On the Complexity of Semantic Integration of OWL Ontologies
Yevgeny Kazakov1, Denis Ponomaryov2
1 Institute of Artificial Intelligence, University of Ulm, Germany
2 Institute of Informatics Systems, Novosibirsk State University, Russia yevgeny.kazakov@uni-ulm.de, ponom@iis.nsk.su
Abstract. We propose a new mechanism for integration of OWL ontologies us- ing semantic import relations. In contrast to the standard OWL importing, we do not require all axioms of the imported ontologies to be taken into account for reasoning tasks, but only their logical implications over a chosen signature. This property comes natural in many ontology integration scenarios. In this paper, we study the complexity of reasoning over ontologies with semantic import relations and establish a range of tight complexity bounds for various fragments of OWL.
1 Introduction and Motivation
Ontology integration is a research topic which, in particular, aims at organizing in- formation on different domains in a modular way so that information from one ontology can be reused in other ontologies. There is a number of approaches known in the liter- ature [1–5, 8], which address this problem from the point of view of ontology linking and importing, see, e.g., an overview in [6]. Integration of multiple ontologies in OWL is organized viaimporting: an OWL ontology can refer to one or several other OWL ontologies, whose axioms must be implicitly present in the ontology. The importing mechanism is simple in that reasoning over an ontology with an import declaration is reduced to reasoning over theimport closureconsisting of the axioms of the ontology plus the axioms of the ontologies that are imported (possibly indirectly). For example, if ontologyO1imports ontologiesO2andO3, each of which, in turn, imports ontology O4, then the import closure ofO1consists of all axioms ofO1− O4.
Although the OWL importing mechanism may work well for simple ontology inte- gration scenarios, it may cause some undesirable side effects if used in complex import situations. To illustrate the problem, suppose that in the above example,O4is an ontol- ogy describing a typical university. It may include concepts such asProfessor,Course, and axioms stating, e.g., that each professor must teach some course: Professor v
∃teaches.Course. Now suppose thatO2andO3are ontologies describing respectively, Oxford and Cambridge universities that useO4as a prototype. For example,O2may includemapping axioms OxfordProfessor≡Professor,OxfordCourse≡Course, from which, due to the axiom inO4, it is now possible to conclude thatOxfordProfessorv
∃teaches.OxfordCourse. Likewise, using similar mapping axioms in O3, it is possi- ble to obtain that CambridgeProfessor v ∃teaches.CambridgeCourse. Finally, sup- pose thatO1is an ontology aggregating information about UK universities, importing, among others, the ontologiesO2andO3for Oxford and Cambridge universities. Al- though the described scenario seems plausible, there will be some undesirable conse- quences inO1due to the mapping axioms ofO2andO3 occurring in the import clo- sure: OxfordProfessor ≡ CambridgeProfessor,OxfordCourse ≡ CambridgeCourse.
The main reason for these consequences is that the ontologiesO2 andO3 happen to reuse the same ontologyO4in two different and incompatible ways. Had they instead used two different ‘copies’ ofO4as prototypes (with concepts renamed apart), no such problem would take place. Arguably, the primary purpose ofO2andO3is to provide semantic description of the vocabulary for Oxford and Cambridge universities, and the means of how it is achieved—either by writing the axioms directly or reusing third party ontologies such asO4—should be an internal matter of these two ontologies and should not be exposed to the ontologies that import them. Motivated by these observations, in this paper we consider a refined mechanism for importing of OWL ontologies called semantic importing. The main difference with the standard OWL importing is that each import is limited only to a subset of symbols. Intuitively, only logical properties of these symbols entailed by the imported ontology should be imported. These symbols can be regarded as the public (or external) vocabulary of the imported ontologies. For example, ontologyO2may declare the symbolsOxfordProfessor,OxfordCourse, and teachespublic, leaving the remaining symbols only for the internal use. Importantly, in our approach every ontology has its own view on ontologies it imports and the views are independent between ontologies unless coordinated by the ‘topology’ of import re- lations.
The main results of this paper are tight complexity bounds for reasoning over on- tologies with semantic imports. We consider ontologies formulated in different frag- ments of OWL starting from the Description Logic (DL)ELand concluding with the DLSROIQ, which corresponds to OWL 2. We also distinguish the case of acyclic im- ports, when ontologies cannot (possibly indirectly) import themselves. Our complete- ness results for ranges of DLs are summarized in the following table, whereaandc denote the case of acyclic/cyclic imports:
DLs Completeness Theorems
EL–EL++ ExpTimea 1, 8 containingEL RE(undecidable)c 2, 9 ALC–SHIQ 2ExpTimea 3, 8 R–SRIQ 3ExpTimea 4, 8 ALCHOIF–SHOIQ coN2ExpTimea 5, 8 ROIF–SROIQ coN3ExpTimea 6, 8
An extended version of the paper containing all proofs and additional results is available at https://arxiv.org/abs/1705.04719
2 Preliminaries
We assume that the reader is familiar with the family of Description Logics from ELtoSROIQ, for which the syntax is defined using a recursively enumerable al- phabet consisting of infinite disjoint sets NC,NR,Ni of concept names(or primitive concepts),roles, andnominals, respectively. The semantics of DLs is given by means of (first-order) interpretations. An ontology is a set of concept inclusions which are called ontologyaxioms. Asignatureis a subset ofNC∪NR∪Ni. InterpretationsIand J are said toagreeon a signatureΣ, written asI =Σ J, if the domains ofI andJ coincide and the interpretation ofΣ-symbols inI is the same as inJ. We denote the reduct of an interpretationIonto a signatureΣasI|Σ. The signature of a conceptC,
denoted assig(C), is the set of all concept names, roles, and nominals occurring inC.
