SHOIQ with transitive closure of roles is decidable

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SHOIQ with transitive closure of roles is decidable

Chan Le Duc, Myriam Lamolle, Olivier Curé

To cite this version:

Chan Le Duc, Myriam Lamolle, Olivier Curé. SHOIQ with transitive closure of roles is decidable.

Description Logics, 2013, Ulm, Germany. �hal-01740557�

SHOIQ with transitive closure of roles is decidable

Chan Le Duc¹, Myriam Lamolle¹, and Olivier Cur´e²

1 LIASD Universit´e Paris 8 - IUT de Montreuil, France {chan.leduc, myriam.lamolle}@iut.univ-paris8.fr

2 LIGM Universit´e Paris-Est, France [email protected]

Abstract. The Semantic Web makes an extensive use of the OWL DL ontology language, underlied by theSHOIQdescription logic, to formalize its resources.

In this paper, we propose a decision procedure for this logic extended with the transitive closure of roles in concept axioms, a feature needed in several applica- tion domains. To address the problem of consistency in this logic, we introduce a new structure for characterizing models which may have an infinite non-tree-like part.

1 Introduction

The ontology language OWL-DL [1] is widely used to formalize data resources on the Semantic Web. This language is mainly based on the description logicSHOINwhich is known to be decidable [2]. Although SHOIN provides transitive rolesto model transitivity of relations, we can find several applications in which thetransitive closure of roles, that is more expressive than transitive roles, is needed. For instance, if we denote byR⁻ andR⁺ the inverse and transitive closure of a roleRrespectively then it is obvious that the concept∃R⁺.∀R⁻.⊥is unsatisfiable w.r.t. an empty TBox. If we now substituteR⁺ for a transitive role R_t such thatR v R_t (i.e. we substitute each occurrence of R⁺ in axioms for Rt) then the concept ∃Rt.∀R⁻.⊥is satisfiable. The point is that an instance ofR⁺represents a sequence of instances ofRbut an instance ofRtcorresponds to a sequence of instances ofitself.

In this paper, we consider an extension ofSHOIQby enabling transitive closure of roles in concept axioms. In the general case, transitive closure is not expressible in the first order logic [3], the logic from which DL is a sublanguage, while the second order logic is sufficiently expressive to do so.

In the DL literature ([4]; [5]), there have been works dealing with transitive closure of roles. Recently, Ortiz [5] has proposed an algorithm for deciding consistency in the logicALCQIb⁺_reg which allows for transitive closure of roles. However, nominals are disallowed in this logic. It is known that reasoning with a DL including number restric- tions, inverse roles, nominals and transitive closure of roles is hard. The reason for this is that there exists an ontology in that DL whose models have aninfinitenon-tree-like part. Calvaneseet al.[6] have presented an automata-based technique for dealing with the logicZOIQthat includes transitive closure of roles, and showed that the sublogics ZIQ,ZOQandZOIare decidable. To obtain this result, the authors have introduced thequasi-forest model propertyto characterize models of ontologies in these sublogics.

Although they are very expressive, none of these sublogics includesSHOIQwith tran- sitive closure of roles, namelySHOIQ₍₊₎. The following example³, notedK₁, shows that there is an ontology inSHOIQ₍₊₎which does not enjoy the quasi-forest model property. We consider the following axioms:

(1){o} vA;AuBv ⊥;Av ∃R.Au ∃R⁰.B;Bv ∃S⁺.{o}

(2){o} v ∀X⁻.⊥;> v ≤1X.>;> v ≤1X⁻.>whereX∈ {R,R⁰,S}

Figure 1 shows an infinite non-tree-like model ofK1. In fact, each individualxthat sat- isfies∃S⁺.{o}must have two distinct paths fromxto the individual satisfying nominal o. Intuitively, we can see that (i) such axmust satisfy∃S⁺.{o}andB, (ii) an individual satisfyingBmust connect to another individual satisfyingAwhich must have aR-path to nominalo, and (iii) two conceptsAandBare disjoint.

R R

R

S⁻ S⁻ S⁻

A A A

{o}, A

B,∃S⁺.{o}

B,∃S⁺.{o}

B,∃S⁺.{o} B,∃S⁺.{o}

R⁰, S⁻ R⁰ R⁰

Fig. 1.An infinite non tree-like model ofK1

This example shows that methods ([7], [8], [6]) based on the hypothesis which says that if an ontology is consistent it has aquasi-forest model, could fail to address the problem of consistency in a DL including simultaneously O (nominals), I (inverse roles),Q(number restrictions) and transitive closure of roles.

In this paper, we propose a decision procedure for the problem of consistency in SHOIQwith transitive closure of roles in concept axioms. The underlying idea of our algorithm is founded on thestar-typeandframenotions introduced by Pratt-Hartmann [9]. This technique uses star-types to represent individuals and “tiles” them together to form a frame for representing a model. For each star-type σ, we maintain a func- tion δ(σ)which stores the number of individuals satisfying this star-type. To obtain termination condition, we introduce two additional structures into a frame : (i) the first one, namelycycles, describes duplicate parts of a model resulting from interactions of logic constructors in SHOIQ, (ii) the second one, namelyblocking-blocked cycles, describes parts of a model bordered by cycles which allow a frame to satisfy transitive closure of roles occurring in concepts of the form∃R⁺.C.

2 The Description Logic SHOIQ

In this section, we present the syntax, the semantics and main inference problems of SHOIQ₍₊₎. In addition, we introduce a tableau structure forSHOIQ₍₊₎, which al- lows us to represent a model of aSHOIQ₍₊₎knowledge base.

