A Tableaux-Based Algorithm for SHIQ with Transitive Closure of Roles in Concept and Role Inclusion Axioms

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A Tableaux-Based Algorithm for SHIQ with Transitive Closure of Roles in Concept and Role Inclusion Axioms

Chan Le Duc, Myriam Lamolle, Olivier Curé

To cite this version:

Chan Le Duc, Myriam Lamolle, Olivier Curé. A Tableaux-Based Algorithm for SHIQ with Transitive Closure of Roles in Concept and Role Inclusion Axioms. 8th Extended Semantic Web Conference (ESWC 2011), May 2011, Heraklion, Crete, Greece. pp.367-381, �10.1007/978-3-642-21034-1_25�.

�hal-00738380�

SHIQ

Transitive Closure of Roles in Concept and Role Inclusion Axioms

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 ocure@univ-mlv.fr

Abstract. In this paper, we investigate an extension of the description logic SHIQ––a knowledge representation formalism used for the Semantic Web––with transitive closure of roles occurring not only in concept inclusion axioms but also in role inclusion axioms. It was proved that adding transitive closure of roles toSHIQwithout restriction on role hierarchies may lead to undecidability. We have identified a kind of role inclusion axioms that is responsible for this undecid- ability and we propose a restriction on these axioms to obtain decidability. Next, we present a tableaux-based algorithm that decides satisfiability of concepts in the new logic.

1 Introduction

The ontology language OWL-DL [1] is widely used to formalize semantic resources on the Semantic Web. This language is mainly based on the description logicSHOIN which is known to be decidable [2]. AlthoughSHOIN is expressive and provides transitive roles to model transitivity of relations, we can find several applications in which the transitive closure of roles, that is more expressive than transitive roles, is necessary. The difference between transitive roles and the transitive closure of roles is clearer when they are involved in role inclusion axioms. For instance, if we denote by R⁻andR⁺the inverse and transitive closure of a roleRrespectively then it is obvious that the concept∃R⁺.(C ∀R⁻.⊥)is unsatisfiable w.r.t. an empty TBox and the trivial axiomRR⁺. If we now substituteR⁺for a transitive roleRtsuch thatRRt(i.e.

we substitute each occurrence ofR⁺in axioms and concepts forRt) then the concept

∃Rt.(C ∀R⁻.⊥)becomes satisfiable. The point is that an instance ofR⁺represents a sequence of instances ofRbut an instance ofRtcorresponds to a sequence of instances of itself.

In several applications, we need to model successive events and relationships be- tween them. An event is something oriented in time, i.e. we can talk about endpoints of an event, or a chronological order of events. When an event of some kind occurs it can trigger an event (or a sequence of events) of another kind. In this situation, it may be suitable to use a role to model an event. If we denote roleseventandeventfor two kinds of events then the axiom(eventevent)expresses the fact that when an event

G. Antoniou et al. (Eds.): ESWC 2011, Part I, LNCS 6643, pp. 367–381, 2011.

© Springer-Verlag Berlin Heidelberg 2011

Fig. 1. Mouse clicks, keystrokes and shortcuts

of the first kind occurs it implies one event or a sequence of events of the second kind.

To express “a sequence of events” we can defineevent to be transitive. However, the semantics of transitive roles is not sufficient to describe this behaviour since the tran- sitive roleeventcan represent a sequence of itself but not a sequence of another role.

Such behaviours can be found in the following example.

Example 1. LetSbe the set of all states of applications running on a computer. We denote byA,B,C⊆Sthe sets of states of applicationsA,B,C, respectively. A user can perform a mouse-click or keystroke to change states. She can type a shortcut (combina- tion of keys) to go fromAtoBor fromBtoC. This action corresponds to a sequence of mouse-clicks or keystrokes. The system’s behaviour is depicted in Figure 1. In such a system, users may be interested in the following question: “from the applicationA, can one go through the applicationBto get directly to the applicationCby a mouse-click or keystroke ?”.

We now use a description logic with transitive closure of roles to express the con- straints as described above. To do this, we use a rolenextto model mouse clicks or keystrokes and a rolejumpto model shortcuts in the following axioms:

(i) start ¬A ¬B ¬C;XY ⊥withX,Y∈ {A,B,C}andX =Y; (ii) A ∃jump.B;A ∃jump.C;B ∃jump.C;

(iii)start ∀next⁻.⊥;jumpnext⁺;

Under some operating systems, users cannot switch directly from an application to a particular one just by one mouse click or keystroke. We can express this constraint with the following axiom:

(iv)C ∃next⁻.B ⊥;

In this case, the concept(A ∃next⁺.(C ∃next⁻.B))capturing the question above is unsatisfiable w.r.t. the axioms presented.

Such examples motivate the study of Description Logics (DL) that allow for the transi- tive closure of roles to occur in both concept and role inclusion axioms. In this work, we introduce a DL that can model systems as described in Example 1 and propose a tableaux-based decision procedure for the concept satisfiability problem in this DL.

To the best of our knowledge, the decidability ofSHIQ with transitive closure of roles, is unknown. [3] and [4] have established decision procedures for concept satisfiability inSHI+ (SHI with transitive closure of roles in concept and role in- clusion axioms) andSHIO+(SHI+with nominals). These decision procedures have used neighborhoods for representing an individual with its neighbors in a model in or- der to build completion graphs. In the literature, many decidability results in DLs can be obtained from their counterparts in modal logics ([5], [6]). However, these counter- parts do not take into account expressive role inclusion axioms. In particular, [6] has shown decidability of a very expressive DL, so-calledCAT S, includingSHIQwith the transitive closure of roles but not allowing it to occur in role inclusion axioms. [6]

has pointed out that the complexity of concept subsumption inCAT S is EXPTIME- complete by translatingCAT S into the logic ConversePDLin which inference prob- lems are well studied.