The signature of a concept inclusion or an ontology is defined identically.
Given a signatureΣ, suppose one wants to import into an ontologyO1the seman- tics ofΣ-symbols defined by some other ontologyO2, while ignoring the rest of the symbols fromO2. Intuitively, importing the semantics ofΣ-symbols means reducing the class of models ofO1 by removing those models that violate the restrictions on interpretation of these symbols, which are imposed by the axioms ofO2:
Definition 1. A(semantic) import relationis a tupleπ=hO1, Σ,O2i, whereO1and O2 are ontologies andΣ a signature. In this case, we say that O1 importsΣ from O2. We say that a modelI |=O1satisfies the import relationπif there exists a model J |=O2such thatI=Σ J.
Example 1. Consider the import relationπ = hO1, Σ,O2i, withO1 = {B v C}, O2 ={Av ∃r.B,∃r.C vD}, andΣ ={A, B, C, D}. It can be easily shown using Definition 1 that a modelI |=O1satisfiesπif and only ifI |=AvD.
Note that ifΣcontains all symbols inO2thenI |=O1satisfiesπ=hO1, Σ,O2i if and only ifI |=O1∪ O2. That is, the standard OWL import relation is a special case of the semantic import relation, when the signature contains all the symbols from the imported ontology. IfOhas several import relationsφi = hO, Σi,Oii,(1≤ i≤n), one can define the entailment fromOby considering only those models ofOthat satisfy all imports: O |= αif I |= α for everyI |= Owhich satisfies all π1, . . . , πn. In practice, however, import relations can be nested: imported ontologies can themselves import other ontologies and so on. The following definition generalizes entailment to such situations.
Definition 2. Anontology networkis a finite setN of import relations between on- tologies. For a DLL, aL-ontology networkis a network, in which every ontology is a set ofL-axioms. Amodel agreementforN (over a domain∆) is a mappingµthat assigns to every ontologyOoccurring inN a classµ(O)of models ofOwith domain
∆such that for everyhO1, Σ,O2i ∈ N and everyI1∈µ(O1)there existsI2∈µ(O2) such thatI1 =Σ I2. An interpretationI is a model ofOin the networkN (notation I |=N O) if there exists a model agreementµforN such thatI ∈ µ(O). An ontol- ogyOentails a concept inclusionϕin the networkN (notationO |=N ϕ) ifI |=ϕ, wheneverI |=N O.
An ontology network can be seen as a labeled directed multigraph in which nodes are labeled by ontologies and edges are labeled by sets of signature symbols. Note that Definition 2 also allows forcyclic networksif this graph is cyclic. Note that ifO |=ϕ thenO |=N ϕ, for every networkN.
In this paper, we are concerned with the complexity ofentailment in ontology net- works, that is, given a networkN, an ontologyO and an axiom ϕ, decide whether O |=N ϕ. We study the complexity of this problem wrt thesizeof an ontology network N, which is defined as the total length of axioms (considered as strings) occurring in ontologies fromN.
3 Expressiveness of Ontology Networks
First, we illustrate the expressiveness of ontology networks by showing that acyclic networks allow for succinctly representing axioms with nested concepts and role chains
of exponential size, while cyclic ones allow for succinctly representing infinite sets of axioms of a special form. The lemmas given in this Section are used as building blocks in the proofs of our hardness results in Section 4.
For a natural numbern > 0, let∃(r, C)n.Dbe a shortcut for the nested concept
∃r.(Cu ∃r.(Cu. . .n times. . .u ∃r.(CuD). . .)), whereC,Dare DL concepts and ra role (in casen = 0the above concept is set to beD). Forn>1, let(r)n denote the role chainr◦. . .n times. . .◦r. For a givenn > 0, let1exp(n)be the notation for2n and fork > 1, let(k+1)exp(n) = 2kexp(n). Then∃(r, C)kexp(n).D(respec- tively,(r)kexp(n)) stands for a nested concept (role chain) of the form above having size exponential inn. In the following, we use abbreviations∃(r, C)n :=∃(r, C)n.>and
∃rn.C:=∃(r,>)n.C. Forn>1, the expression∀rn.Cwill be used as a shortcut for
¬∃r.∃rn−1.¬Cand forn > 2,∀r<n.C will stand for
F 16m<n∀rm.C. LetObe an ontology andN an ontology network.Ois said to beexpressiblebyN if there is an ontologyONinN such that{I|sig(O)| I |=N ON}={J |sig(O)| J |=O}. In case we want to stress the role of ontologyONin the networkN, we say thatOis(N,ON)- expressible. An axiomϕis expressible by a networkN if so is ontologyO={ϕ}. The next two lemmas follow immediately from the definition of expressibility.
