An ExpSpace Tableau-based Algorithm for SHOIQ

An E XP S PACE Tableau-based Algorithm for SHOIQ

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 propose an EXPSPACEtableau-based algorithm for SHOIQ. The construction of this algorithm is founded on the standard tableau- based method forSHOIQand the technique used for designing a NEXPTIME

algorithm for the two-variable fragment of first-order logic with counting quanti- fiersC².

1 Introduction

The ontology language OWL-DL [6] is widely used to formalize semantic resources on the Semantic Web. This language is mainly based on the description logicSHOIQ which is known to be decidable [9]. There have been several works on the consistency problem of aSHOIQknowledge base. These works have not only shown decidability and complexity of the problem but also led to develop and implement efficient systems for reasoning on OWL-based ontologies. Tobies [9] has shown that the consistency problem of aSHOIQknowledge base is NEXPTIME-complete. Horrocks et al.[2]

have proposed a tableau-based algorithm that has been exploited to implement reasoners such as Pellet [8], which inherit from the success of early Description Logic reasoners such as FaCT [1].

It has been shown that when nominals are added to DLs the consistency problem is harder. In fact, the complexity jumps from EXPTIME-complete forSHIQto NEXP- TIME-complete forSHOIQ[9]. Kazakovet al.[4] have indicated that when nominals are allowed inSHIQ, the resolution-based approach yields a triple exponential deci- sion procedure for the consistency problem. The authors have also identified that the interaction between nominals, inverse roles and number restrictions makes termination more difficult to be achieved, and thus, is responsible for this hardness.

Our approach is inspired from the technique that was employed by Pratt-Hartmann [7] to construct a NEXPTIMEalgorithm for the logicC²that almost includesSHOIQ.

Unlike the existing tableau-based algorithms, this technique does not explicitly build a graph for representing a model but it builds a structure, called aframe, fromstar-types each of which represents a set of individuals. Pratt-Hartmann [7] shows that a model of aC²knowledge base can be constructed from a frame tiled bywell selectedstar-types.

The present paper is structured as follows. In the next section, we describe the logic SHOIQand the consistency problem for aSHOIQknowledge base. Section 3 de- scribes a 2EXPSPACEtableau-based algorithm for checking consistency of aSHOIQ

knowledge base. An advantage of this algorithm is that a tree-like structure can be main- tained to obtain termination. Section 4 transfers results fromC²[7] toSHOIQ, and presents an EXPSPACE tableau-based algorithm forSHOIQ. Finally, we discuss the results and future work. For the lack of place, we refer the reader to [5] for examples and full proofs.

2 The Description Logic SHOIQ

In this section, we present the syntax and the semantics ofSHOIQ. We start by defin- ing a role hierarchy and its semantics.

Definition 1 (role hierarchy).LetRbe a non-empty set of role namesandR+ ⊆R be a set of transitive role names. We useRI={P⁻|P ∈R}to denote a set of inverse roles. Each element ofR∪RIis called aSHOIQ-role. We defineR :=R⁻ifR∈R, andR := R ifR ∈ RI. Arole hierarchyRis a finite set ofrole inclusion axioms RvSwhereRandSare twoSHOIQ-roles. A relationv∗ is defined as the transitive- reflexive closure ofvonR∪{R vS |RvS∈ R}. We define a functionTrans(R) which returnstrueiff there is someQ ∈R₊∪ {P | P ∈ R₊}such thatQvR. A∗ roleR is calledsimplew.r.t. RifTrans(Q) =false. AninterpretationI = (∆^I,·^I) consists of a non-empty set∆^I(domain) and a function·^Iwhich maps each role name to a subset of∆^I ×∆^I such that R^−I = {hx, yi ∈ ∆^I ×∆^I | hy, xi ∈ R^I}for allR ∈R, andhx, zi ∈ S^I,hz, yi ∈ S^Iimplieshx, yi ∈ S^I for eachS ∈R₊. An interpretationIsatisfies a role hierarchyRifR^I⊆S^Ifor eachRvS∈ R. Such an interpretation is called amodelofR, denoted byI |=R.