Recently, there have been some works in [7] and [8] which have attempted to aug- ment the expressiveness of role inclusion axioms. A decidable logic, namelySROIQ, resulting from these efforts allows for new role constructors such as composition, dis- jointness and negation. In addition, [9] has introduced a DL, so-calledALCQIb⁺reg, which can captureSRIQ (SROIQwithout nominal), and obtained the worst-case complexity (EXPTIME-complete) of the satisfiability problem by using automata-based technique.ALCQIb⁺reg allows for a rich set of operators on roles by which one can simulate role inclusion axioms. However, transitive closures in role inclusion axioms are expressible neither inSROIQnor inALCQIb⁺reg.

Tableaux-based algorithms for expressive DLs such asSHIQ [10] andSHOIQ [11] result in efficient implementations. This kind of algorithms relies on two structures, the so-called tableau and completion graph. Roughly speaking, a tableau for a concept represents a model for the concept and it is possibly infinite. A tableau translates satisfi- ability of all given concept and role inclusion axioms into the satisfiability of constraints imposed locally on each individual of the tableau by the semantics of concepts in the individual’s label. This feature of tableaux will be called local satisfiability property.

To check satisfiability of a concept, tableaux-based algorithms try to build a comple- tion graph whose finiteness is ensured by a technique, the so-called blocking technique.

It provides a termination condition and guarantees soundness and completeness. The underlying idea of the blocking mechanism is to detect “loops” which are repeated pieces of a completion graph. When transitive closure of roles is added to knowledge bases, this blocking technique allows us to lengthen paths through such loops in order to satisfy semantic constraints imposed by transitive closures. The algorithm in [12] for satisfiability inALCreg (including the transitive closure of roles and other role opera- tors) introduced a method to deal with loops which can hide unsatisfiable nodes. This method detects on so-called concept trees, “good” or “bad” cycles that are similar to those between blocking and blocked nodes on completion trees.

To deal with transitive closure of roles occurring in terms such as∃Q⁺.C, we have to introduce a new expansion rule to build completion trees such that it can generate a path formed from nodes that are connected by edges whose label contains roleQ. In addition, this rule propagates terms∃Q⁺.C to each node along with the path before reaching a node whose label includes conceptC. Such a path may go through blocked

and blocking nodes and has an arbitrary length. To handle transitive closures of roles occurring in role inclusion axioms such asRQ⁺, we use another new expansion rule that translates satisfaction of such axioms into satisfaction of a term∃Q⁺.Φ. From the path generated from∃Q⁺.Φ, a cycle can be formed to satisfy the semantic constraint imposed byR Q⁺. Since the roleQ, which will be defined to be simple, does not occur in number restrictions, the cycle obtained from this method does not violate other semantic constraints.

The contribution of the present paper consists of (i) designing a decidable logic, namelySHIQ₊, with a new definition for simple roles and (ii) proposing a tableaux- based algorithm for satisfiability of concepts inSHIQ₊.

2 The Description Logic SHIQ

The logicSHIQ+ is an extension ofSHIQ introduced in [11] by allowing transi- tive closure of roles to occur in concept and role inclusion axioms. In this section, we present the syntax and semantics of the logicSHIQ+. This includes an extension of the definition of simple roles toSHIQ+and the definition of inference problems that we are interested in. The definitions reuse some notation introduced in [11].

Definition 1. LetRbe a non-empty set of role names. We denoteRI={P⁻|P ∈R}

andR+={Q⁺|Q∈R∪RI}.

∗The set ofSHIQ₊-roles isR∪RI∪R₊. A role inclusion axiom is of the formRS for twoSHIQ₊-rolesRandS. A role hierarchyRis a finite set of role inclusion ax- ioms.

∗An interpretationI= (Δ^I,·^I)consists of a non-empty setΔ^I(domain) and a func- tion·^I which maps each role name to a subset ofΔ^I ×Δ^I such that, forR ∈ R, Q⁺∈R+,

R^−I ={x, y ∈(Δ^I)²| y, x ∈R^I},(Q⁺)^I=

n>0

(Qⁿ)^Iwith(Q¹)^I=Q^Iand (Qⁿ)^I ={x, y ∈(Δ^I)²| ∃z∈Δ^I,x, z ∈(Qⁿ⁻¹)^I,z, y ∈Q^I}.

An interpretationIsatisfies a role hierarchyRifR^I⊆S^Ifor eachRS ∈ R. Such an interpretation is called a model ofR, denoted byI |=R.

∗To simplify notations for nested inverse roles and transitive closures of roles, we de- fine 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 relation ∗ is defined as the transitive-reflexive closureR⁺ ofonR ∪ {R S | R S ∈ R} ∪ {R^⊕ S^⊕ | R S ∈ R} ∪ {QQ^⊕ |Q ∈R∪RI}. We denoteS ≡RiffR∗SandS∗R.

∗A roleRis called simple w.r.t.Riff (i)Q^⊕∗R /∈ R⁺for eachQ∈R∪R_I, and (ii) R∗R,P∗R^⊕∈ R⁺impliesP∗R∈ R⁺.

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⁻)⁺.