Lemma 1. Every ontologyOis(N,O)-expressible, whereN is the network consisting of the single import relationhO,∅,∅i. If an ontologyOiis(Ni,Oi0)-expressible, for a networkNi, ontologyO0i, andi = 1,2, thenO1∪ O2is(N,ON)-expressible, for ontologyON =∅and a network1N =N1∪ N2∪ {hON,sig(Oi),O0ii}i=1,2.
For an axiom ϕ and a set of concepts {C1, . . . , Cn}, n > 1, let us denote by ϕ[C1 7→ D1, . . . , Cn 7→ Dn] the axiom obtained by substituting every concept Ci
with a concept Di inϕ. For an ontology O, let O[C1 7→ D1, . . . , Cn 7→ Dn] be a notation forS
ϕ∈Oϕ[C17→D1, . . . , Cn7→Dn].
Lemma 2. LetLbe a DL and Oan ontology, which is expressible by aL-ontology networkN. LetC1, . . . , Cn beL-concepts and{A1, . . . , An}a set of concept names such thatAi ∈ sig(O), fori = 1, . . . , nandn > 1. Then ontologyO˜ = O[A1 7→
C1, . . . , An 7→Cn]is expressible by aL-ontology network, which is acyclic if so isN and has size polynomial in the size ofN andCi,i= 1, . . . , n.
Proof Sketch. Denote Σ = {A1, . . . , An} and let Σ0 = {A01, . . . , A0n} be a set of fresh concept names. Consider ontologyO0 = O ∪ {Ai ≡ A0i}i=1,...,n. By Lemma 1,O0 is(N0,ON0)-expressible, for an ontologyON0 and an acyclicL-ontology net- workN0 having a linear size (in the size of N). Consider ontology networkN00 = hON00,(sig(O0)\Σ)∪Σ0,ON0i, whereON00=∅. Then obviously, ontologyO00= O[A1 7→A01, . . . , An 7→ A0n]is(N00,ON00)-expressible. Similarly, by Lemma 1, on- tologyO00C = O00∪ {A0i ≡Ci}i=1,...,nis( ˜N,ON˜)-expressible, for an ontologyON˜
and an acyclicL-ontology networkN˜ having a linear size (in the size ofN andCi, for i= 1, . . . , n). Clearly, it holdsON˜ |=N˜ O˜and it is easy to show thatO˜is( ˜N,ON˜)- expressible.
Now we formulate the results on the expressiveness of acyclicEL-,ALC-, andR- ontology networks.
1Note that the size ofNis linear in the sizes ofN1andN2.
Lemma 3. An axiomϕof the formZ ≡ ∃(r, A)1exp(n).B, whereZ, A, B ∈ NC, and n>0, is expressible by an acyclicEL-ontology network of size polynomial inn.
Proof Sketch. We show by induction onn that there exists an acyclic EL-ontology networkNn and ontology On such that ϕis(Nn,On)-expressible. Forn = 0, we defineN0 = {hO0,∅,∅i} and O0 = {Z ≡ ∃r.(AuB)}. In the induction step, let {Z ≡ ∃(r, A)1exp(n−1).B} be (Nn−1,On−1)-expressible, for n > 1. Consider ontologiesOcopy1 ={B ≡ U},Ocopy2 = {U ≡ Z},On = ∅. LetNn be the union ofNn−1 with the set of the following import relations:hOcopyi ,{Z, A, B, r},On−1i, for i = 1,2,hOn,{Z, A, U, r},Ocopy1 i, andhOn,{U, A, B, r},O2copyi. Let us verify that {I|sig(ϕ) | I |=Nn On} = {I|sig(ϕ) | I |= ϕ}. By the induction assumption we haveOn−1 |=Nn Z ≡ ∃(r, A)1exp(n−1).B. Then by the definition ofNn, it holds O1copy |=Nn Z ≡ ∃(r, A)1exp(n−1).U andOcopy2 |=Nn U ≡ ∃(r, A)1exp(n−1).B and thus,On |=Nn {Z ≡ ∃(r, A)1exp(n−1).U, U ≡ ∃(r, A)1exp(n−1).B}, which yields On |=Nn ϕ. To complete the proof, we show that any modelI |=ϕcan be expanded to a modelJn|=Nn Onby settingUJn= (∃(r, A)1exp(n−1).B)I.
The following claim is proved identically to Lemma 3:
Lemma 4. An axiom of the formZv ∃(r, A)1exp(n).B,Z≡ ∀r1exp(n).A, or(r)2exp(n) vs, whereZ, A, B ∈NC,r, sare roles, andn>0, is expressible by an acyclicEL-, ALC-, andR-ontology network, respectively, having size polynomial inn.
We continue with results on the expressiveness of acyclicR-ontology networks.
Lemma 5. An axiomϕof the formZ v ∀r2exp(n).A, whereZ, A∈NCandn>0, is expressible by an acyclicR-ontology network of size polynomial inn.
