunctive queries from [17, 20].
Definition 1 (Logical Difference).TheELran,u,u-subsumption query and con- junctive query difference between T1 and T2 wrt. Σ are the sets cDiffΣ(T1,T2) andqDiffΣ(T1,T2), where
– ϕ∈cDiffΣ(T1,T2)iffϕis an ELran,u,uΣ -inclusion andT1|=ϕandT26|=ϕ;
– (A, q(a))∈qDiffΣ(T1,T2)iffAis aΣ-ABox andq(a)aΣ-conjunctive query such that(T1,A)|=q(a)and(T2,A)6|=q(a).
1 A concept equationA≡Cstands for the inclusionsAvC andCvA.
The following theorem states that ELran,u,u-subsumption queries are suffi- cient to detect the absence of conj. query differences (Lemmas 62 and 63 in [17]).
Theorem 1. cDiffΣ(T1,T2) =∅ iffqDiffΣ(T1,T2) =∅.
If the setcDiffΣ(T1,T2) is not empty, then it typically contains infinitely many concept inclusion. We make use of theprimitive witnesses theorems from [17], which state that if there is a concept inclusion difference in cDiffΣ(T1,T2), then there exists an inclusion in cDiffΣ(T1,T2) of one of the following three types δ1, δ2, δ3, which are built according to the grammar rules below:
δ1::=rvs δ2::=DvA
δ3::=AvE|dom(r)vE |ran(r)vE
where δ1 ranges over role inclusions, δ2 is an ELran-inclusion, and δ3 is an ELran,u,u-inclusion. Note that each of these inclusions has either a simple left- hand or a simple right-hand side.
The set of allELran,u,u-subsumption difference witnesses is defined as WtnΣ(T1,T2) := (roleWtnΣ(T1,T2),lhsWtnΣ(T1,T2),rhsWtnΣ(T1,T2)), where the setroleWtnΣ(T1,T2) consists of all type-δ1inclusions incDiffΣ(T1,T2), and the setslhsWtnΣ(T1,T2)⊆(Σ∩NC)∪ {dom(r)|r∈Σ} ∪ {ran(r)|r∈Σ} and rhsWtnΣ(T1,T2)⊆NC∩Σ of left-hand and right-hand subsumption query difference witnesses consist of the left-hand sides of the type-δ3 inclusions in cDiffΣ(T1,T2) and the right-hand sides of type-δ2 inclusions in cDiffΣ(T1,T2), respectively. Consequently, the set WtnΣ(T1,T2) can be seen as a finite repre- sentation of the setcDiffΣ(T1,T2) [17], which is typically infinite. As a corollary of the primitive witness theorems in [17], we have that the representation is com- plete in the following sense:cDiffΣ(T1,T2) =∅iffWtnΣ(T1,T2) = (∅,∅,∅). Thus, deciding the existence of concept inclusion differences is equivalent to deciding non-emptiness of the three witness sets.
3 Projection Modules
A terminologyT1together with a signatureΣand a query languageQdetermine a set Φ of queries from Q formulated using only symbols from Σ that follow from T1. Intuitively,Φcaptures theQ-knowledge ofT1 aboutΣ. In this paper, we consider conjunctive queries or, equivalently,ELran,u,u-subsumption queries as a query language. A projection module for Φof another terminology T2 is a subset ofT2 that entails the queries inΦ. TheQ-knowledge inΦofT1 aboutΣ is captured by the projection module of T2 and it is represented using axioms fromT2. The projection can be seen as transforming the finite representation of Φwith axioms fromT1into a finite representation ofΦwith axioms fromT2. As an application, the projection of Φonto T2 allows us to determine the axioms that implementΦinT2. For instance, projection modules enable the verification
of compliance to certain modelling guidelines and standards by the axioms ofT2 that implementΦ. In particular, projection modules that are minimal w.r.t. set inclusion are relevant for this task. Note that existing module notions suggest to compute a module of T2 for the given signature,Σ, to obtain theQ-knowledge ofT2 aboutΣ, whereas we are only interested inΦ, i.e. theQ-knowledge ofT1
aboutΣ. Consequently, extracting modules ofT2 forΣwill yield modules that generally contain irrelevant axioms and that are, therefore, likely too large for manual inspection.