Notice that a transitive roleS (i.e.x, y ∈ S^I,y, z ∈ S^I impliesx, z ∈ S^I whereI is an interpretation) can be expressed by using a role axiom S^⊕ S. In addition, a roleRwhich is simple according to Definition 1 is simple according to [10]

as well. In fact, ifQ^⊕∗R /∈ R⁺for eachQ∈R∪RIthen there is no transitive role S such that S∗R ∈ R⁺. Finally, if R∗S ∈ R⁺ andR is not simple according to Definition 1 thenSis not simple according to Definition 1.

Definition 2. LetCbe a non-empty set of concept names.

∗The set ofSHIQ₊-concepts is inductively defined as the smallest set containing all CinC,,CD,CD,¬C,∃R.C,∀R.C,(≤n S.C)and(≥n S.C)whereCand DareSHIQ₊-concepts,Ris anSHIQ₊-role andS is a simple role. We denote⊥ for¬.

∗An interpretationI= (Δ^I,·^I)consists of a non-empty setΔ^I(domain) and a func- tion·^Iwhich maps each concept name to a subset ofΔ^Isuch that

^I =Δ^I, (CD)^I=C^I∩D^I, (CD)^I=C^I∪D^I, (¬C)^I=Δ^I\C^I, (∃R.C)^I={x∈Δ^I| ∃y∈Δ^I,x, y ∈R^I∧y∈C^I},

(∀R.C)^I={x∈Δ^I| ∀y∈Δ^I,x, y ∈R^I⇒y∈C^I}, (≥n S.C)^I ={x∈Δ^I|card{y∈C^I | x, y ∈S^I} ≥n}, (≤n S.C)^I ={x∈Δ^I |card{y∈C^I| x, y ∈S^I} ≤n}

wherecard{S}denotes the cardinality of a setS.

∗ C D is called a general concept inclusion (GCI) where C, D are SHIQ+- concepts (possibly complex), and a finite set of GCIs is called a terminologyT. An interpretationIsatisfies a GCIC DifC^I ⊆ D^I andIsatisfies a terminologyT ifIsatisfies each GCI inT. Such an interpretation is called a model ofT, denoted by I |=T.

∗A conceptCis called satisfiable w.r.t. a role hierarchyRand a terminologyT iff there is some interpretationIsuch thatI |=R,I |=T andC^I =∅. Such an interpretation is called a model ofCw.r.t.RandT. A pair(T,R)is called aSHIQ+ knowledge base and said to be consistent if there is a modelI of bothT andR, i.e.,I |=T and I |=R.

∗A conceptDsubsumes a conceptCw.r.t.RandT, denoted byCD, ifC^I ⊆D^I holds in each modelIof(T,R).

Notice that we can reduce subsumption and consistency problems inSHIQ₊to con- cept satisfiability w.r.t. a knowledge base(T,R). Thanks to these reductions, it suffices to study the concept satisfiability problem inSHIQ₊.

For the ease of construction, we assume all concepts to be in negation normal form (NNF) i.e. negation occurs only in front of concept names. AnySHIQ+-concept can be transformed to an equivalent one in NNF by using DeMorgan’s laws and some equiv- alences as presented in [10]. For a conceptC, we denote thennfofCbynnf(C)and thennfof¬Cby¬C˙

LetDbe anSHIQ+-concept in NNF. We definesub(D)to be the smallest set that contains all sub-concepts ofDincludingD. For a knowledge base(T,R),R_(T,R)is used to denote the set of all role names occurring inT,Rwith their transitive closure and inverses. We denote byR⁺_(T,R)the set of transitive closure of roles occurring in

R_(T,R). Finally, we define setssub(T,R)andsub(T ,R)as follows:

sub(T,R) =

CD∈T

sub(nnf(¬CD),R)where (1)

sub(E,R) =sub(E)∪ {¬C˙ |C∈sub(E)} ∪ (2) {∀S.C|(∀R.C∈sub(E), S∗R)or( ˙¬∀R.C ∈sub(E), S∗R)

whereSoccurs inT orR} ∪ {∃P.β|β∈ {C,∃P^⊕.C},∃P^⊕.C∈sub(E)}

Φ^σ=

C∈σ∪ {¬˙D|D∈sub(T,R)\σ}

C for eachσ⊆sub(T,R) (3)

Ω={Φσ |σ⊆sub(T,R)} (4)

sub(T ,R) =Ω∪ {α.β|α∈ {∃P.∃P^⊕,∃P^⊕,∃P}, P^⊕∈R⁺_(T,R), β∈Ω} (5)

3 Tableaux for SHIQ

Basically, a tableau structure is used to represent a model of aSHIQ₊knowledge base.

Properties in such a tableau definition express semantic constraints resulting directly from the logic constructors inSHIQ₊. Considering the tableau definition forSHIQ presented in [10], Definition 3 forSHIQ₊ adopts two additional properties, namely P8andP9. In particular,P8imposes a global constraint on a set of individuals of a tableau. This causes the tableaux to lose the local satisfiability property. A tableau has the local satisfiability property if each property of the tableau is related to only one node and its neighbors. This means that, for a graph with a labelling function, checking each node of the graph and its neighbors for each property is sufficient to prove whether this graph is a tableau. The tableau definition forSHIQin [10] has the local satisfiability property althoughSHIQincludes transitive roles. The propagation of value restrictions on transitive roles by∀⁺-rule (i.e. the rule for ∀R.C ifR is transitive or includes a transitive role) and the absence of number restrictions on transitive roles help to avoid global properties that impose a constraint on an arbitrary set of individuals in a tableau.