Proof. Consider ontologyO = {Z v ∀s.A, (r)2exp(n) v s}. Clearly, it holds O |= ϕ and any model I |= ϕcan be expanded to a model J |= O by setting sJ = ((r)2exp(n))I. By Lemmas 1, 4,Ois expressible by an acyclicR-ontology net- work of size polynomial inn, from which the claim follows.
Similarly, we demonstrate that an axiom of the formZ v ∀r<2exp(n).A, where Z, A∈NCandn>0, is expressible by an acyclicR-ontology network of size polyno- mial innand show the next statement, which is an analogue of Lemma 4 for the case of double exponent.
Lemma 6. An axiomϕ of the form Z v ∃(r, A)2exp(n).B, whereZ, A ∈ NC and n>0, is expressible by an acyclicR-ontology network of size polynomial inn.
Proof. Consider ontologyO¯ ={Z v ∃s.>, Z v ∀s<2exp(n).X, Z v ∀s2exp(n).Y, s v r}. By Lemmas 1, 5, and the claim above, we show thatO¯ is expressible by an acyclicR-ontology network of size polynomial inn. Then by Lemma 2, so is ontology O = ¯O[X 7→ Au ∃s.>, Y 7→ AuB]. Clearly, we have O |= ϕ. Now letI be an arbitrary model ofϕ and for m = 2exp(n), let x0, . . . , xm be arbitrary domain elements such thatx0∈ZI,hx0, x1i ∈rI, andhxi, xi+1i ∈rI,xi ∈AI, for16i <
m, andxm∈AIuBI. LetJ be an expansion ofIin whichsJ ={hxi, xi+1i}06i<m. Then we haveJ |=O, from which the claim follows.
Lemma 7. An axiomϕof the formZ ≡ ∀r2exp(n).A, whereZ, A∈NCandn>0, is expressible by an acyclicR-ontology network of size polynomial inn.
Proof. Consider ontologyO¯ ={Z v ∀r2exp(n).A, Z¯ v ∃r2exp(n).A}. By Lemmas¯ 1, 5, 6,O¯is expressible by an acyclicR-ontology network of size polynomial innand by Lemma 2, so is ontologyO = ¯O[ ¯Z 7→ ¬Z, A¯ 7→ ¬A]. It remains to note thatO and{ϕ}are equivalent, so the claim is proved.
We conclude this section with lemmas formulating the expressibility of substitutions by ontology networks.
Lemma 8. LetLbe a DL and Oan ontology, which is expressible by aL-ontology networkN. LetC1, . . . , CmbeL-concepts and{A1, . . . , Am}a set of concept names such thatAi ∈sig(O), fori = 1, . . . , mandm >1. Then fork= 1,2andn>0, ontologyO˜ = O[A1 7→ ∀rkexp(n).C1, . . . , Am 7→ ∀rkexp(n).Cm]is expressible by a L0-ontology network, which is acyclic if so isN and has size polynomial in the size of N,n, and Ci, fori = 1, . . . , m, where:L0 = L ifLcontainsALC andk = 1, or L0 =LifLcontainsRandk= 2.
Proof. The proof uses Lemmas 4, 7 and is identical to the proof of Lemma 2.
The next statement can be shown similarly by using Lemma 3:
Lemma 9. In the conditions of Lemma 8, ontologyO˜ = O[A1 7→ ∃r1exp(n).C1, . . . , Am 7→ ∃r1exp(n).Cm], forn > 0, is expressible by aL0-ontology network, which is acyclic if so isNand has size polynomial in the size ofN,n, andCi, fori= 1, . . . , m, whereL0 =LifLcontainsEL.
Lemma 10. LetLbe a DL andOan ontology, which is(N,ON)-expressible, for aL- ontology networkN and an ontologyON. Let{A1, . . . , An},n>1, be concept names such thatAi∈sig(O), fori= 1, . . . , n, and let{C1, . . . , Cn}beL-concepts, where everyCiis of the form∃(r, D)p.Ai, for some roler, concept nameD, andp>1. Then ontologyO˜ = S
m>0Om, whereO0 =Oand Om+1 = Om[A1 7→ C1, . . . , An 7→
Cn], for allm>0, is expressible by a cyclicL-ontology network.
Proof Sketch.Letσ={B1, . . . , Bk}=S
i=1,...,n(sig(Ci)∩NC)andσ0 ={B10, . . . , Bk0}be a set of fresh concept names disjoint withσandsig(O). Let{C10, . . . , Cn0} be ‘copy’ concepts obtained fromC1, . . . , Cn by replacing everyBiwithB0i, fori= 1, . . . , k. Consider ontologiesO˜N0 ={Bi ≡Bi0}i=1,...,k,O0={Ai ≡Ci0}i=1,...,n, and an ontology networkN0given by the union ofN with the set of import relations hO˜N0,sig(O),ONi,hO˜N0, Σ0,O0i,hO0, Σ,O˜N0i, whereΣ0 = (Σ\σ)∪σ0andΣ= sig(O)∪S
i=1,...,nsig(Ci). We claim that ontology O˜ is (N0,O˜N0)-expressible.