3This example is initially proposed by Sebastian Rudolph from an informal discussion

Definition 1. LetRbe a non-empty set of role namesandR₊ ⊆Rbe a set of tran- sitive role names. We useR_I = {P⁻ | P ∈R}to denote a set of inverse roles, and R_⊕={Q⁺ |Q∈R∪R_I}to denote a set of transitive closure of roles. Each element ofR∪RI∪R_⊕ is called aSHOIQ₍₊₎-role. Arole inclusion axiomis of the form R vS for twoSHOIQ₍₊₎-rolesRandS such thatR /∈ R_⊕ andS /∈R_⊕. Arole hierarchy Ris a finite set of role inclusion axioms. An interpretationI = (∆^I,·^I) consists of a non-empty set∆^I(domain) and a function·^Iwhich maps each role name to a subset of∆^I×∆^Isuch that

R^−I={hx, yi ∈∆^I×∆^I | hy, xi ∈R^I}for allR∈R, hx, zi ∈S^I,hz, yi ∈S^Iimplieshx, yi ∈S^Ifor eachS∈R+, and

(Q⁺)^I = [

n>0

(Qⁿ)^Iwith(Q¹)^I =Q^I,

(Qⁿ)^I={hx, yi ∈(∆^I)²| ∃z∈∆^I,hx, zi ∈(Qⁿ⁻¹)^I,hz, yi ∈Q^I}forQ⁺∈R_⊕

∗An interpretationI satisfies a role hierarchyRifR^I ⊆S^I for eachR vS ∈ R.

Such an interpretation is called amodelofR, denoted byI |=R. To simplify notations for nested inverse roles and transitive closures of roles, we define two functions· and

·^⊕as follows:

R =











R⁻ ifR∈R;

S ifR=S⁻andS∈R;

(S⁻)⁺ ifR=S⁺,S∈R, S⁺ ifR= (S⁻)⁺,S∈R

R^⊕=











R⁺ ifR∈R;

S⁺ ifR= (S⁺)⁺andS∈R;

(S⁻)⁺ ifR=S⁻andS ∈R;

(S⁻)⁺ ifR= (S⁺)⁻andS∈R

∗A relationv∗ is defined as the transitive-reflexive closureR⁺ ofvonR ∪ {R v S |R vS ∈ R} ∪ {R^⊕ vS^⊕ |R vS ∈ R} ∪ {Q vQ^⊕ | Q∈R∪R_I}. We define a functionTrans(R)which returnstrueiff there is someQ∈R₊∪ {P |P ∈ R+} ∪ {P^⊕ |P ∈R∪RI}such thatQvR∗ ∈ R⁺. A roleRis calledsimplew.r.t.R ifTrans(R) =false.

The reason for the introduction of two functions· and·^⊕ in Definition 1 is that they avoid usingR⁻⁻ andR⁺⁺. Moreover, it remains a unique nested case(R⁻)⁺. According to Definition 1, axiomR vQ^⊕is not allowed in a role hierarchyRsince this may lead to undecidability [10]. Notice that the closureR⁺may containRvQ^⊕ ifRvQbelongs toR⁺.

Definition 2 (terminology).LetCbe a non-empty set of concept nameswith a non- empty subset Co ⊆ C of nominals. The set of SHOIQ(+)-concepts is inductively defined as the smallest set containing allCinC,>,CuD,CtD,¬C,∃R.C,∀R.C, (≤n S.C)and(≥n S.C)wherenis a positive integer, C andD are SHOIQ(+)- concepts,Ris anSHOIQ(+)-role andS is a simple role w.r.t. a role hierarchy. We denote⊥for¬>. The interpretation function·^Iof an interpretationI= (∆^I,·^I)maps each concept name to a subset of∆^I such that>^I = ∆^I, (CuD)^I =C^I∩D^I, (CtD)^I=C^I∪D^I,(¬C)^I=∆^I\C^I,|{o^I}|= 1for allo∈C_o,(∃R.C)^I={x∈

∆^I | ∃y ∈∆^I,hx, yi ∈R^I∧y ∈C^I},(∀R.C)^I ={x∈∆^I | ∀y ∈∆^I,hx, yi ∈ R^I ⇒ y ∈ C^I}, (≥n S.C)^I = {x ∈ ∆^I | |{y ∈ C^I | hx, yi ∈ S^I| ≥ n}, (≤n S.C)^I ={x∈∆^I | |{y ∈C^I | hx, yi ∈S^I| ≤n}where|S|is denoted for the cardinality of a setS. An axiomC v Dis called a general concept inclusion (GCI)

whereC, DareSHOIQ₍₊₎-concepts (possibly complex), and a finite set of GCIs is called a terminologyT. An interpretationIsatisfies a GCICvDifC^I⊆D^IandI satisfies a terminologyT ifIsatisfies each GCI inT. Such an interpretation is called amodelofT, denoted byI |= T. A pair(T,R)is called aSHOIQ₍₊₎knowledge base whereRis aSHOIQ₍₊₎role hierarchy andT is aSHOIQ₍₊₎terminology. A knowledge base(T,R)is said to be consistent if there is a modelI of bothT andR, i.e.,I |=T andI |=R. A conceptCis calledsatisfiablew.r.t.(T,R)iff there is some interpretationIsuch thatI |=R,I |=T andC^I6=∅. Such an interpretation is called amodelofCw.r.t.(T,R). A conceptDsubsumesa conceptCw.r.t.(T,R), denoted byCvD, ifC^I⊆D^Iholds in each modelIof(T,R). C Since unsatisfiability, subsumption and consistency w.r.t. aSHOIQ₍₊₎knowledge base can be reduced to each other, it suffices to study knowledge base consistency. For the ease of construction, we assume all concepts to be innegation normal form(NNF), i.e., negation occurs only in front of concept names. AnySHOIQ(+)-concept can be transformed to an equivalent one in NNF by using DeMorgan’s laws and some equiva- lences as presented in [11]. According to [12],nnf(C)can be computed in polynomial time in the size ofC. For a conceptC, we denote thennfofCbynnf(C)and thennfof