Definition 3. Let(T,R)be aSHIQ+knowledge base. A tableauTfor a conceptD w.r.t(T,R)is defined to be a triplet(S,L,E)such thatSis a set of individuals,L:S

→2^{sub(T,R)∪sub(T ,R)}andE:R_(T,R)→2^S×S, and there is some individuals∈Ssuch thatD ∈ L(s). For alls∈ S,C, C1, C2 ∈sub(T,R)∪sub(T ,R),R, S ∈R_(T,R)

andQ^⊕∈R⁺_(T,R),Tsatisfies the following properties:

P1IfC1C2∈ T thennnf(¬C₁C2)∈ L(s), P2IfC∈ L(s)then¬C /˙ ∈ L(s),

P3IfC1C2∈ L(s)thenC1∈ L(s)andC2∈ L(s), P4IfC1C2∈ L(s)thenC1∈ L(s)orC2∈ L(s), P5If∀S.C ∈ L(s)ands, t ∈ E(S)thenC∈ L(t),

P6If∀S.C ∈ L(s),Q^⊕∗Sands, t ∈ E(Q)then∀Q^⊕.C∈ L(t),

P7If∃P.C∈ L(s)withP ∈R_(T,R)\R⁺_(T,R)then there is somet∈Ssuch that s, t ∈ E(P)andC∈ L(t),

P8If∃Q^⊕.C∈ L(s)then(∃Q.C ∃Q.∃Q^⊕.C)∈ L(s), and there ares1,· · ·, sn

∈Ssuch that∃Q.C∈ L(s₀)∪ L(sn−1)andsi, si+1 ∈ E(Q)with0≤i < n, s0=sand∃Q^⊕.C∈ L(sj)for all0≤j < n.

P9Ifs, t ∈ E(Q^⊕)then∃Q^⊕.Φ^σ∈ L(s)withσ=L(t)∩sub(T,R)and

Φ^σ =

C∈σ∪ {¬˙D|D∈sub(T,R)\σ}

C,

P10s, t ∈ E(R)ifft, s ∈ E(R),

P11Ifs, t ∈ E(R)andR∗Sthens, t ∈ E(S), P12If(≤nS.C)∈ L(s)thencard{S^T(s, C)} ≤nwhere

S^T(s, C) :={t∈S|s, t ∈ E(S)∧C∈ L(t)}, P13If(≥nS.C)∈ L(s)thencard{S^T(s, C)} ≥n,

P14If(≤nS.C)∈ L(s)ands, t ∈ E(S)thenC∈ L(t)or¬C˙ ∈ L(t).

P8in Definition 3 expresses not only the semantic constraint imposed by the transitive closure of roles occurring in concepts such as∃Q^⊕.C(i.e. a path including nodes are connected by edges containingQand the label of the last node containsC) but also the non-determinism of transitive closure of roles (i.e. the term∃Q.Cmay be chosen at any node of such a path to satisfy∃Q^⊕.C). Additionally,P8andP9in Definition 3 enable to satisfy each transitive closureQ^⊕occurring in the label of an edges, twith simple roleQ. In fact,P9makesΦ^σ belong to the label of a nodetandsconnected tot by edges containingQdue toP8. The definition ofΦ^σallowst to be combined withtwithout causing contradiction. Moreover, this combination does not violate num- ber restrictions sinceQis simple. For this reason, the new definition for simple roles presented in Definition 1 is crucial to decidability ofSHIQ+.

In addition,P8andP9defined in this way do not require explicitly cycles to satisfy role inclusion axioms such asR Q^⊕. This makes it possible to design of tableaux- based algorithm forSHIQ+that aims to build tree-like structure i.e. no cycle is explic- itly required to be embedded within this structure. The following lemma affirms that a tableau represents exactly a model for the concept.

Lemma 1. Let(T,R)be aSHIQ₊knowledge base. LetDbe aSHIQ₊concept.D is satisfiable w.r.t.(T,R)iff there is a tableau forDw.r.t.(T,R).

For a proof of Lemma 1, we refer the reader to [13].

4 A Tableaux-Based Decision Procedure for SHIQ

As mentioned, a tableau for a concept represents a model that is possibly infinite. How- ever, the goal of a tableaux-based algorithm is to find a finite structure that must imply a tableau. Conversely, the existence of a tableau can guide us to build such a structure.

We introduce in Definition 4 such a finite structure, namely, completion tree.

Definition 4. Let(T,R)be aSHIQ+knowledge base. LetDbe aSHIQ+concept.

A completion tree forDand(T,R)is a treeT= (V, E,L, xT, =)^· where

∗V is a set of nodes containing a root nodexT∈V. Each nodex∈V is labelled with a functionLsuch thatL(x)⊆sub(T,R)∪sub(T ,R). In addition, =^· is a symmetric binary relation overV.

∗E is a set of edges. Each edgex, y ∈ E is labelled with a functionLsuch that L(x, y)⊆R_(T,R).

∗Ifx, y ∈Ethenyis called a successor ofx, denoted byy∈succ¹(x), orxis called the predecessor ofy, denoted byx=pred¹(y). In this case, we say thatxis a neighbor ofyoryis a neighbor ofx. Ifz∈succⁿ(x)(resp.z=predⁿ(x)) andyis a successor ofz(resp.yis the predecessor ofz) theny ∈succ⁽ⁿ⁺¹⁾(x)(resp.y =pred⁽ⁿ⁺¹⁾(x)) for alln≥0wheresucc⁰(x) ={x}andpred⁰(x) =x.