Definition 2 (Projection Module). Let ρ=hT1, Σ,T2i be a projection set- ting. A set M ⊆ T2 is a conjunctive query projection module under ρ iff for every Σ-ABox A and every Σ-conjunctive query q(a): (T1,A) |=q(a) implies (M,A)|=q(a).
There may exist several (even exponentially many) minimal projection modules.
Example 1. Let T1 = {A1 v A4}, T2 = {A1 ≡ A2uA3, A2 v A4, A3 v A4} and Σ ={A1, A4}. Then the conjunctive query projection module under ρ= hT1, Σ,T2iisT2.
The notion of projection module is not symmetric, i.e., a projection mod- ule under hT1, Σ,T2i is not necessarily the same as a projection module un- der hT2, Σ,T1i. When the reference terminology equals the terminology from which axioms are to be extracted, a reflexive projection setting of the form ρ =hT, Σ,T iis used. We call a projection module underρ also anautomor- phic projection module.
An interesting application of the projection module is to compare entailment capacities of two terminologies, as shown in the following example.
Example 2. LetT1={AivAi+1|1≤i≤n}, T2={A1vAn, B1vB2}, Σ= {A1, An, B1, B2}. Intuitively,T1contains less information aboutΣthanT2. The minimal projection module underρ=hT1, Σ,T2iisM={A1vAn}. By using
|M|/|T2|= 1/2 as a measure, we see that only half ofT2is about the information ofT1 w.r.t.Σ.
4 Representing Projection Modules using Justifications
In this section, we introduce justification notions for sets of inclusions and we show how they can be combined to obtain minimal projection modules. We start with defining the notion of role subsumption justifications for a set ofELran,u,u- inclusions of the formrvs, wherer, s∈Σ (cf. Section 2).
Definition 3 (Role Subsumption Justification). Letρ=hT1, Σ,T2i. A set M is called a role subsumption module under ρ iff M ⊆ T2 and for every r, s∈NR∩Σ,T1|=rvs impliesM |=rvs. A role subsumption justification under ρis the role subsumption module underρthat is minimal w.r.t. (.
We denote the set of all role subsumption justifications underρas JρR. Lemma 1. Let J ∈ JρR. ThenroleWtnΣ(T1, J) =∅.
We continue with defining the notion of subsumption justifications for in- clusions of ELran,u,u that are of the form D v F, where D ranges over ELran- concepts andF overELu,u-concepts.
Definition 4 (Subsumption Justification).A subsumption setting is a tuple χ=hT1, X1, Σ,T2, X2i, whereT1 andT2 are normalisedELHr-terminologies,Σ is a signature,X1, X2∈NC∪ {dom(r),ran(r)|r∈NR}.
A set M is called a subsumer module under χ iff M ⊆ T2 and for every C ∈ ELran,u,uΣ , T1 |=X1 vC implies M |=X2 vC. M is called a subsumee module under χiff M ⊆ T2 and for every C∈ ELran,u,uΣ ,T1 |=CvX1 implies M |=CvX2.
M is called a subsumption module under χ iff M is a subsumer module, a subsumee module under χ and role subsumption module under hT1, Σ,T2i. A subsumee (resp. subsumer, subsumption) justification under χ is a subsumee (resp. subsumer, subsumption) module underχ that is minimal w.r.t. (.
We denote the set of all subsumee (resp. subsumer, subsumption) justifica- tions underχas Jχ←(resp.Jχ→,Jχ), whereχ=hT1, X1, Σ,T2, X2i.
Using Definition 1 and 4, we obtain the following proposition stating the ab- sence of certain concept names, and domain and range restrictions of role names as left-hand and right-hand difference witnesses between a reference terminol- ogyT1 and a subsumer and subsumee justification of a second terminologyT2.
For a signatureΣ, letΣdom={dom(t)|t∈NR∩Σ} andΣran={ran(t)| t∈NR∩Σ}be the sets consisting of concepts of the formdom(t) andran(t) for every role nametinΣ, respectively. Furthermore, letΣζ =Σ∪Σdom∪Σranfor ζ∈ CΣ.