¬Cby¬C. Let˙ Dbe aSHOIQ(+)-concept in NNF. We definecl(D)to be the smallest set that contains all sub-concepts ofDincludingD. For a knowledge base(T,R), we reusecl(T,R), which was introduced by Horrockset al.[7], to denote all sub-concepts occurring in the axioms of(T,R). We havecl(T,R)is bounded byO(|(T,R)|)[7].

To translatestar-typeandframestructures presented by Pratt-Hartmann (2005) forC² into those forSHOIQ, we need to add new sets of concepts, denotedcl1(T,R)and cl2(T,R), to the signature of aSHOIQ₍₊₎knowledge base(T,R).

cl1(T,R) ={≤mS.C| {(≤nS.C),(≥nS.C)} ∩cl(T,R)6=∅,1≤m≤n} ∪ {≥mS.C| {(≤nS.C),(≥nS.C)} ∩cl(T,R)6=∅,1≤m≤n}

For a generating concept(≥nS.C)and a setI ⊆ {0,· · ·, log n+ 1}, we denote C_(≥nS.C)^I = l

i∈I

C_(≥nS.C)ⁱ ul

j /∈I

¬C_(≥nS.C)^j whereC_(≥nS.C)ⁱ are new concept names for0≤i≤log n+ 1. We definecl₂(T,R)as follows:

cl2(T,R) ={C_(≥S.C)ⁱ |(≥nS.C)∈cl(T,R)∪cl1(T,R),0≤i≤log n+ 1}∪

{C_(≥nS.C)^I |(≥nS.C)∈cl(T,R)∪cl1(T,R), I⊆ {0,· · ·, log n+ 1}}

Remark 1. If numbers are encoded in binary then the number of new concept names C_(≥nS.D)ⁱ for0 ≤i ≤ log n+ 1, is bounded byO(|(T,R)|)sincenis bounded by O(2^|(T,R)|). This implies that|cl2(T,R)|is bounded byO(2^|(T,R)|). Note that two conceptsC_(≥nS.C)^I andC_(≥nS.C)^J are disjoint for allI, J⊆ {0,· · · , log n+ 1},I6=J. The conceptsC(∃S.C)andC_(≥nS.C)^I will be used for building chromatic star-types. This notion will be clarified after introducing the frame structure (Definition 5).

Finally, we denoteCL(T,R) =cl(T,R)∪cl₁(T,R)∪cl₂(T,R), and useR(T,R) to denote the set of all role names occurring inT,Rwith their inverse. The definition ofCL(T,R)is inspired from the Fischer-Ladner closure that was introduced in [13].

The closureCL(T,R)contains not only sub-concepts syntactically obtained fromT

but also sub-concepts that are semantically derived from T w.r.t.R. For instance, if

∀S.Cis a sub-concept fromT andRvS∗ ∈ Rthen∀R.C∈CL(T,R).

To describe a model of aSHOIQ₍₊₎knowledge base in a more intuitive way, we use a tableau structure that expresses semantic constraints resulting directly from the logic constructors inSHOIQ₍₊₎. A tableau definition forSHOIQ₍₊₎can be found in [14].

3 A Decision Procedure For SHOIQ

This section starts by translating star-type and frame structures presented by Pratt- Hartmann (2005) forC²into those forSHOIQ₍₊₎.

Definition 3 (star-type). Let (T,R)be a SHOIQ(+) knowledge base. A star-type is a pair σ = hλ(σ), ξ(σ)i, where λ(σ) ∈ 2^CL(T,R) is called core label,ξ(σ) = (hr1, l1i,· · · ,hrd, ldi)is ad-tuple over2^R(T,R)×2^CL(T,R). A pairhr, liis arayof σifhr, li=hri, liifor some1≤i≤d. We usehr(ρ), l(ρ)ito denote a rayρ=hr, li wherer(ρ) =randl(ρ) =l.

– A star-typeσisnominalifo∈λ(σ)for someo∈Co.

– A star-typeσischromaticifρ 6=ρ⁰ impliesl(ρ) 6= l(ρ⁰)for two raysρ, ρ⁰ ofσ.

When a star-typeσis chromatic,ξ(σ)can be considered as a set of rays.

– Two star-typesσ, σ⁰ are equivalentif λ(σ) = λ(σ⁰), and there is a bijection π betweenξ(σ)andξ(σ⁰)such thatπ(ρ) =ρ⁰impliesr(ρ⁰) =r(ρ)andl(ρ⁰) =l(ρ).

We denoteΣfor the set of all star-types for(T,R). C Note that for a chromatic star-typeσ,ξ(σ)can be considered as a set of rays since rays are distinct and not ordered. We can think of a star-typeσas the set of individuals xsatisfying all concepts in λ(σ), and each rayρ of σ corresponds to a “neighbor”

individual xi of xsuch thatr(ρ) is the label of the link betweenxand xi; andxi

satisfies all concepts inl(ρ). In this case, we say thatxsatisfiesσ.