∗A nodeyis called aR-successor ofx, denoted byy ∈ succ¹R(x)(resp.yis called theR-predecessor ofx, denoted byy = pred¹R(x)) if there is some roleRsuch that R ∈ L(x, y)(resp.R ∈ L(y, x)) andR∗R. A nodeyis called aR-neighbor ofx ifyis either aR-successor orR-predecessor ofx. Ifzis aR-successor ofy(resp.zis theR-predecessor ofy) andy∈succⁿR(x)(resp.y=predⁿR(x)) thenz∈succ⁽Rⁿ⁺¹⁾(x) (resp.z=pred⁽Rⁿ⁺¹⁾(x)) forn≥0withsucc⁰R(x) ={x}andx=pred⁰R(x).

∗For a nodexand a roleS, we define the setS^T(x, C)ofx’sS-neighbors as follows:

S^T(x, C) ={y∈V |yis aS-neighbor ofxandC∈ L(x)}

∗A nodexis called blocked byy, denoted byy=b(x), if there are numbersn, m >0 and nodesx, y, ysuch that

1. xT=predⁿ(y),y=pred^m(x), and 2. x=pred¹(x),y=pred¹(y), and 3. L(x) =L(y),L(x) =L(y), and 4. L(x, x) =L(y, y), and

5. if there arez, zsuch thatz=pred¹(z),predⁱ(z) =xT,L(z) =L(y),L(z) = L(y)andL(z, z) =L(y, y)thenn≤i.

∗We define an extended functionsucc fromsuccoverTas follows:

– ifxhas a successory(resp.xhas aR-successory) that is not blocked theny ∈ succ¹(x)(resp.y∈succ¹R(x)),

– ifxhas a successorz (resp.xhas aR-successorz) that is blocked byb(z)then b(z)∈succ¹(x)(resp.b(z)∈succ¹R(x)).

– ify∈succⁿR(x)andz∈succ¹R(y)thenz∈succ⁽Rⁿ⁺¹⁾(x)forn≥0.

∗A nodezis called a∃R^⊕.C-reachable ofxwith∃R^⊕.C∈ L(x)if there arex1,· · ·, x^k+n ∈ V with x^k+n = z, x0 = x andk+n ≥ 0 such that xⁱ = predⁱR(x0),

∃R^⊕.C ∈ L(xi)withi ∈ {0,· · ·, k}, andx^j+k ∈ succ^jR(xk),∃R^⊕.C ∈ L(xj+k),

∃R.C ∈ L(x₍k+n))withj∈ {0,· · ·, n}.

∗Clashes:Tis said to contain a clash if one of the following conditions holds:

1. There is some nodex ∈ V such that{A,¬A} ⊆ L(x)˙ for some concept name A∈C,

2. There is some node x ∈ V with(≤nS.C) ∈ L(x) and there are(n+ 1)S- neighborsy1,· · · , yⁿ+1ofxsuch thatyⁱ =˙ y^jandC∈ L(xi)for all1≤i < j≤ (n+ 1),

3. There is some nodex∈V with∃R^⊕.C∈ L(x)such that there does not exist any

∃R^⊕.C-reachable nodeyofx,

Algorithm 2 builds a completion tree for aSHIQ+concept by applying the expansion rules in Figure 2 and 3. The expansion rules in Figure 2 were given in [10]. We introduce two new expansion rules that correspond toP8andP9in Definition 3.

In comparison withSHIQ, there is a new source of non-determinisms that could augment the complexity of an algorithm for satisfiability of concepts inSHIQ₊. This source comes from the presence of transitive closure of role in concepts. This means that for each occurrence of a term such as∃Q^⊕.Cin the label of a node of a completion tree we have to check the existence of a sequence of edges such that the label of each edge containsQand the label of the last node containsC. The process for checking the existence of paths whose length is arbitrary must be translated into a process that works for a finite structure. To do this, we reuse the blocking condition introduced in [10]

and introduce a functionsucc(x) that returns the set ofx’s successors in a completion tree. An infinite path over a completion tree can be defined thanks to this function. The

∃+-rule in Figure 3 generates all possible paths. The clash-freeness of the third kind in Definition 4 ensures that a “good” path has to be picked from this set of all possible paths.

The functioncheckReachability^QC(x, d,B)depicted in Algorithm 1 represents an al- gorithm for checking the clash-freeness of the third kind for a completion tree. It returns trueiff there exists a∃Q^⊕.C-reachable node ofx. In this function, the parameterxrep- resents a node of the tree to be checked i.e. there is a term such as∃Q^⊕.C∈ L(x). The parameterdindicates the direction to search fromx. Depending ond = 1ord = 0, the algorithm goes up to ancestors ofxor goes down to descendants ofxrespectively.

When the algorithm goes down, it never goes up again. The subsetB ⊆V represents the set of all blocked nodes among the nodes that the algorithm has visited. The func- tion checkReachability^QC(x,1,∅) would be called for each non-blocked nodex of a completion tree and for each term of the form∃Q^⊕.C∈ L(x).

Lemma 2 (Termination). Let (T,R) be a SHIQ+ knowledge base. Let D be a SHIQ+-concept w.r.t.(T,R). Algorithm 2 terminates.