Definition 2 (terminology).LetCbe a non-empty set of concept nameswith a non- empty subsetCo⊆Cof nominals. The set ofSHOIQ-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,CandDareSHOIQ-concepts,Ris an SHOIQ-role andSis 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∆^Isuch that>^I =∆^I, (CuD)^I=C^I∩D^I, (CtD)^I =C^I∪D^I, (¬C)^I = ∆^I\C^I,card{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 | card{y ∈ C^I | hx, yi ∈ S^I} ≥ n}, (≤n S.C)^I={x∈∆^I|card{y∈C^I | hx, yi ∈S^I} ≤n}

wherecard{S}is denoted for the cardinality of a setS.

∗ C v D is called a general concept inclusion (GCI) where C, D are SHOIQ- concepts (possibly complex), and a finite set of GCIs is called a terminologyT.

∗An interpretationIsatisfies a GCICvDifC^I ⊆D^IandIsatisfies a terminology T ifI satisfies each GCI inT. Such an interpretation is called amodelofT, denoted byI |=T.

Definition 3 (knowledge base).A pair(T,R)is called aSHOIQknowledge base whereRis aSHOIQrole hierarchy andT is aSHOIQterminology. A knowledge

base(T,R)is said to be consistent if there is a modelIof bothT andR, i.e.,I |=T andI |=R. A conceptC is calledsatisfiablew.r.t.(T,R)iff there is some interpre- tationI such that I |= R,I |= T andC^I 6= ∅. Such an interpretation is called a modelofCw.r.t.(T,R). A conceptDsubsumesa conceptCw.r.t.(T,R), denoted by CvD, ifC^I⊆D^Iholds in each modelIof(T,R).

Thanks to the reductions between unsatisfiability, subsumption of concepts and knowl- edge base consistency, 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. Any SHOIQ-concept can be transformed to an equivalent one in NNF by using DeMorgan’s laws and some equivalences as presented in [3]. For a conceptC, we denote thennfofCbynnf(C) and thennfof¬Cby¬C. Let˙ Dbe anSHOIQ-concept in NNF. We definecl(D)to be the smallest set that contains all sub-concepts ofDincludingD. For a knowledge base(T,R), we can define a setcl(T,R). For the sake of brevity, we refer the reader to [2] for a more complete definition.

To prove soundness and completeness of our algorithms, we need a tableau structure that represents a model of a SHOIQ knowledge base. Regarding the definition of tableaux forSHOIQpresented in [2], we add a new property that imposes an exact number ofS-neighbour individualstofsif(≤ nS.C) ∈ L(s). This property makes explicit non-determinism implied from the semantics of(≤nS.C)and requires an extra expansion rule for the tableau-based algorithm.

3 A 2E

S

Decision Procedure for SHOIQ

In this section, we introduce a structure, calledSHOIQ-forest. We will show that such a forest is sufficient to represent a model of aSHOIQ-knowledge base.

Definition 4 (tree).Let(T,R)be aSHOIQknowledge base. For each o ∈ C_o, a SHOIQ-tree for(T,R), denoted byT_o= (V_o, E_o,L_o,xb_o,6=^·_o), is defined as follows:

∗Vois a set of nodes containing a root nodexbo ∈Vo. Each nodex∈Vois labelled with a functionLosuch that Lo(x) ⊆ cl(T,R)and o ∈ Lo(xb_o). A nodex ∈ V_o is callednominalifo⁰ ∈ Lo(x)for someo⁰∈C_o. In addition, the inequality relation6=^·o

is a symmetric binary relation overV_o.