Definition 4 (valid star-type).Let(T,R)be aSHOIQ₍₊₎knowledge base. Letσbe a star-type for(T,R)whereσ=hλ(σ), ξ(σ)i. The star-typeσisvalidifσis chromatic and the following conditions are satisfied:

1. IfCvD∈ T thennnf(¬CtD)∈λ(σ);

2. {A,¬A} 6⊆ λfor every concept nameAwhereλ = λ(σ)orλ = l(ρ)for each ρ∈ξ(σ);

3. IfC1uC2∈λ(σ)then{C1, C2} ⊆λ(σ);

4. IfC1tC2∈λ(σ)then{C1, C2} ∩λ(σ)6=∅;

5. If∃R.C∈λ(σ)then there is some rayρ∈ξ(σ)such thatC∈l(ρ)andR∈r(ρ);

6. If(≤ nS.C) ∈ λ(σ)and there is some rayρ ∈ ξ(σ)such thatS ∈ r(ρ)then C∈l(ρ)or¬C˙ ∈l(ρ);

7. If(≤nS.C)∈λ(σ)and there is some rayρ∈ξ(σ)such thatC ∈l(ρ)andS ∈ r(ρ)then there is some1≤m≤nsuch that{(≤mS.C),(≥mS.C)} ⊆λ(σ);

8. For each rayρ∈ξ(σ), ifR∈r(ρ)andRvS∗ thenS∈r(ρ);

9. If∀R.C∈λ(σ)andR∈r(ρ)for some rayρ∈ξ(σ)thenC∈l(ρ);

10. If ∀R.D ∈ λ(σ),SvR,∗ Trans(S)and R ∈ r(ρ)for some ray ρ ∈ ξ(σ)then

∀S.D∈l(ρ);

11. If∀Q^⊕.C∈λ(σ),RvQ∗ andR∈r(ρ)for some rayρ∈ξ(σ)then∀Q^⊕.C∈l(ρ);

12. If∃Q^⊕.C∈λ(σ)then(∃Q.Ct ∃Q.∃Q^⊕.C)∈λ(σ);

13. If(≥ nS.C) ∈ λ(σ)then there arendistinct raysρ1,· · · , ρn ∈ ξ(σ)such that {C,C_(≥nS.C)^Ii } ⊆l(ρ_i),S∈r(ρ_i)for all1≤i≤n; andI_j, I_k⊆ {0,· · · , log n+

1},Ij6=Ikfor all1≤j < k≤n.

14. If(≤nS.C)∈λ(σ)and there do not existn+ 1raysρ₀,· · · , ρ_n∈ξ(σ)such that

C∈l(ρi)andS∈r(ρi)for all0≤i≤n. C

Roughly speaking, a star-typeσis valid if each individualxsatisfiessemantically all concepts inλ(σ). In fact, each condition in Definition 4 represents the semantics of a constructor inSHOIQ₍₊₎except for transitive closure of roles. From valid star-types, we can “tile” a model instead of using expansion rules for generating nodes as described in tableau algorithms. Before presenting how to “tile” a model from star-types, we need some notation that will be used in the remainder of the paper.

Notation 1 We call P = h(σ1, ρ₁, d₁),· · · ,(σ_k, ρ_k, d_k)ia sequencewhereσ_i ∈ Σ, ρ_i∈ξ(σ_i)andd_i∈Nfor1≤i≤k.

– tail(P) = (σk, ρk, dk),tailσ(P) =σk,tailρ(P) =ρk,tailδ(P) =dkand|P|=k.

We denoteL(P) =λ(tailσ(P)).

– pⁱ(P) = (σ_i, ρ_i, d_i),pⁱ_σ(P) =σ_i,pⁱ_ρ(P) =ρ_iandpⁱ_δ(P) =d_ifor each1≤i≤k.

– an operationadd(P,(σ, ρ, d))extendsPto a new sequence withadd(P,(σ, ρ, d)) = hP,(σ, ρ, d)i.

Definition 5 (frame).Let(T,R)be aSHOIQ(+)knowledge base. Aframefor(T,R) is a tupleF=hN,No, Ω, δi, where

1. N is a set of valid star-types such thatσis not equivalent toσ⁰for allσ, σ⁰∈ N; 2. No⊆ N is a set of nominal star-types;

3. Ωis a function that maps each pair(σ, ρ)withσ∈ N andρ∈ξ(σ)to a sequence Ω(σ, ρ) =h(σ1, ρ1, d1),· · ·,(σm, ρm, dm)iwithσi ∈ N,ρi∈ξ(σi),di ∈Nfor 1 ≤ i ≤ msuch that for eachσi with1 ≤ i ≤ m, it holds thatl(ρ) = λ(σi), l(ρi) =λ(σ)andr(ρi) =r⁻(ρ)wherer⁻(ρ) ={R |R∈r(ρ)}.

4. δis a functionδ:N →N. By abuse of notation, we also useδto denote a function which maps each pair(σ, ρ)withσ ∈ N andρ∈ ξ(σ)into a number inN, i.e.,

δ(σ, ρ)∈N. C

The frame structure, as introduced in Definition 5, allows us to compress individuals of a model into star-types. For each star-typeσand each rayρ∈ξ(σ), a listΩ(σ, ρ)of triples(σi, ρi, di)withρi ∈ξ(σi)is maintained whereσiis a “neighbor” star-type of σviaρ∈ξ(σ), anddiindicates thedi-th “layer” of rays ofσi. We can think a layer of rays ofσias an individual that connects to its neighbor individuals via the rays ofσi. The following definition presents how to connect such layers to form paths in a frame.