Proof. The termination of Algorithm 2 is a consequence of the following claims:

1. Applications of rules in Figure 2 and 3 do not remove concepts from the label of nodes. Moreover, applications of rules in Figure 2 and 3 do not remove roles from the label of edges except that they may set the label of edges to an empty set. However, when the label of an edge becomes empty it remains to be empty forever. Therefore, we can compute a upper bound of the completion tree’s height from the blocking condition. This upper bound equalsK = 2^2m+k wherem = card{sub(T,R)∪sub(T ,R)}andk is the number of roles occurring inT and Rplus their inverse and transitive closure. Moreover, the number of neighbors of any node is bounded byM = mⁱwheremⁱoccurs in a number restriction term (≥mⁱR.C)that appears inT.

-rule: if CD∈ T andnnf(¬CD)∈ L(x)/ thenL(x)←− L(x)∪ {nnf(¬CD)}

-rule: if C1C2∈ L(x)and{C1, C2} ⊆ L(x) thenL(x)←− L(x)∪ {C1, C2}

-rule: if C1C2∈(x)and{C1, C2} ∩ L(x) =∅ thenL(x)←− L(x)∪ {C}for someC∈ {C1, C2}

∃-rule: if 1.∃S.C∈ L(x),xis not blocked, and 2.xhas noS-neighbourywithC∈ L(y)

then create a new nodeywithL(x, y)={S}andL(y)={C}

∀-rule: if 1.∀S.C∈ L(x), and

2. there is aS-neighbouryofxsuch thatC /∈ L(y) thenL(y)←− L(y)∪ {C}

∀+-rule: if 1.∀S.C∈ L(x), and

2. there is someQwithQ^⊕S∗ , and

3. there is anQ-neighbouryofxsuch that∀Q^⊕.C /∈ L(y) thenL(y)←− L(y)∪ {∀Q^⊕.C}

ch-rule: if 1.(≤n S.C)∈ L(x), and

2. there is anS-neighbouryofxwith{C,¬C} ∩ L(y) =˙ ∅ thenL(y)←− L(y)∪ {E}for someE∈ {C,¬C}˙

≥-rule: if 1.(≥n S.C)∈ L(x)andxis not blocked, and

2. there are non S-neighborsy1, ..., ynsuch thatC∈ L(yi), andyi=y· jfor 1≤i < j≤n,

then creatennew nodesy1, ..., ynwithL(x, yi)={S}, L(yi)={C}, andyi=y· jfor1≤i < j≤n.

≤-rule: if 1.(≤n S.C)∈ L(x), and

2.card{S^T(x, C)}> nand there are twoS-neighborsy, zofxwith C∈ L(y)∩ L(z),yis not an ancestor ofz, and noty=z^·

then 1.L(z)←− L(z)∪ L(y)andL(x, y)←− ∅ 2. Ifzis an ancestor ofx

thenL(z, x)←− L(z, x)∪ {R|R∈ L(x, y)}

elseL(x, z)←− L(x, z)∪ L(x, y) 4. Addu=z^· for allusuch thatu=y^·

Fig. 2. Expansion rules forSHIQpresented in [10]

∃+-rule: if ∃S^⊕.C∈ L(x)and(∃S.C ∃S.∃S^⊕.C)∈ L(x)/ thenL(x)←− L(x)∪ {∃S.C ∃S.∃S^⊕.C}

⊕-rule: if xhas aP^⊕-neighboryand∃P^⊕.Φσ ∈ L(x)/ withσ=L(y)∩sub(T,R) thenL(x) =L(x)∪ {∃P^⊕.Φσ}

Fig. 3. New expansion rules forSHIQ₊

checkReachability^Q_C(x, d,B)

1

if∃Q.C∈ L(x)then

2

returntrue;

3

ifd= 1then

4

if there ispred¹_Q(x)with∃Q^⊕.C∈ L(pred¹_Q(x))then

5

checkReachability^Q_C(pred¹_Q(x),1,B);

6

foreachx∈succ¹_Q(x)such that∃Q^⊕.C∈ L(x)do

7

if∃Q.C∈ L(x)then

8

returntrue;

9

ifxis not blocked then

10

checkReachability^Q_C(x,0,B);

11

else

12

ifx∈ B/ then

13

B=B ∪ {x};

14

checkReachability^Q_C(b(x),0,B);

15

returnfalse;

16

Algorithm 1.checkReachability^QC(x, d,B)for checking the existence of a ∃Q^⊕.C- reachable node of x ∈ V where d ∈ {1,0}, B ⊆ V, ∃Q^⊕.C ∈ L(x) and T = (V, E,L, xT, =)˙ is a completion tree. As shown in Lemma 2, the complexity of Algorithm 1 is bounded by an exponential function in size of a completion tree. This implies that the complexity of the tableaux algorithm forSHIQ+ (Algorithm 2) is bounded by a double exponential function in size of inputs.

Input : ASHIQ₊knowledge base(T,R)and aSHIQ₊-conceptD Output: IsDsatisfiable w.r.t.(T,R)?

LetT= (V, E,L, xT,=)^· be an initial tree such thatV ={xT},L(xT) ={D}, and

1

there is nox, y∈V such thatx=y^· ;

while there is a non-empty setSof expansion rules in Figure 2 and 3 such that eachr∈S

2

can be applied to a nodex∈V do Applyr;

3

if there is a clash-free treeTwhich is built by Line 2 to 2 then

4

YES;

5

else

6

NO;

7

Algorithm 2. Algorithm for building a completion tree inSHIQ+

2. Algorithm 1 checks the clash-freeness of the third kind for each x ∈ V with

∃Q^⊕.C ∈ L(x). To do this, it starts fromxand go up to an ancestorx ofx, and go down to a descendant ofxthrough the functionsucc(x). The length of such a path is bounded byK×LwhereKis given above andLis the number of blocked nodes of the completion tree. Algorithm 1 may consider all paths which go though all possible blocked nodes. The cardinality of this set is bounded by the number of all permutations of the blocked nodes. Therefore, the complexity of Algorithm 1 is bounded by(K×L)×L!. Algorithm 1 would be called for each occurrence of each term such as∃Q^⊕.Cthat occurs in each nodev∈V.