DenoteO˜0 =S
m>0O0m, whereOm0 =Om[B1 7→ B10, . . . , Bk 7→ Bk0], for allm >
0. First, we show by induction that O˜N0 |=N0 Om, for all m > 0. The induction base for n = 0 is trivial, since we have O0 = O andO is(N,ON)-expressible, hO˜N0,sig(O),ONi ∈ N0, and thus,O˜N0 |=N0 O. SupposeO˜N0 |=N0 Om, for some m>0. SincehO0, Σ,O˜N0i ∈ N0, we haveO0|=N0 Omand thus, by the equivalences inO0, it holdsO0 |=N0 O0m+1. SincehO˜N0, Σ0,O0i, we haveO˜N0 |=N0 O0m+1and hence, by the equivalences inO˜N0, it holds thatO˜N0 |=N0 Om+1. Finally, in the full proof we show that any modelIofO˜can be expanded to a modelJ |=N0 O˜N0, which shows thatO˜is(N0,O˜N0)-expressible.
4 Hardness Results
We use reductions from the word problem for Turing machines (TMs) and alternat- ing Turing machines (ATMs) to obtain most of the hardness results. We define a Turing Machine (TM) as a tupleM =hQ,A, δi, withqh∈ Qbeing the accepting state, and assume w.l.o.g. that configuration ofMis a word in the alphabetQ∪ A. An initial con- figuration is a word of the formb. . .bq0b. . .b, whereq0∈Qandb∈ Ais the blank symbol. It is a well-known property of the transition functions of Turing machines that the symbolc0i at positioniof a successor configurationc0 is uniquely determined by a 4-tuple of symbolsci−2, ci−1, ci, ci+1at positionsi−2,i−1,i, andi+ 1of a config- urationc. We assume that this correspondence is given by the (partial) functionδ0and use the notationci−2ci−1cici+1 δ
0
7→c0i.
Theorem 1. Entailment in acyclicEL-ontology networks isExpTime-hard.
Proof Sketch.We reduce the word problem for TMs making exponentially many steps to entailment inEL-ontology networks. LetMbe a TM andn=1exp(m)an exponential, form>0. Consider an ontologyOdefined forM andnby the following axioms:
Av ∃rn·(2n+3).(q0u ∃(r,b)2n+2), whereA6∈Q∪ A (1)
∃r2n(Xu ∃r.(Y u ∃r.(Uu ∃r.Z)))vW, forXY U Z δ
0
7→W (2)
∃r.qhvH, ∃r.H vH, whereH 6∈Q∪ A (3)
The first axiom gives ar-chain containingn+ 1segments of length2n+ 3, which are used to store fragments of consequent configurations ofM. We assume the following enumeration of segments in the r-chain: . . .
|{z}sn
.. . . .
|{z}s1
q0b. . .b
| {z }
s0
, i.e.,s0 represents a fragment of the initial configurationc0ofM. For06i < n, everyi-th and(i+ 1)-st segments in ther-chain are reserved for a pair of configurationsci,ci+1such thatci+1
is a successor ofci. Axioms (2), withX, Y, U, Z, W ∈ Q∪ A, represent transitions of M and define the ‘content’ of (i+ 1)-st segment based on the ‘content’ of i-th segment. Finally, axioms (3) are used to initialise the halting markerH and propagate it to the ‘left’ of ther-chain. We show thatM accepts the empty word in nsteps iff O |=AvH. For the ‘only if’ direction we assume there is a sequence of configurations c0, . . . ,cn such that for all0 6 i < n,ci+1is a successor configuration ofciandqh is the state symbol in cn. LetI be a model of O anda domain element such that a ∈ AI. Then by axiom (1), there is an r-chain outgoing from x, which contains segments s0, . . . , sn of length 2n+ 3, wheres0 represents a fragment of c0. It can be shown by induction that due to axioms (2), every segmentsirepresents a fragment of ci, for1 6 i 6 n, and contains the state symbol fromci. Then by axiom (3), it follows thata∈HI. For the ‘if’ direction, one can provide a modelI ofOsuch that AI ={a}is a singleton,qhI = HI =∅, andI gives anr-chain outgoing froma, which containsn+ 1segments representing fragments of consequent configurations of M, neither of which containsqh. To complete the proof of the theorem we show that ontologyOis expressible by an acyclicEL-ontology network of size polynomial inm.
Consider axiom (1) and a concept inclusionϕof the formAv ∃rn·(2n+3).B, where B is a concept name. Observe that it is equivalent toA v ∃rp.∃rp
| {z }
2times
.∃rn.∃rn.∃rn
| {z }
3times
.B,
wherep=1exp(2m). Consider axiomψof the formAvB. By iteratively applying Lemma 9 we obtain thatψ[B 7→ ∃rp.∃rp.B]is expressible by an acyclicEL-ontology network of size polynomial inm. By repeating this argument we obtain the same for ϕ. Further, by Lemma 2, the axiomθ =ϕ[B 7→ q0uB]is expressible by an acyclic EL-ontology network of size polynomial inm. Again, by iteratively applying Lemma 9 together with Lemma 2 we conclude thatθ[B 7→ ∃(r,b)2n+2]is expressible by an acyclicEL-ontology network of size polynomial inmand thus, so is axiom (1). The expressibility of axioms (2) is shown identically. The remaining axioms of ontologyO are EL-axioms whose size does not depend onm. By applying Lemma 1 we obtain that there exists an acyclicEL-ontology network N of size polynomial inmand an ontologyON such thatOis(N,ON)-expressible and thus, it holdsON |=N AvH iffMaccepts the empty word in1exp(m)steps.