∗Eois a set of edges. Each edgehx, yi ∈Eois labelled with a functionLosuch that Lo(hx, yi) ⊆ R_(T,R). If hx, yi ∈ E_o theny is called a successor ofx, denoted by y ∈ succ¹(x), orxis called the predecessor ofy, denoted byx = pred¹(y). In this case, we say that xis a neighbour of y or y is a neighbour of x. If z ∈ succⁿ(x) (resp.z =predⁿ(x)) andy is a successor ofz (resp.y is the predecessor ofz) then y ∈succ⁽ⁿ⁺¹⁾(x)(resp.y =pred⁽ⁿ⁺¹⁾(x)) for alln ≥0wheresucc⁰(x) ={x}and pred⁰(x) =x. A nodeyis called adescendantofxify ∈succⁿ(x)for somen > 0.

A nodey is called anancestorofxify = predⁿ(x)for somen > 0. To ensure that Tois a tree, it is required that (i)xis a descendant ofxbofor allx∈Vowithx6=bxo, and (ii) each nodex∈ Vowithx6= bxohas a unique predecessor. A nodey is called anR-successor ofx, denoted by y ∈ succ¹_R(x)(resp.y is called theR-predecessor

ofx, denoted byy = pred¹_R(x)) if there is some role R⁰ such thatR⁰ ∈ L_o(hx, yi) (resp. R⁰ ∈ L_o(hy, xi)) and R⁰vR. A node∗ y is called a R-neighbour of xif y is either aR-successor orR-predecessor ofx. Ifzis anR-successor ofy(resp.zis the R-predecessor ofy) andy ∈ succⁿ_R(x)(resp.y = predⁿ_R(x)) thenz ∈succ⁽ⁿ⁺¹⁾_R (x) (resp.z=pred⁽ⁿ⁺¹⁾_R (x)) forn≥0withsucc⁰_R(x) ={x}andx=pred⁰_R(x).

∗For a nodex, a roleSando∈Co, we define the setS^To(x, C)ofx’sS-neighbours as follows:S^To(x, C) ={y∈Vo|yis aS-neighbour ofxandC∈ Lo(x)}.

∗A nodexis callediteratedbyyw.r.t. a nodex_oifxhas no nominal ancestor except forbxoand there are integersn, m >0and nodesx⁰, y⁰ such that : (i)xo=predⁿ(y), y = pred^m(x), (ii)x⁰ = pred¹(x),y⁰ = pred¹(y), (iii)Lo(x) = Lo(y),Lo(x⁰) = Lo(y⁰), (iv)Lo(hx⁰, xi) = Lo(hy⁰, yi), and (v) if there arez, z⁰ andi > 0 such that z⁰ = pred¹(z),predⁱ(z⁰) =xo,Lo(z) = Lo(y),Lo(z⁰) =Lo(y⁰)andLo(hz⁰, zi) = Lo(hy⁰, yi)theni≥n.

A nodexis called 1-iterated by y if xis iterated by y w.r.t. xbo. A node xis called blockedbyy, denoted byy=b(x), ifxis iterated byyw.r.t. a 1-iterated nodexo.

∗ In the following, we often useL(x),L(hx, yi),S^T(x, C)and6=^· instead ofLo(x), Lo(hx, yi),S^To(x, C)and6=^·o, respectively. This does not cause any confusion since Vo∩Vo⁰ =∅andEo∩Eo⁰ =∅ifo6=o⁰. In addition,x6=^·oyis never defined forx∈Vo

andy∈Vo⁰ witho6=o⁰.

We remark that the definition of 1-iterated nodes in Definition 4 forSHOIQ-trees is very similar to the standard definition of blocked nodes forSHIQcompletion trees (see [3]). Moreover, if we consider the sub-tree rooted at a 1-iterated node as aSHIQ completion tree then blocked nodes according to Definition 4 are also blocked nodes according to the standard definition for thisSHIQcompletion tree.

ASHOIQ-tree consists of two layers : the first layer is formed of nodes from the root to 1-iterated nodes or nominal nodes, and the second layer consists of nodes from each 1-iterated node to blocked or nominal nodes. In addition, each nodexin the layer 2 has a unique 1-iterated node, denotedbb(x), such thatbb(x)is an ancestor ofx.