Lemma 3 (Soundness). Let(T,R)be aSHIQ₊knowledge base. LetDbe aSHIQ₊- concept w.r.t.(T,R). If Algorithm 2 can build a clash-free completion tree forDw.r.t.

(T,R)then there is a tableau forDw.r.t.(T,R).

Proof sketch. Assume thatT= (V, E,L, xT, =)^· is a clash-free completion tree forD w.r.t.(T,R). First, we build an extended treeT = (V , E, L, x_T, =)^· fromTwith help of functionssucc andb(x)as follows:

We definexT =xT. Ifx∈V andx ∈ succ(x) then we add toV a successorx ofx. In particular, ifz, zare two distinct successors ofxsuch thatb(z) =b(z)then there are two distinct nodes that are added toV. We define a tableauT= (S,L,E)for Das follows:

– We defineS=V =

n≥0

succⁿ(xT),

– For each s ∈ V there is a uniquexs ∈ V such thatxs ∈ succ^k(x_T)ands ∈ succ^l(xs)withn=k+l. We defineL(s) =L(xs).

– E(R) =E1(R)∪ E2(R)where

E1(R) ={s, t ∈S²|R∈ L(xs, xt)∨R∈ L(xt, xs)}, and

E₂(R) = {s, t ∈S² |(R∈ L(xs, z)∧(b(z) = xt) )∨(R ∈ L(xt, z)∧ (b(z) =xs) )}

We now show thatT satisfiesP8in Definition 3, which is the most problematical prop- erty. For the other properties, we refer the reader to [13].

Assume thats∈Swith∃Q^⊕.C∈ L(s). SinceTis clash-free (third kind),xshas a

∃Q^⊕.C-reachablexni.e. there arex1,· · · , xnsuch thatxi+1is aQ-neighbor ofxior xⁱ+1 blocks aQ-successor ofxⁱwithx^s =x0and∃Q.C ∈ L(xn),∃Q^⊕.C ∈ L(xi) for alli∈ {0,· · ·, n−1}.

Assume that∃Q.C ∈ L(s). This implies that x^s has a Q-neighbory such that C ∈ L(y)due to the non-applicable of∃-rule. By the definition of T, there is some t∈Switht∈succ¹(s)ors∈succ¹(t)such thats, t ∈ E(Q). Thus,P8holds.

Assume that∃Q.C /∈ L(s). According to the definition of∃Q^⊕.C-reachable nodes, there is some0 ≤k < nsuch thatxk is an ancestor ofx0 andxk+1 is a (extended) successor of xk. If k = 0 then there are s1,· · ·, sn with xs_i = xi,s0 = s and si, si+1 ∈ E(Q),∃Q.C∈ L(sn),∃Q^⊕.C ∈ L(si)for alli∈ {0,· · ·, n}. Thus,P8 holds. Assume thatk > 0. We define a functionpred^j(t)as follows:pred^j(t) = xT

ifft ∈ succ^j(xT)for allt ∈ S. This implies that for eacht ∈ Sthere is a unique

jsuch thatpred^j(t) = xT. LetxT =pred^l(s),xT = pred^m(x0) = pred^m(x0)and xT =pred^p(xk) =pred^p(xk). We consider the following cases :

Assumem=l. By the definition ofTthere ares0,· · ·, sⁿ ∈Ssuch thatx^si =xⁱ andsi, sⁱ+1 ∈ E(Q),∃Q^⊕.C ∈ L(si)for alli∈ {0,· · ·, n−1}withs0 =sand

∃Q.C∈ L(sn). Thus,P8holds.

Assumem < l. Let0≤K≤lbe the least number such thatx_pred^K_(s)has a∃Q^⊕.C- reachableywithy∈succ^K(x_pred^K_(s)). We can pickK=l−pwithxT=pred^p(xk) if there is no suchKwithK < l−p. IfK= 0thenk= 0, which was considered. For K >0, we show thatpred^j+1(s),pred^j(s) ∈ E(Q)and∃Q^⊕.C ∈ L(pred^j+1(s)) for allj∈ {0,· · ·, K−1}(***).

Forj = 0, we have s,pred¹(s) ∈ E(Q) and ∃Q^⊕.C ∈ L(pred¹(s)), since s,pred¹(s)∈ E/ (Q)or∃Q^⊕.C /∈ L(pred¹(s))impliesK= 0.

Assume that∃Q^⊕.C ∈ L(pred^j(s))withj < K. Due to the clash-freeness (third kind) ofT,x_pred^j_(s)has a∃Q^⊕.C-reachable nodewi.e. there are nodesw1,· · · , wⁿ and somek ≥0 such thatwk is an ancestor ofx_pred^j_(s),wk+1 is a (extended) suc- cessor ofwk,wi is a Q-neighbor ofwi−1 and∃Q^⊕.C ∈ L(wi),∃Q.C ∈ L(wn) for alli ∈ {1,· · · , n} withw0 = x_pred^j_(s). Due toj < K andL(pred^j+1(s)) = L(x_pred^j+1_(s)), we havek>0,pred^j+1(s),pred^j(s) ∈ E(Q)and

∃Q^⊕.C∈ L(pred^j+1(s)). Thus, (***) holds.