Theorem 2. Entailment in cyclicEL-ontology networks isRE-hard.
Proof Sketch.For a TMM, we define an infinite ontologyO, which contains variants of axioms (1-2) from Theorem 1 and additional axioms for a correct implementation of transitions ofM:
Av ∃rk.(∃vl.Luεu ∃r.(q0u ∃(r,b)2l+2)), whereA, L, ε6∈Q∪ A (4)
∃r.∃vk.Lv ∃vk.L, k>0 (5)
∃vk.Lu ∃r2k+4.εvε, k>0 (6)
∃vk.Lu ∃r2k+1.(Xu ∃r.(Y u ∃r.(Uu ∃r.Z)))vW, forXY U Z δ
0
7→W (7)
∃vk.Lu ∃r2k.(bu ∃r.(εu ∃r.(Y u ∃r.(Uu ∃r.Z))))vW, forbY U Z δ
0
7→W (8)
∃vk.Lu ∃r2k.(bu ∃r.(bu ∃r.(εu ∃r.(Uu ∃r.Z))))vW, forbbU Z δ
0
7→W (9)
∃vk.Lu ∃r2k.(Xu ∃r.(bu ∃r.(bu ∃r.(εu ∃r.Z))))vW, forXbbZ δ
0
7→W (10)
qhvH, ∃r.HvH, whereH 6∈Q∪ A (11)
Axioms (4) give an infinite family ofr-chains, each having a ‘prefix’ of lengthk+ 1, fork>0(reserved for fragments of consequent configurations ofM), and a ‘postfix’
containing a chain of length2l+ 3, forl>0, which represents a fragment of the initial configurationc0. Propagated to the ‘left’ by axioms (5), concept ∃vk.Lindicates the length of the ‘postfix’ forc0 on everyr-chain given by axioms (4). The conceptεis used to separate fragments of consequent configurations ofM and is propagated by axioms (6). The family of axioms, (7)-(10), withX, Y, U, Z, W ∈Q∪ Aandk >0, implements transitions of M. Concept ∃vk.L guarantees that transitions have effect only along r-chains which represent a fragment ofc0 of length 2k+ 3. Since ε 6∈
Q∪ A, the transitions involvingεare implemented separately by axioms (8)-(10). The more involved implementation of transitions (in comparison to Theorem 1) allows to prevent defect situations, when there are two consequent segments si,si+1 of an r- chain, which represent fragments of configurationsci,ci+1ofM, respectively, butci+1 is not a successor ofci. In Theorem 1, the prefix of lengthn·(2n+ 3)given by axiom (1) guarantees a correct implementation of up tontransitions of the TM. The situation is different in the infinite case, since the prefix reserved for fragments of consequent configurations ofMcan be of any length, due to axioms (4).
We prove thatM halts iffO |= A vH. Suppose thatc0 is an accepting config- uration andM halts innsteps; w.l.o.g. we assume thatn > 1. LetI be a model of O anda ∈ AI a domain element. Due to axioms (4),I is a model of the concept inclusion:A v ∃rn·(2n+4).(∃vn.Luεu ∃r.(q0u ∃(r,b)2n+2))and thus,I gives a r-chain containingn+ 1 segments of length 2n+ 3 separated byε. By using argu- ments from the proof of Theorem 1, it can be shown that due to axioms (5) - (10), these segments represent fragments of consequent configurations ofM, starting with c0, and there is an elementbin ther-chain such thatb ∈ qhI. Then by axiom (11), it holdsa∈HI. For the ‘if’ direction, one can show that ifM does not halt, then there exists a modelIofOsuch thatqhI =HI =∅,AI ={a}is a singleton and there are infinitely many disjoint r-chains{Rm,n}m,n>1 outgoing from a, such that every Rm,nrepresents a fragment ofc0of lengthnand has a prefix of lengthm+ 1repre- senting fragments of consequent configurations ofM, each having length62n+ 3.
To complete the proof of the theorem we show that ontology O is expressible by a cyclic EL-ontology network. Let us demonstrate that so is the family of axioms (4).
Letϕ = A v B be a concept inclusion andB, B1, B2 concept names. By Lemma 10, ontologyO1 ={ϕ[B 7→ ∃rk.B] |k>0}is expressible by a cyclicEL-ontology network. Then by Lemma 2, ontologyO2 =O1[B 7→ B1uεu ∃r.(q0uB2)]is ex- pressible by a cyclicEL-ontology network. By applying Lemma 10 again, we conclude that so is ontologyO3 = S
l>0O2[B1 7→ ∃vl.B1, B2 7→ ∃(r,b)2l.B2], i.e., the on- tology given by axiomsAv ∃rk.(∃vl.B1uεu ∃r.(q0u ∃(r,b)2l.B2)), fork, l >0.