Definition 6 (path).LetF =hN,No, Ω, δibe a frame for aSHOIQ₍₊₎knowledge base(T,R). Apathis inductively defined as follows:

1. A sequenceh∅,(σ, ρ,1)iis a path ifσ∈ N_oandρ∈ξ(σ);

2. A sequencehP,(σ, ρ, d)i withP 6= ∅ and tail(P) = (σ₀, ρ₀, d₀), is a path if (σ, ρ, d) =p^d0(Ω(σ₀, ρ⁰))for eachρ⁰ 6=ρ₀. In this case, we say thathP,(σ, ρ, d)i is theρ⁰-neighborofP, and two pathsP,hP,(σ, ρ, d)iare neighbors.

Additionally, ifhP,(σ, ρ, d)iis aρ⁰-neighbor ofPandQ∈r(ρ⁰)thenhP,(σ, ρ, d)i is aQ-neighbor ofP. In this case, we say thathP,(σ, ρ, d)iis aQ-neighbor ofP, orPis aQ -neighbor ofhP,(σ, ρ, d)i.

We define P ∼ P⁰ iftailσ(P) = tailσ(P⁰)and tailδ(P) = tailδ(P⁰). Since∼is an equivalence relation over the set of all paths, we useP to denote the set of all equiva- lence classes[P]of paths inF. For[P],[Q]∈P, we define:

1. [P]is a neighbor (ρ⁰-neighbor) of[Q]if there areP⁰ ∈[P]andQ⁰ ∈[Q]such that Q⁰is a neighbor (ρ⁰-neighbor) ofP⁰;

2. [Q]is a reachable path of[P]if there are[P1],· · · ,[Pn]∈Psuch that[Pi+1]is a neighbor of[Pi]for all1≤i < nwhere[P1] = [P]and[Q] = [Pn].

3. [Q]is aQ-neighbor of[P]if there areP⁰ ∈ [P]andQ⁰ ∈[Q]such thatQ⁰ is a Q-neighbor ofP⁰, orP⁰is aQ -neighbor ofQ⁰;

4. [Q]is aQ-reachable path of[P]if there are[P1],· · ·,[Pn]∈Psuch that[Pi+1] is aQ-neighbor of[Pi]for all1≤i < nwhere[P1] = [P]and[Q] = [Pn]. C Note that for two paths P,P⁰ withtailρ(P) 6= tailρ(P⁰), we have P ∼ P⁰ if tail_σ(P) = tail_σ(P⁰) andtail_δ(P) = tail_δ(P⁰). This does not allow for extending tail_ρ(P)totail_ρ([P]). As a consequence, there may be several “predecessors” of an equivalence class [P]. However, we can define tail_σ([P]) = tail_σ(P), tail_δ([P]) = tail_δ(P)andL([P]) = L(P). In the sequel, we useP instead of[P]whenever it is clear from the context.

Definition 7 (cycle).LetF =hN,N_o, Ω, δibe a frame for aSHOIQ₍₊₎knowledge base(T,R)with a setPof paths inF.

1. Acycleis a setΘof triples(P, ρ,Q)withP,Q ∈ P andρ ∈ξ(tailσ(P))such that for each(P, ρ,Q)∈Θthe following conditions are satisfied:

(a) tailδ(P)>1andtailδ(Q)>1;

(b) IfP⁰is theρ-neighbor ofPthentailσ(P⁰) =tailσ(Q);

(c) for each sequence P1,· · ·,Pn ∈ P such that P1 = P, P2 is not the ρ- neighbor ofP, and Pi+1 is a neighbor of Pi for1 ≤ i < n, there is some (P⁰⁰, ρ⁰⁰,Q⁰⁰)∈Θsuch that

i. eitherQ⁰⁰ =Pj,tailσ(Pj+1) =tailσ(P⁰⁰),tailδ(Pj+1)≥tailδ(P⁰⁰)and Pjis theρ-neighbor ofPj+1for some1< j < n,

ii. or there arePn+1,· · ·,Pn+m∈PwithQ⁰⁰=P_n+m−1,tailσ(Pn+m) = tailσ(P⁰⁰),tailδ(Pn+m) ≥ tailδ(P⁰⁰)and Pi+1 is a neighbor ofPi for n≤i < n+m.

In this case, we say thatQiscycledbyPviaρ.

2. A cycleΘ⁰is areachablecycle ofΘif for each(P, ρ,Q)∈Θand for each sequence P1,· · · ,Pn ∈ Psuch thatP1 =P,P2 is not aρ-neighbor ofP, andPi+1is a neighbor ofPifor1≤i < n, there is some(P⁰⁰, ρ⁰⁰,Q⁰⁰)∈Θ⁰such that

(a) eitherQ⁰⁰=P_j,tail_σ(P_j+1) =tail_σ(P⁰⁰),tail_δ(P_j+1)≥tail_δ(P⁰⁰)andP_jis aρ-neighbor ofP_j+1for some1< j < n,

(b) or there are Pn+1,· · ·,Pn+m ∈ P withQ⁰⁰ = P_n+m−1,tailσ(Pn+m) = tailσ(P⁰⁰),tailδ(Pn+m)≥tailδ(P⁰⁰)andPi+1is a neighbor ofPiforn≤i <

n+m. C

Note that cycles may encapsulate loops if tailδ(Pj+1) = tailδ(P⁰⁰)holds in Condi- tions 1(c)i and 1(c)ii, Definition 7. LetΘ be a cycle in a frame. Definition 7 ensures that each reachable path of some pathP with(P, ρ,Q)∈Θgoes through a star-type σ=tail_σ(Q⁰)with some(P⁰, ρ⁰,Q⁰)∈Θ. Such a cycle, which is similar to blocking- blocked nodes in completion graphs forSHOIQ[7], allows for “unravelling” infinitely the frame to obtain a model of a KB inSHOIQ(without transitive closure of roles).