From (***), it follows that there aresi = predⁱ(s)for all i ∈ {0,· · ·, K} and s^K+j = succ^j(pred^K(s))for allj ∈ {1,· · ·, K}such thatsh, s^h+1 ∈ E(Q)and

∃Q^⊕.C ∈ L(sh),∃Q.C ∈ L(sK+K)for allh ∈ {0,· · ·, K +K}withs0 = s.

Thus,P8holds.

Lemma 4 (Completeness). Let (T,R)be a SHIQ₊ knowledge base. Let D be a SHIQ₊-concept w.r.t.(T,R). If there is a tableau forDw.r.t.(T,R)then Algorithm 2 can build a clash-free completion tree forDw.r.t.(T,R).

Proof sketch. LetT = (S,L,E)be a tableau forDw.r.t.(T,R). We show that there exists a sequence of expansion rule applications such that it generates a clash-free com- pletion treeT = (V, E,L, xT, =)^· (**). We define a functionπfromV toSprogres- sively over the construction ofTsuch that it satisfies the following conditions, denoted by (*):

1. L(x)⊆ L(π(x))forx∈V,

2. ifyis aS-neighbor ofxinTthenπ(x), π(y) ∈ E(S), 3. x =y^· impliesπ(x) =π(y),

4. if∃Q^⊕.C∈ L(x)and∃Q.C∈ L(π(x))then∃Q.C∈ L(x)forx∈V,

To prove (**), we have to show that (i) we can apply expansion rules such that the conditions in (*) are preserved, and (ii) if the conditions (*) are satisfied when constructing a completion tree by expansion rules then the obtained completion tree is

clash-free. SinceT is a tableau there is a nodes ∈ Ssuch thatD ∈ L(s). A node x∈ V is created withπ(x) = sandL(x) ={D}. Applications of-rule,-rule,∃- rule,-rule,∀-rule,∀+-rule,≤-rule,≥-rule andch-rule preserve the conditions in (*).

The proof is similar to that in [10]. It is not hard to check that applications of∃+-rule,

⊕-rule preserve the conditions in (*) as well.

We now show that if a completion treeTcan be built with a functionπsatisfying (*) thenTis clash-free.

1. If the condition 1 in (*) is satisfied then there is no nodexinTsuch thatA,¬A˙ ∈ L(x)due toP2and the condition 1. That means thatT does not contain a clash of the first kind as described in Definition 4.

2. There is no clash of the second kind inTif the conditions 1 to 3 in (*) are satisfied withP12.

3. Assume that ∃Q^⊕.C ∈ L(x). Due to the condition 1 in (*), we have∃Q^⊕.C ∈ L(π(x)). According toP8andP4, there ares1,· · · , sⁿ ∈Ssuch thatsi, sⁱ+1 ∈ E(Q),∃Q^⊕.C∈ L(si)and{∃Q.∃Q^⊕.C,∃Q.C}∩L(si) =∅fori∈ {0,· · ·, n−

1}withs0=π(x), and∃Q.C∈ L(s)∪ L(sn−1).

Assume∃Q.C∈ L(s). Due to the condition 4 in (*), we have∃Q.C∈ L(x). This implies thatTdoes not have a clash of the third kind.

Assume ∃Q.C /∈ L(s)andn > 1. Without loss of the generality, assume that

∃Q.C /∈ L(si)for alli ∈ {0,· · ·, n−2}and∃Q.C ∈ L(sn−1)(otherwise, if there is some0≤k < n−1such that∃Q.C∈ L(sk)then we pickn=k+ 1).

By applying successively∃-rule,∃+-rule and-rule, there are nodesx1,· · ·, x^l∈ V such thatπ(xi) =si,Q ∈ L(xi−1, xi)and{∃Q^⊕.C,∃Q.∃Q^⊕.C} ⊆ L(xi) for alli ≤ lwith somel ≤ n−1. Ifl =n−1thenxhas a∃Q^⊕.C-reachable nodexlsuch that∃Q.C∈ L(xl)due to∃Q^⊕.C ∈ L(xl),∃Q.C∈ L(π(xl))and the condition 4 in (*). Ifl < n−1andxlis blocked byzthen we restart from the nodezwith∃Q^⊕.C ∈ L(z)(sinceL(z) =L(xl)) findingx₁,· · ·, xl ∈V which have the same properties as those ofx1,· · ·, xl. This process can be repeated until finding a nodew∈V such thatwis a∃Q^⊕.C-reachable node ofx.

Therefore,Tdoes not have a clash of the third kind.

The following theorem is a consequence of Lemmas 2, 3 and 4.

Theorem 1. Algorithm 2 is a decision procedure for satisfiability ofSHIQ₊-concepts w.r.t. aSHIQ₊knowledge base.

5 Conclusion and Discussion

We have presented a tableaux-based decision procedure forSHIQ₊ concept satisfi- ability. In order to define tableaux for SHIQ+ we have introduced new properties that allow to represent semantic constraints imposed by transitive closure of roles and to avoid expressing explicitly cycles for role inclusion axioms with transitive closure.

These new tableaux properties are translated into new non-deterministic expansion rules which cause the complexity of the tableaux-based algorithm presented in this paper to jump up to double exponential. An open issue consists in investigating whether this

complexity is worst-case optimal. To the best of our knowledge, this problem has not addressed yet. Another future work concerns the extension of our tableaux-based algo- rithm toSHIQ+with nominals.

Acknowledgements. Thanks to Ulrike Sattler for helpful discussions and to the anony- mous reviewers for their comments.

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