Definition 5 (forest). Let(T,R)be a SHOIQknowledge base. ASHOIQ-forest for(T,R)is a pairG= hT, ϕi, whereT = {To | o ∈ Co}is a set ofSHOIQ- trees for(T,R)withTo = (Vo, Eo,Lo,bxo,6=^·o), andϕis apartitioningfunctionϕ : V → 2^V withV =S

o∈CoVo. We denoteL⁰(hx, yi) =Lo(hx, yi)ifhx, yi ∈Eo, and L⁰_o(hx, yi) ={S |S∈ Lo(hy, xi)}ifhy, xi ∈Eofor someo∈Co. The partitioning functionϕsatisfies the following conditions:

1. For eachx∈ V,ϕ(x)is thepartitionofxwithx∈ϕ(x). There arex0,· · ·, xn∈ V such thatϕ(xi)∩ϕ(xj) =∅with0≤i < j≤nandS

0≤i≤nϕ(xi) =V;

2. For allx, x⁰ ∈ V, ifx⁰ ∈ ϕ(x)thenϕ(x) = ϕ(x⁰)andL(x) = L(x⁰). We de- noteΛ(ϕ(x)) = L(x). In addition, an inequality relation over partitions can be described as follows : forx, x⁰ ∈ V we defineϕ(x)6=ϕ(x^{· 0})if there are two nodes y∈ϕ(x)andy⁰ ∈ϕ(x⁰)such thaty6=^·oy⁰for someo∈Co;

3. For allϕ(x)andϕ(x⁰), if there are two edgeshy, y⁰i ∈E_oandhw, w⁰i ∈E_o⁰with o, o⁰∈Cosuch thaty, w∈ϕ(x),y⁰, w⁰∈ϕ(x⁰),L⁰(hy, y⁰i)6=∅,L⁰(hw, w⁰i)6=∅ thenL⁰(hy, y⁰i) =L⁰(hw, w⁰i).

We define a functionΛ(h·,·i)for labelling edges ended by two partitions as follows:

Λ(hϕ(x), ϕ(x⁰)i) =L⁰(hz, z⁰i)wherez∈ϕ(x),z⁰ ∈ϕ(x⁰),L⁰(hz, z⁰i)6=∅, and {hz, z⁰i,hz⁰, zi} ∩E_o⁰ 6= ∅ for someo⁰ ∈ C_o. We say ϕ(x⁰)is aS-neighbour partition ofϕ(x)ifS∈Λ(hϕ(x), ϕ(x⁰)i).

4. For allx, x⁰∈ V, ifo∈ L(x)∩ L(x⁰)for someo∈Coandϕ(x)6=ϕ(x^{· 0})does not hold thenϕ(x) =ϕ(x⁰);

5. If(≤nR.C)∈Λ(ϕ(x))for somex∈ Vand there exist(n+1)nodesx0,· · ·, xn∈ Vsuch that (i)ϕ(xi)∩ϕ(xj) =∅for all0≤i < j ≤n, and (ii)C ∈Λ(ϕ(xi)), R∈Λ(hϕ(x), ϕ(xi)i)for alli∈ {0,· · · , n}, thenϕ(xl)6=ϕ(x^· m)for all0≤l <

m≤n; and

6. If(≥nR.C)∈Λ(ϕ(x))for somex∈ Vthenϕ(x)hasn R-neighbour partitions ϕ(x₁),· · ·, ϕ(x_n)such thatϕ(x_i)∩ϕ(x_j) =∅andC∈Λ(ϕ(x_i))for all1≤i <

j≤n.

∗Clashes:Tis said to contain aclashif one of the following conditions holds:

1. There is some nodex∈ Vsuch that{A,¬A} ⊆˙ Λ(ϕ(x))for some concept name A∈C;

2. There are nodesx, y ∈ Vsuch thatϕ(x)6=ϕ(y)^· ando ∈Λ(ϕ(x))∩Λ(ϕ(y))for someo∈C_o;

3. There is a nodex ∈ V with(≤ nR.C) ∈ Λ(ϕ(x))and there are (n+ 1)nodes x₀,· · · , x_n ∈ Vsuch thatϕ(x_i)∩ϕ(x_j) =∅,ϕ(x_i)6=ϕ(x^· _j)with0≤i < j≤n, andC∈Λ(ϕ(xi)),R∈Λ(hϕ(x), ϕ(xi)i)fori∈ {0,· · ·, n}.