Further, by Lemma 2, we obtain thatO2[B1 7→ L, B2 7→ ∃(r,b)2]is expressible by a cyclicEL-ontology network and hence, so is the family of axioms (4). A similar ar- gument shows the expressibility of ontologies given by axioms (5)-(10). The remaining subset of axioms (11) ofOis finite. By Lemma 1, there exists a cyclicEL-ontology networkN and an ontologyON such thatOis(N,ON)-expressible and thus, it holds ON |=N AvHiffM halts.
Theorem 3. Entailment inALC-ontology networks is2ExpTime-hard.
Proof Sketch.The proof is by reduction of the word problem for1exp(n)-space bounded ATMs to entailment inALC-ontology networks. We demonstrate that under a minor modification the construction from Theorem 7 in [7] shows that there exists aALC- ontologyOand a concept nameAsuch thatO 6|= A v ⊥iff a given1exp(n)-space bounded ATMM accepts the empty word. The ontology contains axioms with concepts of the form∃(r, C)1exp(n).Dand∀r1exp(n).D. By using Lemmas 4, 8, we show that ev- ery axiom ofOcontaining concepts of size exponential innis expressible by an acyclic ALC-ontology networkN of size polynomial inn. Then by applying Lemma 1 we ob- tain that there exists an acyclicALC-ontology networkN of size polynomial innand an ontologyONsuch thatOis(N,ON)-expressible and thus, it holdsON |=N Av ⊥ iffM accepts the empty word. Since AExpSpace = 2ExpTime, we obtain the re- quired statement.
Theorem 4. Entailment inR-ontology networks is3ExpTime-hard.
Proof Sketch.Given a2exp(n)-space bounded ATMM and a numbern>0, we con- sider ontologyOfrom the proof of Thm. 3 forM and letO0 be the ontology obtained fromOby replacing every concept of the form∃(r, C)1exp(n).Dand∀r1exp(n).Dwith
∃(r, C)2exp(n).D and∀r2exp(n).D, respectively. A repetition of the proof of Thm. 3
using Lemmas 1, 6, 8 shows that there is an acyclic R-ontology networkN of size polynomial innand an ontologyON such thatO0 is(N,ON)-expressible and thus, ON |=N Av ⊥iffM accepts the empty word. SinceA2ExpSpace =3ExpTime, we obtain the required statement.
Theorem 5. Entailment inALCHOIF-ontology networks iscoN2ExpTime-hard.
Proof Sketch.The theorem is proved by reduction of theN2ExpTime-hard problem of domino tiling of size2exp(n)×2exp(n)to (non-)entailment inALCHOIF-ontology networks. Under a minor modification the construction from Theorem 5 in [7] shows that there exists aALCHOIF-ontologyOand a concept nameAsuch thatO 6|=Av
⊥iff a given domino system admits a tiling of size2exp(n)×2exp(n), forn > 0.
OntologyOcontains axioms with concepts of the form∃r1exp(n).C and∀r1exp(n).C.
By using the same arguments as in the proof of Theorem 3 we show that there is a ALCHOIF-ontology networkN of size polynomial innand an ontologyON such thatON |=N Av ⊥iff the domino system does not admit a tiling.
Theorem 6. Entailment inROIF-ontology networks iscoN3ExpTime-hard.
Proof Sketch.The proof is by reduction of theN3ExpTime-hard problem of domino tiling of size 3exp(n)×3exp(n)to (non-)entailment inROIF-ontology networks.
The proof employs a modification of the ontologyOfrom the proof of Thm. 5 which is obtained like in the proof sketch to Thm. 4.
5 Membership Results
As a tool for proving upper complexity bounds, we demonstrate that entailment in a networkN can be reduced to entailment from (a possibly infinite) union of ‘copies’
of ontologies appearing inN. For an ontology networkN, let us denotesig(N) = S
hO1,Σ,O2i∈N (sig(O1)∪Σ∪sig(O2)). Animport pathinN is a sequencep= {O0, Σ1,O1, . . . ,On−1, Σn,On},n ≥ 0, such thathOi−1, Σi,Oii ∈ N for eachi, with1≤i≤n. We denote bylen(p) =nandlast(p) =Onthelengthofpand the lastontology on the pathp. Bypaths(N,O)we denote the set of paths that originate inO. For every symbolX ∈ sig(N)and every import path pinN, take a distinct symbolXpof the same type (concept name, role name, or individual) not occurring in sig(N). For each import path pinN, define arenamingθpof symbols insig(N) inductively as follows. Iflen(p) = 0, we setθp(X) = X for everyX ∈ sig(N).
Otherwise,p=p0∪ {On−1, Σn,On}for some pathp0and we defineθp(X) =θp0(X) ifX ∈Σnandθp(X) =Xpotherwise. Arenamed import closureof an ontologyOin N is defined byO˜=S
p∈paths(N,O)θp(last(p)).