This means that we can extend the setP of paths by adding infinitely paths which lengthenQsuch that(P, ρ,Q)∈ΘandPis not a neighbor ofQ. However, such a cy- cle structure is not sufficient to represent models of a KB with transitive closure of roles since a concept such as∃Q^⊕.D∈ L(P)can be satisfied by aQ-reachable pathP⁰of Pwhich is arbitrarily far fromP. There are the following possibilities for an algorithm which builds a frame: (i) the algorithm stops building the frame as soon as a cycleΘ is detected such that each concept of the form∃Q^⊕.Doccurring inL(P)for each cy- cling pathPofΘis satisfied, i.e.,Phas aQ-reachable pathP⁰with∈ ∃Q.D∈ L(P), (ii) despite of several detected cycles, the algorithm continues building the frame un- til each concept of the form ∃Q^⊕.Doccurring in L(P)is satisfied for each cycling pathP ofΘ. If we adopt the first possibility, the completeness of such an algorithm cannot be established since there are models in which paths satisfying concepts of the form∃Q^⊕.Dcan spread over several “iterative structures” such as cycles. For this rea- son, we adopt the second possibility by introducing into frames an additional structure, namelyblocking-blocked cycles, which determines a sequence of cyclesΘ₁,· · ·, Θ_k such thatΘ_i+1is a reachable cycle ofΘ_i. Reachability of cycles allows for “unravel- ling” the frame between cycled pathsQ⁰ with(P⁰, ρ⁰,Q⁰) ∈Θk and cycling pathsP with(P, ρ,Q)∈Θ1.

Definition 8 (blocking).LetF =hN,No, Ω, δibe a frame for aSHOIQ(+)knowl- edge base(T,R)with a setPof all paths inF.

1. A cycle Θ⁰ is blockable by a cycle Θ if Θ⁰ is a reachable cycle of Θ, and for each(P⁰, ρ⁰,Q⁰) ∈ Θ⁰ there is some (P, ρ,Q) ∈ Θ such thatL(P) = L(P⁰), L(Q) =L(Q⁰)andr(ρ) =r(ρ⁰). In this case, we say thatQ⁰ is blockable byP viaρ.

2. A cycleΘ⁰ isblockedby a cycleΘif there areΘ1,· · ·, Θk withΘ = Θ1,Θ⁰ = Θk such that Θi+1 is blockable byΘi for1 ≤ i < k, and for each(λ, s, λ⁰) ∈ 2^CL(T,R)×2^R(T,R)×2^CL(T,R)with∃Q^⊕.D∈λ,

– if there is some pathPk with(Pk, ρ_k,Qk)∈ Θ_k,L(Pk) = λ,L(Qk) = λ⁰, s=r(ρ_k)such thatPkhas noQ-reachable pathP_k⁰ with∃Q.D∈ L(P_k⁰), – then there is some P1 with (P1, ρ1,Q1) ∈ Θ1,L(P1) = λ,L(Q1) = λ⁰,

r(ρ1) =ssuch that for each∃P^⊕.C∈ L(P1),

• there is aP-reachable pathQ⁰⁰ofP1with∃P.C∈ L(Q⁰⁰), and

• there are two triples(P_j, ρ_j,Q_j)∈Θ_jand(P_j+1, ρ_j+1,Q_j+1)∈Θ_j+1 for some1≤j < k,

which satisfy thatQ⁰⁰is a reachable path ofQ_jandQ_j+1is a reachable path ofQ⁰⁰.

In this case, we say thatP1blocksPkviaρk. C

According to Definition 8, there are a sequentially reachable cycles between a blocking cycle Θ₁ and a blocked cycle Θ_k, which allows for unravelling the frame betweenΘ_k andΘ₁. Condition 2 Definition 8 says that if(P, ρ,Q) ∈Θ_k andL(P) contains concepts of the form∃Q^⊕.Dwhich are not satisfied by reachable paths ofP then there exists someP⁰with(P⁰, ρ⁰,Q⁰)∈Θ₁which allows for satisfying these con- cepts∃Q^⊕.DinL(P)by unravelling. We would like to note that a pathP is blocked if there is some blocked cycleΘksuch that(P, ρ,Q)∈Θk.