We now describe the tableau-based algorithm whose goal is to construct from a knowledge base(T,R)aSHOIQ-forestG= hT, ϕi. To do this, the algorithm ap- plies expansion rules (as described in Fig. 1 and Fig. 2 in [5]), and terminates when none of the rules is applicable. The obtainedGis calledcomplete, and ifGcontains no clash thenGis calledclash-free. In this case, we also sayTois complete and clash-free for allTo∈T.

The expansion rules maintain the tree-like structure ofSHOIQ-forest and they are similar to those in [2] except that if a conceptCis added to the label of a nodexdue to application of these rules thenCis propagated to the label of each nodey ∈ϕ(x).

Moreover, all rules in Fig. 1 in [5] except for∃- and≥-rule update only the label of nodes or edges and do not change the partitioning functionϕ. In particular, when the

≤-rule is applied to a nodexwith twoS-neighboursy, z of x, it must propagate the label ofhx, yito that of allhx⁰, z⁰i(orhz⁰, x⁰i) wherex⁰ ∈ϕ(x)andz⁰ ∈ ϕ(z), and set the label ofhx, yito empty set. This may changeϕonly ifϕ(y)is singleton. By a new rule, namely the./-rule, each nodexcontaining a term(≤nS.C)has exactlym S-neighbours containingCwith somem≤n. As a result, this rule and≥-rule ensure that if there are two nodesy, y⁰ ∈ ϕ(x)theny andy⁰ have exactlym S-neighbours which containC in their label. Finally, we can avoid infinite sequences of “merging- and-generating” without pruning nodes since all merges due to number restrictions or nominals are performed by updating the partitioning function. The following lemma establishes correctness and completeness of the algorithm.

Lemma 1. Let(T,R)be aSHOIQknowledge base.

1. The tableau algorithm terminates and builds aSHOIQ-forest whose the size is bounded by a doubly exponential function in the size of(T,R).

2. If the tableau algorithm yields a clash-free and completeSHOIQ-forest for(T,R) then there is a tableau for(T,R).

3. If there is a tableau for(T,R)then the tableau algorithm yields a clash-free and completeSHOIQ-forest for(T,R).

To prove soundness of the tableau algorithm, we can devise a model from a clash-free and completeSHOIQ-forest by considering a partition as an individual and unravel- ling blocked nodes since we can show that each blocking nodeb(x)has no “core path”

fromb(x)to each nominal descendanty, i.e., there do not exist terms(≤miRi.Ci)∈ predⁱ(y), rolesRi ∈ L(hpredⁱ⁻¹(y),predⁱ(y)i)and conceptsCi ∈ L(predⁱ⁺¹(y))for k < i≤0,b(x) =pred^k(y). The following theorem is a consequence of Lemma 1.

Theorem 1. Let (T,R)be a SHOIQknowledge base. The tableau algorithm is a decision procedure for consistency of(T,R)and it runs in2NEXPTIMEin the size of (T,R).

4 An E

S

Tableau-based Algorithm for SHOIQ

This section starts by translating some results presented in [7] forC² into those for SHOIQ.

Definition 6 (star-type).Let(T,R)be aSHOIQknowledge base. Astar-typeis a tripletσ = hλσ,ν¯σ,µ¯σi, whereλσ ∈ 2^cl(T,R),µ¯σ = (hr1, l1i,· · ·,hrd, ldi)is ad- tuple over2^R(T,R)×2^cl(T,R), andν¯σ = (hr⁰, l⁰i)withhr⁰, l⁰i ∈ 2^R(T,R)×2^cl(T,R). A pairhr, liis arayofσifhr, liis a component ofµ¯σ orν¯σ. In particular,hr, liis a predecessor rayif(hr, li) = ¯νσ, andhr, liis a successor rayif hr, liis a component ofµ¯σ. We denote ξ¯σ = (hr1, l1i,· · ·,hrd, ldi,hrd+1, ld+1i)if ν¯σ = (hr⁰, l⁰i)where r⁰ =rd+1,l⁰ =ld+1, andξ¯σ = ¯µσifν¯σis empty.