Lemma 2. Let ϕ∈(Σ∩NC)∪Σdom∪Σran and letA∈Σ∩NC. Additionally, letχ=hT1, ϕ, Σ,T2, ϕiandχ0=hT1, A, Σ,T2, Ai. Then:
– ϕ6∈lhsWtnΣ(T1, Jχ)for every Jχ∈ Jχ→; – A6∈rhsWtnΣ(T1, Jχ0)for every Jχ0∈ Jχ←0.
To obtain subsumption modules we can use an operator⊗to combine sets of role, subsumer and subsumee justifications, one justification for each potential difference witness that needs to be prevented; cf. Lemmas 1 and 2. Given a set S andS1,S2⊆2S,S1⊗S2:={S1∪S2|S1∈S1, S2∈S2}. For instance, ifS1= {{α1, α2},{α3}} and S2 = {{α1, α3},{α4, α5}}, then S1⊗S2 ={{α1, α2, α3}, {α1, α2, α4, α5},{α3, α4, α5},{α1, α3}}. For a setMof sets, we define a function Minimise⊆(M) as follows: M ∈Minimise⊆(M) iffM ∈M and there does not exist a set M0 ∈ M such that M0 ( M. Continuing the previous example, Minimise⊆(S1⊗S2) ={{α2, α3},{α1, α2, α4, α5},{α3, α4, α5}}.
We now use⊗and Minimise⊆(·) to combine sets of role, subsumer and sub- sumee justifications to obtain the set of all minimal projection modules.
Theorem 2. Let Mρ be the set of all projection modules underρ=hT1, Σ,T2i that are minimal w.r.t.(. Then the following holds, whereχ(ψ) =hT1, ψ, Σ,T2, ψi:
Mρ =M inimize⊆ JρR ⊗ O
ϕ∈(Σ∩NC)∪Σdom∪Σran
Jχ(ϕ)→ ⊗ O
A∈Σ∩NC
Jχ(A)←
5 Computing Projection Modules
In this section, we present algorithms for computing role, subsumer and sub- sumee justifications. The algorithms use the following notion of a cover of a set of sets. For a finite setSand a setT⊆2S, we say that a setM⊆2S is acover of TiffM⊆Tand for everyM ∈T, there existsM0 ∈Msuch that M0⊆ M. In other words, a cover is a subset ofTcontaining all sets fromTthat are minimal w.r.t.(. Therefore, a cover of the set of all subsumption modules also contains all subsumption justifications. We will use covers to characterise the output of our algorithms to ensure that all justifications have been computed.
Algorithm 7 shows how to collect relevantΣ-role inclusions for role subsump- tion justifications (cf. Def. 3). The following proposition states its correctness.
Proposition 1. JρR=CoverR(T1, Σ,T2), whereρ=hT1, Σ,T2i.
5.1 Computing Subsumer Justifications
We now present our algorithm for computing subsumer justifications. The al- gorithm relies on the notion of a subsumer simulation between terminologies from [11, 23]. For defining the simulation notion, we need a specific notion of reachability. For ϕ∈NC∪ {dom(r),ran(r)|r ∈NR} and a normalisedELHr- terminologyT, letFT(ϕ) be the smallest set closed under the following three con- ditions:ϕ∈FT(ϕ);Y ∈FT(ϕ) ifX∈FT(ϕ),T |=X vX0andX0./∃r.Y ∈ T; anddom(r)∈FT(ϕ) ifran(r)∈FT(ϕ).
Definition 5 (Subsumer Simulation).A relation S⊆N(T1, Σ)×N(T2, Σ), whereN(T, Σ) ={X,dom(r),ran(r)|X, r∈(sig(T)∪Σ), X∈NC, r∈NR}, is a Σ-subsumer simulation fromT1 toT2iff the following conditions are satisfied:
(SN→C) if (X1, X2)∈ S, then for every ϕ∈Σ∪Σdom with T1 |=X1 vϕ, it holds thatT2|=X2vϕ;
(S→∃) if (X1, X2)∈ S and X10 ./1 ∃r.Y1 ∈ T1 with ./1 ∈ {v,≡} such that T1 |= X1 vX10 andT1 |=r vs for some s∈Σ, there exists X20 ./2 ∃r0.Y2 ∈ T2
with ./2 ∈ {v,≡} such that T2 |= X2 v X20 and for every s ∈ Σ with T1|=rvs, it holds that T2|=r0 vs and(Y1, Y2)∈S.