Definition 9 (valid frame).Let(T,R)be aSHOIQknowledge base. A frameF = hN,No, Ω, δiisvalidif the following conditions are satisfied:

1. For eacho∈C_othere is a uniqueσ_o∈ Nosuch thato∈λ(σ_o)andδ(σ_o) = 1;

2. For eachσ∈ N,σis valid;

3. If∃Q^⊕.C ∈λ(tail_σ(P₀))for someP₀∈ Pthen there areP,P⁰ ∈ Psuch that one of the following conditions is satisfied:

(a) P0=P =P⁰and∃Q.C∈ L(P0);

(b) P⁰is aQ-reachable ofP, and∃Q.C∈ L(P⁰)whereP=P0orPblocksP0; (c) P is aQ -reachable ofP⁰, and∃Q.C ∈ L(P⁰)whereP =P0orP blocks

P0. C

Conditions 1-3 in Definition 9 ensure satisfaction of tableau properties [14]. In par- ticular, Condition 3 takes into account satisfaction of transitive closure of roles. In fact, for each blocking pathP with∃Q^⊕.D ∈ L(P), this condition says that P must be satisfied by aQ-reachable pathP⁰ of P between a blocking cycleΘ1 and a blocked cycleΘk. If there is some pathQbetweenΘ1andΘkwith∃S^⊕.C∈ L(Q)that is not satisfied, due to the construction of star-types and frames, a concept∃S^⊕.C is prop- agated alongS-reachable paths ofQ. This implies thatQhas a S-reachable pathQ⁰ such that Q⁰ is blocked and ∃S^⊕.C ∈ L(Q⁰). By unravelling (details will be given in soundness proof), we can build an (extended)S-reachable pathQ⁰⁰ofQ⁰ such that

∃S.C ∈ L(Q⁰⁰).

We now present Algorithm 1 for building a frame which is valid if the conditions in Definition 9 are satisfied. This algorithm starts by adding nominal star-types to the frame. For each non blocked pathPwith a rayρ∈ξ(tailσ(P))such thatδ(tailσ(P))is minimal and there is a difference betweenδ(tail_σ(P))andδ(tail_σ(P), ρ), the algorithm picks in a nondeterministic way a valid star-typeω that matchestail_σ(P)via ρ, and updatesΩ(tail_σ(P), ρ),Ω(ω, ρ⁰),δ(tail_σ(P), ρ),δ(ω, ρ⁰), eventually,δ(tail_σ(P))and δ(ω)by callingupdateFrame(· · ·)[14]. The algorithm terminates when a blocked cycle is detected.

Figure 2 depicts a frame when executing Algorithm 1 forK₁in the example pre- sented in Section 1. The algorithm builds a frame F = hN,No, Ω, δiwhereN = {σ0, σ1, σ2, σ3, σ4} andNo = {σ0}. The dashed arrows indicate how the function

Require: ASHOIQ(+)knowledge base(T,R) Ensure: A framehN,No, Ω, δifor(T,R)

1: LetΣbe the set of all star-types for(T,R) 2: for allo∈Codo

3: ifthere is noσ∈ N such thato∈λ(σ)then 4: Choose a star-typeσo ∈Σsuch thato∈λ(σo) 5: Setδ(σo) = 1,N =N ∪ {σo}andNo=No∪ {σo} 6: Setδ(σo, ρ) = 0,Ω(σo, ρ) =∅for allρ∈ξ(σo) 7: end if

8: end for

9: whilethere is a pathPthat is not blocked and a rayρ∈ξ(tailσ(P))such that tailδ(P) =δ(tailσ(P), ρ) + 1andδ(tailσ(P))≤δ(ω)for allω∈ N do 10: Choose a star-typeσ⁰∈Σsuch that there is a rayρ⁰∈ξ(σ⁰)satisfying

l(ρ) =λ(σ⁰),l(ρ⁰) =λ(σ),r(ρ⁰) =r⁻(ρ), and σ⁰∈ N impliesδ(σ⁰) =δ(σ⁰, ρ⁰) + 1

11: updateFrame(σ, ρ, σ⁰, ρ⁰) 12: end while

Algorithm 1:An algorithm for building a frame

Ω(σ, ρ)can be built. For example,Ω(σ0, ρ0) ={(σ1, ν0,1)},Ω(σ0, ρ1) ={(σ2, ρ⁰₀,1)}

whereρ0andρ1are the respective horizontal and vertical rays ofσ0;ν0is the left ray of σ1;ρ⁰₀is the vertical ray ofσ2. Moreover, the directed dashed arrow fromσ0toσ1indi- cates that the rayρ0ofσ0can match the rayν0on the left ray ofσ1sincel(ρ0) =λ(σ1), r(ν0) =λ(σ0),r(ν0) =r⁻(ρ0).

Then, the algorithm generatesδ(σ0) = 1,δ(σ1) = 1,δ(σ2) = 1and forms a cycle Θ consisting of the following triples :((σ3,2), ρ1,(σ3,3))(ρ1 is the left ray ofσ3)), ((σ₃,2), ρ₃,(σ₄,1))(ρ₃is the vertical ray ofσ₃),((σ₄,1), ρ₄,(σ₄,2))(ρ₄is the left ray ofσ₄) and((σ₄,1), ρ₅,(σ₃,2))(ρ₅is the vertical ray ofσ₄). Note that for the sake of brevity, we use justtail_σ(P)andtail_δ(P)to denote a path in the triples. We can check that any path that is an extension of a pathP gets through a pathQwherePis the first component of a triple andQis the third component of a triple.

The algorithm may add some more paths that go throughσ₃ andσ₄ to form a blocked cycle. A model of the ontology can be built by starting fromσ0 and getting (i)σ4viaσ1, (ii)σ3viaσ1, and (iii)σ3viaσ2. Fromσ3andσ4, the model goes through σ3andσ4infinitely. Note that from any individualxsatisfyingσ3(orσ4), i.e. the “la- bel” ofxcontains∃Q⁺.{o}, there is a path containingSwhich goes back the individual satisfyingσ0. Thus, the concept∃Q⁺.{o}is satisfied for each individual whose label contains∃Q⁺.{o}.