– A rayhr⁰, l⁰iofσisprimaryw.r.t. a term(≤mR.C)if(≤mR.C)∈λσ,R∈ r⁰ andC ∈ l⁰. For a term(≤ mR.C) ∈ λσ, we denoteC^σ_h≤mR.Cifor the set of all rayshr⁰, l⁰iofσsuch thatR∈r⁰,C∈l⁰.

– A star-typeσisnominalifo∈λσfor someo∈Co.

– A star-typeσischromaticif there is a term(≥ nS.D) ∈ λ_σ andσhasnrays hr₁⁰, l⁰₁i,· · ·,hr⁰_n, l⁰_nisuch thatS∈l⁰_i,D∈l⁰_ifor all1≤i≤n, andl_i⁰6=l⁰_jfor all 0≤i < j≤nwithl⁰₀=λ_σ.

– A star-typeσishomomorphic(resp.isomorphic) to a star-typeσ⁰ ifλσ = λσ⁰, and for each term(≤ mR.C) ∈ λσ, there is an injection (resp. a bijection)π : C_h≤mR.Ci^σ → C_h≤mR.Ci^σ0 such thatπ(hr, li) =hr⁰, l⁰iimpliesr⁰=randl⁰=l.

– Two star-typesσ, σ⁰areequivalentifλσ =λσ⁰, and there is a bijectionπbetween ξ¯σandξ¯σ⁰such thatπ(hr, li) =hr⁰, l⁰iimpliesr⁰=randl⁰=l.

We denoteΣfor the set of all star-types for(T,R). C

In the context of aSHOIQ-forest, we can think of a star-typeσas the set of nodes xsuch thatL(x) =λ_σ, and each rayhr_i, l_iiofσcorresponds to a neighbourx_iofx such thatL⁰(hx, x_ii) =r_iandL(x_i) =l_i. In this case, we say thatxsatisfiesσ.

Remark 1. The notion of chromaticity introduced in Definition 6 implies an inequality relation6=^· over nodes. That stronger notion is needed to prevent “distinct” star-types from including nodesx, ywhich are neighbours or x6=y. In order to make star-types^· chromatic, it is necessary to add to knowledge bases some new concepts and axioms as follows. Let (T,R)be a SHOIQknowledge base. For each term(≥ nS.D) ∈ cl(T,R), we add tocl(T,R)nnew concept namesC_(≥nS.D)⁰ ,· · · , C_(≥nS.D)ⁿ , and to T the following axioms: C_(≥nS.D)ⁱ uC_(≥nS.D)^j v ⊥ for all0 ≤ i < j ≤ n. It is straightforward to prove that the terminology(T⁰,R)is consistent iff(T,R)is con- sistent whereT⁰is obtained fromT by adding these new axioms. Thanks to these new concepts and axioms, the following definition points out how to build chromatic star- types.

Definition 7 (valid star-type).Let (T,R)be aSHOIQknowledge base. Letσ be a star-type for (T,R)where σ = hλσ,¯ν,µi¯ with µ¯ = (hr1, l1i,· · · ,hrd, ldi)and λσ = l0,ν¯ ={hrd+1, ld+1i}.σisvalidwith an inequality relation6=^· overC^σ if the following conditions are satisfied:

1. IfCvD∈ T thennnf(¬CtD)∈lifor all0≤i≤d+ 1;

2. {A,¬A} 6⊆lifor every concept nameAwith0≤i≤d+ 1;

3. IfC1uC2∈lithen{C1, C2} ⊆lifor all0≤i≤d+ 1;

4. IfC1tC2∈lithen{C1, C2} ∩li6=∅for all0≤i≤d+ 1;

5. If∃R.C∈λσthen there is some1≤i≤d+ 1such thatC∈liandR∈ri; 6. If(≤nS.C)∈λσand there is some1≤i≤d+ 1such thatS ∈rithenC ∈li

or¬C˙ ∈li;