We writeT1∼→Σ T2 iff there exists aΣ-subsumer simulationS fromT1 toT2 such that for every ϕ ∈ (Σ∩NC)∪Σdom∪Σran: (ϕ, ϕ) ∈ S, and for every ψ1∈FT1(ϕ), there exists aψ2∈FT2(ϕ)such that(ψ1, ψ2)∈S.
For X1, X2 ∈ NC, we write hT1, X1i ∼→Σ hT2, X2i iff there exists a Σ- subsumer simulationS fromT1 toT2 with (X1, X2)∈S for whichT1∼→Σ T2.
A subsumer simulation conveniently captures the set of subsumers in the following sense: If a Σ-subsumer simulation from T1 to T2 contains the pair (X1, X2), thenX2entails w.r.t.T2all subsumers ofX1w.r.t.T1that are formu- lated in the signature Σ. Formally, we obtain the following theorems from [23].
Theorem 3. It holds that T1∼→Σ T2 ifflhsWtnΣ(T1,T2) =∅.
Theorem 4. Let hT1, X1i ∼→Σ hT2, X2i. Then for every D ∈ ELran,u,uΣ : T1 |= X1vD impliesT2|=X2vD.
Algorithm 2 collects all the axioms necessary to satisfy the conditions of Def- inition 5. Observe that Cover→(T1, X1, Σ,T2, X2) may be called several times during the execution of the algorithm. A possible optimisation is to store return values in memory in order to retrieve them more quickly for subsequent calls.
The following theorem shows that Algorithm 2 indeed computes the set of subsumer modules, thus producing a cover of subsumer justifications.
Theorem 5. Letχ=hT1, X1, Σ,T2, X2iandM:=Covercq→(T1, X1, Σ,T2, X2).
If T1∼→Σ T2, thenM is a cover of the set of subsumer justifications underχ.
5.2 Computing Subsumee Justifications
Next we present the algorithm for computing subsumee justifications based on a notion of a subsumee simulation. The basic idea of the algorithm is to collect as few axioms fromT2as possible to maintain the subsumee simulation between Σ-concept names.
First we present some auxiliary notions for handling conjunctions on the left-hand side of subsumptions. We define for each concept name X a so-called definitorial forest consisting of sets of axioms of the form Y ≡ Y1u. . .uYn which can be thought of as forming trees. Any subsumee justification under hT1, X1, Σ,T2, X2icontains the axioms of a selection of these trees, i.e., one tree for every conjunction formulated overΣ that entailsX1 w.r.t.T1. Formally, we define a set of a DefForestTu(X) ⊆2T to be the smallest set closed under the following conditions: ∅ ∈ DefForestTu(X); {α} ∈ DefForestTu(X) for α = X ≡ X1u. . .uXn ∈ T; andΓ ∪ {α} ∈DefForestTu(X) forΓ ∈DefForestTu(X) with Z ≡Z1u. . .uZk∈Γ andα=Zi≡Zi1u. . .uZin∈ T. GivenΓ ∈DefForestTu(X), we set leaves(Γ) := sig(Γ)\ {X ∈ sig(C) | X ≡ C ∈ Γ} if Γ 6= ∅; and {X}otherwise. We denote the maximal element of DefForestTu(X) w.r.t.⊆with max-treeTu(X). Finally, we set non-conjT(X) := leaves(max-treeTu(X)).
For example, letT ={α1, α2, α3}, whereα1:=X ≡YuZ,α2:=Y ≡Y1uY2, and α3 := Z ≡ Z1uZ2. Then DefForestTu(X) = {∅,{α1},{α1, α2},{α1, α3}, {α1, α2, α3}}. We have that leaves({α1, α3}) = {Y, Z1, Z2}, max-treeTu(X) = {α1, α2, α3}, and non-conjT(X) ={Y1, Y2, Z1, Z2}.
We say that a concept nameA is Σ-entailed w.r.t.T iff there is an ELranΣ - conceptC such thatT |=CvA; and we say that a role names isΣ-entailed in T iff there existss0∈NR∩Σ such thatT |=s0vs.