Lemma 11. IfI |= ˜OthenI |=N Oand for everyI |=N Othere existsJ |= ˜Osuch thatJ =sig(N)I.
Theorem 7. LetN be an ontology network,Oan ontology inN, andαan axiom such thatsig(α)⊆sig(N). ThenO |=N αiffO |˜ =α.
Theorem 7 provides a method for reducing the entailment problem in ontology net- works to entailment from ontologies. Note that, in general, the renamed closureO˜of an
ontologyOin a (cyclic) networkN can be infinite (even ifN and all ontologies inN are finite). There are, however, special cases whenO˜is finite. For example, if all import signatures inN include all symbols insig(N), then it is easy to see thatO˜is equiva- lent to the union of ontologies appearing inN.O˜is also finite ifpaths(N,O)is finite, e.g., ifN is acyclic. In this case, the size ofO˜is at most exponential inO. If there is at most one import path between every pair of ontologies (i.e., ifN is tree-shaped) then the size ofO˜is the same as the size ofN. This immediately gives the upper complexity bounds on deciding entailment in acyclic networks.
Theorem 8. LetL be a DL with the complexity of entailment in[co][N]TIME(f(n)) ([co]and[N]denote possible co- and N-prefix, respectively). LetN be an acyclic on- tology network andOan ontology inNsuch thatO˜is aL-ontology. Then forL-axioms α, the entailmentO |=N αis decidable in[co][N]TIME(f(2n)). IfN is tree-shaped then decidingO |=N αhas the same complexity as entailment inL.
The next theorem is proved by the compactness theorem for First-Order Logic by showing that the renamed import closure of an ontology is recursively enumerable.
Theorem 9. LetLbe a DL, which can be translated to FOL,N an ontology network, andOan ontology inN such thatO˜is aL-ontology. Then forL-axiomsα, the entail- mentO |=N αis semi-decidable.
6 Conclusions
We have introduced a new mechanism for ontology integration based on semantic import relations between ontologies. Reasoning over ontologies with semantic imports can be reduced to reasoning over the union of ontologies obtained as ‘copies’ of ontolo- gies from the import closure under an injective renaming of signature symbols. We have shown that this gives an upper bound on the complexity of reasoning with acyclic se- mantic imports, which is one exponential harder than entailment in the underlying DLs, fromELtoSROIQ. Our hardness results demonstrate that the increased complexity of reasoning is unavoidable. When cyclic imports are allowed, the complexity jumps to undecidability, even if every ontology in a combination is given in the DLEL. These complexity results are shown for situations when the imported symbols include roles. It is natural to ask whether the complexity drops when only concept names are imported.
The second parameter which may influence the complexity of reasoning is the semantics which is ‘imported’. In the proposed mechanism, importing the semantics of symbols is implemented via agreement of models of ontologies. One can consider refinements of this mechanism, e.g., by carefully selecting the classes of models of ontologies which must be agreed. The third way to decrease the complexity is to restrict the language in which ontologies are formulated. We conjecture that reasoning with cyclic imports is decidable for ontologies formulated in the family of DL-Lite.
Acknowledgements
The second author acknowledges support of German Research Foundation within the Transregional Collaborative Research Center SFB/TRR 62 Companion-Technology for Cognitive Technical Systems, while working at the Institute of Artificial Intelli- gence, University of Ulm, and support of RSF Grant No. 17-11-01176, while working at the Institute of Informatics Systems, Novosibirsk State University.
References
1. Bao, J., Voutsadakis, G., Slutzki, G., Honavar, V.: Package-based description logics. In: Mod- ular Ontologies, pp. 349–371 (2009)
2. Borgida, A., Serafini, L.: Distributed description logics: Assimilating information from peer sources. J. Data Semantics pp. 153–184 (2003)
3. Euzenat, J., Zimmermann, A., de Freitas, F.L.G.: Alignment-based modules for encap- sulating ontologies. In: Proceedings of the 2nd International Workshop on Modular On- tologies, WoMO 2007, Whistler, Canada, October 28, 2007 (2007), http://ceur-ws.org/Vol- 315/paper4.pdf
4. Grau, B.C., Motik, B.: Reasoning over ontologies with hidden content: The import-by-query approach. J. Artif. Intell. Res. (JAIR) pp. 197–255 (2012)
5. Grau, B.C., Parsia, B., Sirin, E.: Ontology integration using epsilon-connections. In: Modular Ontologies, pp. 293–320 (2009)
6. Homola, M., Serafini, L.: Towards formal comparison of ontology linking, mapping and im- porting. In: Description Logics (2010)
7. Kazakov, Y.:RIQand SROIQ are harder thanSHOIQ. In: Proc. 11th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR’08). pp. 274–284. AAAI Press (2008)
8. Pan, J.Z., Serafini, L., Zhao, Y.: Semantic import: An approach for partial on- tology reuse. In: Haase, P., Honavar, V., Kutz, O., Sure, Y., Tamilin, A. (eds.) WoMO. CEUR Workshop Proceedings, vol. 232. CEUR-WS.org (2006), http://dblp.uni- trier.de/db/conf/semweb/womo2006.html