Lemma 1. Let(T,R)be aSHOIQ₍₊₎knowledge base.

1. Algorithm 1 terminates.

2. If Algorithm 1 can build a valid frame for(T,R)then(T,R)is consistent.

3. If(T,R)is consistent then Algorithm 1 can build a valid frameFfor(T,R).

Proof (sketch).Since the functionsδ(σ)andδ(σ,hr, li)is increased monotonously by Algorithm 1, termination of the algorithm can be proved if we can show that : (i) the

{o}, A R R⁰, S⁻

σ₀

A

R

B,∃S⁺.{o}

{o}, A A

S⁻

B,∃S⁺.{o} R⁰⁻, S S

A σ1

σ2

σ3

R A

A A

R⁻ A {o}, A

B,∃S⁺.{o}

B,∃S⁺.{o}

B,∃S⁺.{o}

B,∃S⁺.{o}

B,∃S⁺.{o}

R⁻

S⁻

R⁰ B,∃S⁺.{o}

σ4

R⁰

R⁰⁻

Fig. 2.A frame obtained by Algorithm 1 forK1in the example in Section 1

number of different star-types is bounded; (ii) the detection of a blocked cycle accord- ing to Definition 8 terminates. For the soundness of Algorithm 1, we can extend the set Pof paths to a setPcof extended paths by “unravelling” the frame between blocking- blocked cycles. A tableau [14] can be built fromPc. The main argument is that when extending a pathP if tailσ(P) 6= tailσ(P⁰)for all blocked path P⁰ then this exten- sion process can be continued up to a star-typeσ⁰ =tailσ(P⁰⁰)for some blocked path P⁰⁰. This holds due to the definition of cycles and blockable cycles. Otherwise, i.e., tailσ(P) =tailσ(P⁰)for some blocked pathP⁰ byQthenP can be extended by get- ting throughtail_σ(Q).

Regarding completeness, a tableau can guide the algorithm (i) to choose valid star- types, (ii) to ensure thatδ(σ) = 1for each nominal star-typeσ, and (iii) to detect a pair (Θ₁, Θ_k)of blocking and blocked cycles as soon as each concept of the form∃Q^⊕.D inΘ₁is satisfied. We refer the readers to [14] for a complete proof of Lemma 1.

The following theorem is a consequence of Lemma 1.

Theorem 1. SHOIQ(+)is decidable.

4 Conclusion

In this paper, we have presented a decision procedure for the description logicSHOIQ with transitive closure of roles in concept axioms, whose decidability was not known.

The most significant feature of our contribution is to introduce a structure for charac- terizing models which have an infinite non-tree-like part. This structure would provide an insight into regularity of such models which would be enjoyed by a more expressive DL, such asZOIQ[6], whose decidability remains open. In future work, we aim to improve the algorithm by making it more goal-directed and aim to investigate another open question about the hardness ofSHOIQ₍₊₎.

References

1. Patel-Schneider, P., Hayes, P., Horrocks, I.: OWL web ontology language semantics and abstract syntax. In: W3C Recommendation. (2004)

2. Tobies, S.: The complexity of reasoning with cardinality restrictions and nominals in expres- sive description logics. Journal of Artificial Intelligence Research12(2000) 199–217 3. Aho, A.V., Ullman, J.D.: Universality of data retrieval languages. In: Proceedings of the 6th

of ACM Symposium on Principles of Programming Language. (1979)

4. Baader, F.: Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles. In: Proceedings of the Twelfth International Joint Conference on Artificial Intelligence. (1991)

5. Ortiz, M.: An automata-based algorithm for description logics aroundSRIQ. In: Proceed- ings of the fourth Latin American Workshop on Non-Monotonic Reasoning 2008, CEUR- WS.org (2008)

6. Calvanese, D., Eiter, T., Ortiz, M.: Regular path queries in expressive description logics with nominals. In: IJCAI. (2009) 714–720

7. Horrocks, I., Sattler, U.: A tableau decision procedure forSHOIQ. Journal Of Automated Reasoning39(3) (2007) 249–276

8. Motik, B., Shearer, R., Horrocks, I.: Hypertableau reasoning for description logics. J. of Artificial Intelligence Research36(2009) 165–228

9. Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Jour- nal of Logic, Language and Information14(3) (2005) 369–395

10. Le Duc, C., Lamolle, M.: Decidability of description logics with transitive closure of roles. In: Proceedings of the 23rd International Workshop on Description Logics (DL 2010), CEUR-WS.org (2010)

11. Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for expressive description logics.

In: Proceedings of the International Conference on Logic for Programming, Artificial Intel- ligence and Reasoning (LPAR 1999), Springer (1999)

12. Baader, F., Nutt, W.: Basic description logics. In: The Description Logic Handbook: Theory, Implementation and Applications (2nd edition), Cambridge University Press (2007) 47–104 13. Fischer, M.J., Ladner, R.I.: Propositional dynamic logic of regular programs. Journal of

Computer and System Sciences18(18) (1979) 174–211

14. Le Duc, C., Lamolle, M., Cur´e, O.: A decision procedure for SHOIQ with tran- sitive closure of roles in concept axioms. In: Technical Report, http://www.iut.univ- paris8.fr/∼leduc/papers/RR-SHOIQTr.pdf (2013)