We now define the notion of asubsumee simulationfromT1toT2as a subset of sigNC(T1)×sigNC(T2)× CTΣ
1, whereCTΣ
1 :={} ∪(NR∩(Σ∪sig(T1))) is the range of role contexts.
Definition 6 (Subsumee Simulation).A relationS⊆sigNC(T1)×sigNC(T2)×
CTΣ1 is a Σ-subsumee simulation from T1 toT2 iff the following conditions hold:
(SN←C) if(X1, X2, ζ)∈S, then for everyϕ∈Σζ and for everyX20 ∈non-conjT2(X2) withT26|=ran(ζ)vX20,T1|=ϕvX1 impliesT2|=ϕvX20;
(S←∃) if (X1, X2, ζ) ∈ S and X1 ≡ ∃r.Y1 ∈ T1 such that T1 |= s v r for s ∈ Σ and Y1 is Σ-entailed w.r.t. T1, then for every X20 ∈ non-conjT
2(X2) not entailed by dom(s) or ran(ζ) w.r.t. T2, there exists X20 ≡ ∃r0.Y2 ∈ T2 such thatT2|=svr0 and(Y1, Y2, s)∈S;
(S←u) if (X1, X2, ζ) ∈ S and X1 ≡ Y1 u. . .uYn ∈ T1, then for every Y2 ∈ non-conjT2(X2)not entailed byran(ζ)inT2, there existsY1∈non-conjT1(X1) not entailed byran(ζ)w.r.t. T2 such that(Y1, Y2, )∈S.
We write T1 ∼←Σ T2 iff there is a Σ-subsumer simulation S from T1 to T2
such that for everyA∈Σ∩NC:(A, A, )∈S.
Forζ ∈Σ∩NR, we write hT1, X1i ∼←Σ,ζ hT2, X2i iff there is aΣ-subsumer simulationS fromT1 toT2 with (X1, X2, ζ)∈S for whichT1∼←Σ T2.
Analogously to subsumer simulations, a subsumee simulation captures the set of subsumees as it is made precise in the following theorems.
Theorem 6. T1∼←Σ T2 iffrhsWtnΣ(T1,T2) =∅.
Theorem 7. Let hT1, X1i ∼←Σ,ζ hT2, X2i. Then for every C∈ ELranΣ :T1|=Cv X1 implies thatT2|=CvX2.
Before introducing the algorithms, we first extend the notion ofΣ-entailment.
We say that a concept name X is complex Σ-entailed w.r.t. T iff for every Y ∈non-conjT(X) one of the following conditions holds:
– there existsB∈Σ such thatT |=BvY and T 6|=BvX; or – there existsY ≡ ∃r.Z∈ T andr, Z areΣ-entailed in T.
Otherwise, X is said to be simply Σ-entailed. For example, let T = {X ≡ X1uX2, B1vX1, X2≡ ∃r.Z, B2 vZ, svr}. We have that non-conjT(X) = {X1, X2}, then r is Σ-entailed w.r.t. T; X is complexΣ-entailed w.r.t. T for Σ={B1, B2, s}; butXis not complexΣ0-entailed w.r.t.T, whereΣ0ranges over {B1, B2},{B1, s},{B2, s}. Additionally,X is not complexΣ-entailed w.r.t.T ∪ {B1vX}.
Algorithm 3 is responsible for computing a cover of all subsumee justifica- tions. It orchestrates three further algorithms in order to collect the axioms nec- essary to satisfy the three conditions of Definition 6: Algorithm 4 for Case (S←N
C), Algorithm 5 for Case (S←∃ ) and Algorithm 6 for Case (Su←). While Algorithm 4 can readily be understood, we provide some additional explanation for the re- maining algorithms.