7. If(≤nS.C)∈λσand there is some1 ≤i≤d+ 1such thatC ∈li andS ∈ri

then there is some1≤m≤nsuch that{(≤mS.C),(≥mS.C)} ⊆λ;

8. For each1≤i≤d+ 1, ifR∈riandRvS∗ thenS∈ri; 9. If∀R.C∈λ_σandR∈r_ifor some1≤i≤d+ 1thenC∈l_i;

10. If∀R.D∈λ_σ,SvR,∗ Trans(S)andR∈r_ifor some1≤i≤d+1then∀S.D∈l_i; 11. If(≥nS.C)∈λ_σ thenC_(≥nS.C)⁰ ∈λ_σand there are1≤i₁<· · ·< i_n ≤d+ 1

such that{C, C_(≥nS.C)^j } ⊆l_ij,S∈r_ij for all1≤j ≤n.

12. If(≤nS.C)∈λσand there are no1≤i1<· · ·< in+1≤d+ 1such thatC∈li_j

andS∈ri_j for all1≤j≤n+ 1. C

Notice that a valid star-type according to Definition 7 is chromatic. If we think of a star-typeσas a nodexsatisfyingσin aSHOIQ-forest thenσis valid if no expansion rule is applicable tox. Moreover, due to the conditions 7, 11 and 12 in Definition 7, if there is a term(≤nS.D)∈λσfor a valid star-typeσthenσhas exactlynprimary rays hri, lii,· · · ,hrn, lniw.r.t.(≤nS.D).

Definition 8 (frame).Let(T,R)be aSHOIQknowledge base. Aframefor(T,R)is a tupleF=h(N0,· · · ,NH), δ, Φ,bδi, whereH ∈Nis thedimensionofF;Ni⊆Σfor

all0≤i≤H, and all star-types inN₀are nominal. We denoteN=S

i∈{1,···,H}N_i; δis a functionδ:N→N;Φis a functionΦ: ¯N→ΣwhereN¯ is denoted for the set of all star-typesσ∈Nsuch that one of the three condition holds : (i)σis nominal; (ii)σ has a rayhr, lisuch that there is a term(≤mR.C)with(≤mR.C)∈λ_σ,C∈land R∈r; (iii)σhas a rayhr, lisuch that there is a term(≤mR.C)with(≤mR.C)∈l, C∈λσandR∈r;δbis a functionbδ:Φ( ¯N)→Nwhich is defined as follows:

For two star-typesσ, σ⁰ ∈ N,¯ Φ(σ) = Φ(σ⁰)iff eitherσis isomorphic toσ⁰, or there is a star-typeω∈N\ N_Hsuch thatσandσ⁰are homomorphic toω.

Additionally, a star-typeσ∈ Nk(0 < k < H) islinkablewith a star-typeσ⁰ ∈ Nk−1

by a rayhr, liofσifσ⁰has a rayhr⁰, l⁰isuch thatλ_σ⁰ =l,r⁰ =r⁻andl⁰ =λ_σwhere r⁻ ={R |R∈r}.

Remark 2. Forσ, σ⁰ ∈ N/ H andσ, σ⁰ are valid, ifσis homomorphic toσ⁰ thenσis isomorphic toσ⁰. In fact, if there is a term(≤ mR.C) ∈ λσ then bothσ, σ⁰ have exactlymprimary rays w.r.t.(≤mR.C). If there is a homomorphism between the two sets of primary rays w.r.t.(≤mR.C)then it is an isomorphism as well.

The frame structure, as introduced in Definition 8, allows us to tile a forest structure by star-types. Such a structure is crucial to obtain termination when designing a tableau- based algorithm. An important difference between a frame and aSHOIQ-forest is that a frame does not represent nodes corresponding to individuals but stores the number of individuals satisfying a star-type. The functionδ(σ)is used for this purpose. In the context of a SHOIQ-forest, we can think of Φ(σ) as a star-type which is satisfied by nodes forming a set of partitions. In fact, the functionΦmaps star-types forming a SHOIQ-forest into another set of star-types that regroups non-neighbour partitions.