Algorithm 1:Computing a Cover of all Sub- sumer Justifications for Conjunctive Queries
1 functionCovercq→(T1, X1, Σ,T2, X2)
2 M→(X1,X2):={∅}
3 for everyψ1∈FT1(X1)do
4 M→ψ1:={∅}
5 for everyψ2∈FT2(X2)such that hT1, ψ1i ∼→Σ hT2, ψ2ido
6 M→ψ1,ψ2:=Cover→(T1, ψ1, Σ,T2, ψ2)
7 M→ψ1:=M→ψ1∪M→ψ1,ψ2 8 M→(X1,X2)=M→(X1,X2)⊗M→ψ1 9 returnMinimise⊆(M→(X1,X2))
Algorithm 2:Computing a Cover of all Sub- sumer Justifications (Recursive)
1 functionCover→(T1, X1, Σ,T2, X2)
2 M→(X1,X2):={∅}
3 for everyB∈(Σ∩NC)∪ {dom(r)|r∈Σ} such thatT1|=X1vBdo
4 M→(X1,X2):=M→(X1,X2)⊗JustT2(X2vB)
5 for everyY ./1∃r.Z∈ T1(./1∈ {v,≡}) withT1|=X1vY,T1|=rvsfor some s∈Σ∩NRdo
6 M→∃r.Z:={∅}
7 for everyY0./2∃r0.Z0∈ T2(./2∈ {v,≡}) withT2|=X2vY0andT2|=r0vsfor everys∈ {s0∈Σ∩NR| T1|=rvs0} andhT1, Zi ∼→ΣhT2, Z0ido
8 M→r0:={∅}
9 for everys∈Σ∩NRwithT1|=rvs do
10 M→r0:=M→r0⊗JustT2(r0vs)
11 M→Z0:=Cover→(T1, Z, Σ,T2, Z0)
12 M→∃r.Z:=M→∃r.Z∪ JustT2(X2vY0)
⊗ {{Y0./2∃r0.Z0}} ⊗M→r0⊗M→Z0
13 M→(X1,X2):=M→(X1,X2)⊗M→∃r.Z 14 returnM→(X1,X2)
Algorithm 3:Computing a Cover of all Sub- sumee Justifications
1 functionCovercq←(T1, X1, Σ,T2, X2, ζ)
2 ifX1is notΣ-entailedw.r.t.T1then
3 return{∅}
4 M←(X1,X2):=CoverN←C(T1, X1, Σ,T2, X2, ζ)
5 ifX1is not complexΣ-entailed inT1then
6 returnM←(X1,X2)
7 ifX1≡ ∃r.Y∈ T1, andr, Y areΣ-entailed w.r.t.T1then
8 M←(X1,X2):=
M←(X1,X2)⊗Cover∃←(T1, X1, Σ,T2, X2, ζ)
9 else ifX1≡Y1u. . .uYm∈ T1then
10 M←(X1,X2):=
M←(X1,X2)⊗Coveru←(T1, X1, Σ,T2, X2, ζ)
11 returnMinimise⊆(M←(X1,X2))
Algorithm 4:Computing a Cover of all Sub- sumee Projection Justifications (S←NC)
1 functionCoverN←C(T1, X1, Σ,T2, X2, ζ)
2 M←(X1,X2):={∅}
3 foreveryB∈Σζsuch thatT1|=BvX1do
4 foreveryX2∈non-conjT2(X1)such that ζ=εorT2|=ran(ζ)vX2do
5 M←(X1,X2):=M←(X1,X2)⊗JustT2(BvX2)
6 returnM←(X1,X2)
Algorithm 5:Computing a Cover of all Sub- sumee Projection Justifications (S←∃)
1 functionCover∃←(T1, X1, Σ,T2, X2, ζ)
2 letαX1:=X1≡ ∃r.Y1∈ T1 3 M←(X1,X2):={max-treeTu2(X2)}
4 for everys∈Σ∩NRsuch thatT1|=svr do
5 for everyX20∈non-conjT2(X2) such that ζ6=εimpliesT26|=ran(ζ)vX20and T26|=dom(s)vX20 do
6 letαX0
2:=X20≡ ∃r0.Y20∈ T2
M←Y20:=Covercq←(T1, Y1, Σ,T2, Y20, s) M←(X1,X2):=M←(X1,X2)
⊗ {{αX02}} ⊗JustT2(svr)⊗M←Y20
7 returnM←(X1,X2)
Algorithm 6:Computing a Cover of all Sub- sumee Justifications (S←u)
1 functionCoveru←(T1, X1, Σ,T2, X2, ζ)
2 letαX1:=X1≡Y1u. . .uYm∈ T1 3 M←(X1,X2):=∅
4 for everyΓ∈DefForestTu2(X2)do
5 letδΓ:={defTu2(X0)|X0∈ leaves(Γ)∩defTu2}
6 M←Γ :={Γ}
7 for everyX20∈leaves(Γ) such thatζ=ε orT26|=ran(ζ)vX02do
8 M←X20:=∅
9 for everyX10∈non-conjT1(X1) such thatζ=εorT26|=ran(ζ)vX01do
10 ifhT1, X10i ∼←Σ,ζhT2\δΓ, X20ithen
11 M←X20:=M←X20∪
Covercq←(T1, X10, Σ,T2\δΓ, X02, )
12 M←Γ :=M←Γ ⊗M←X20 13 M←(X1,X2):=M←(X1,X2)∪M←Γ 14 returnM←(X1,X2)
Algorithm 7:Computing a Cover of all Sub- sumption Justifications for Role Inclusions
1 functionCoverR(T1, Σ,T2)
2 M={∅}
3 for everyr, s∈Σ∩NRsuch that T1|=rvsdo
4 M:=M⊗JustT2(rvs)
5 returnMinimise⊆(M)
Fig. 1.Algorithms of computing all subsumer and subsumee justifications
The existence of axiomαX1 :=X1≡ ∃r.Y1∈ T1 in Line 2 of Algorithm 5 is guaranteed by Line 7 of Algorithm 3. The axiomαX0
2 :=X20 ≡ ∃r0.Y20 ∈ T2 in Line 6 of Algorithm 5 exists as we assume thatX2inT2“subsumee-simulates”X1 inT1. Moreover, there is at most one axiomαX1 ∈ T1and at most oneαX0
2 ∈ T2
as T1 and T2 are terminologies. The concept name X2 may be defined as a conjunction inT2 whose conjuncts in turn may also be defined as a conjunction inT2and so forth. In Line 3 all axioms forming the maximal resulting definitorial conjunctive tree are collected.
For the next algorithm, we define defTu:={X ∈sigNC(T)|X≡Y1u. . .uYn∈ T }to be the set of concept names that are conjunctively defined inT. For every X ∈defTu, we set defTu(X) :=α, whereα=X ≡Y1u. . .uYn∈ T.
The axiom αX1 := X1 ≡ Y1 u. . .uYm ∈ T1 in Line 2 of Algorithm 6 is guaranteed by Line 9 of Algorithm 3. In case X2 is defined as a conjunc- tion in T2, the pair consisting of T2 containing only a partial conjunctive tree rooted atX2andX2needs to be considered to be sufficient to “subsumee simu- late”X1inT1. Therefore Algorithm 3 considers every partial conjunctive treeΓ from DefForestTu
2(X2) in Line 4 and removes the axioms inδΓ connecting the leaves ofΓ with the remaining conjunctive tree fromT2 in lines 10 and 11.
The following theorem shows that Algorithm 3 indeed computes a cover of the set of subsumee modules. Thus every subsumee justification is guaranteed to be among the computed sets of axioms.
Theorem 8. Let χ=hT1, ϕ1, Σ,T2, ϕ2i andϕ1, ϕ2 ∈(Σ∩NC)∪Σdom∪Σran. Additionally, letM:=Covercq←(T1, ϕ1, Σ,T2, ϕ2, ). If T1∼←Σ T2, thenMis the set of all subsumee justifications underχ.
The number of (minimal) projection justifications depends onT1,T2andΣ.
In general, this number is bounded by an exponential in the size of the termi- nologies. The simulation checks can be performed in polynomial time [11, 23].
Our algorithm for computing projection justifications, therefore, runs in time exponential in the size of the input.
6 Conclusion
We introduce the notion of projection module for conjunctive queries as a subset of an ontology capturing the knowledge about a given signature as specified in a reference ontology. Here knowledge about a signature means the set of entailed conjunctive queries about the signature. This allows comparing ontologies in a more fine-grained fashion compared to merely extracting modules. Projection modules enable us to check how knowledge is implemented in terms of axioms in different ontologies. In particular, we can verify that and how specifications as defined in reference ontologies have been realised. We have presented algo- rithms for computing projection modules of acyclic ELHr-terminologies w.r.t.
conjunctive queries. Similar algorithms can be used to deal with projection mod- ules for concept subsumption queries and instance queries. We expect that the algorithms can be extended to deal with cyclic terminologies and even general ELHr-TBoxes.
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