Notice that the functionΦintroduced in Definition 8 does not transfer the relation of linkability fromN¯ toΦ( ¯N), and that chromaticity of star-types prevents chromati- cally linkable star-types from collapsing into a unique star-type by the functionΦ. The functionδ(Φ(σ))b counts the number of partitions that are mapped to a star-type byΦ.

Such a function can be defined if all star-types “covering” a partition must be mapped to a unique star-type byΦ. This is a consequence of theΦ’s definition.

Definition 9 (valid frame).Let(T,R)be aSHOIQknowledge base. A frameF = h(N0,· · · ,NH), δ, Φ,bδiisvalidif the following conditions are satisfied:

1. For eachσ∈N, ifδ(σ)≥1thenσis valid;

2. For eachσ∈N, if¯ δ(σ)≥1thenbδ(Φ(σ))≥1;

3. For eacho∈C_othere is a uniqueσ_o∈ N₀such thato∈λ_σoandδ(σ_o) = 1;

4. For eachσ∈N, ifo∈λ_σwith someo∈C_othenΦ(σ) =Φ(σ_o)andδ(Φ(σb _o)) = 1withσ_o∈ N₀such thato∈λ_σo andδ(σ_o) = 1;

5. For each0 ≤k < H andhλ, r, λ⁰i ∈2^cl(T,R)×2^R(T,R)×2^cl(T,R)withr⁻ = {R |R∈r},

X

σ∈Nk

δ(σ)|µ¯_σ|_hλ,r,λ0i= X

σ⁰∈Nk+1

δ(σ⁰)|¯ν_σ⁰|_hλ0,r⁻,λi

where|¯ν_σ|_hλ,r,λ0iand|¯µ_σ|_hλ,r,λ0iare denoted for the number of componentshr⁰, l⁰i of respectiveν¯_σandµ¯_σsuch thatλ_σ =λ,r⁰=randl⁰=λ⁰;

6. For eachhλ, r, λ⁰i ∈2^cl(T,R)×2^R(T,R)×2^cl(T,R)withr⁻={R |R∈r}, X

σ∈N¯

bδ(Φ(σ)) ξ¯_Φ(σ)

_hλ,r,λ0i= X

σ⁰∈N¯

δ(Φ(σb ⁰)) ξ¯_Φ(σ⁰₎

_{hλ0,r−,λi}

where ξ¯_Φ(σ)

_hλ,r,λ0i=

¯ν_Φ(σ)

_hλ,r,λ0i+

¯µ_Φ(σ)

_hλ,r,λ0i C

Remark 3. It is not required that star-typesΦ(σ)are valid. We will use functionΦto trim rayshr, liof star-types such that (i)hr, liorhr⁻, liis not a primary ray of every star-type. The images of star-typesσbyΦ, i.e. trimmed star-typesΦ(σ), are employed to represent partitions obtained from merge processes. As described in Section 3, in order to govern partitions, it suffices to deal with nodesxcontaining a term(≤mR.C) and the R-neighbours of xcontainingC. For this reason, the functionΦmaps only star-types inN¯ by trimming non-primary rays.

The notion of validity for a frame is crucial to establish a connection with the tableau-based algorithm presented in Section 3, i.e., how to build a SHOIQ-forest from a valid frame, and inversely. The condition 1 in Definition 9 requires that every star-type satisfied by at least one node must be valid. The condition 2 implies that each valid star-type including a primary ray will be mapped byΦ. The condition 3 ensures that each nominal is counted exactly once while the condition 4 imposes that all nomi- nal star-types containing someo∈Coare mapped into a unique star-type byΦ. In the context of aSHOIQ-forest, these conditions imply that for each nominalo∈Cothere is exactly one tree whose root containsoand there is exactly one partition containing o. The condition 5 allows for linking star-types at levelkwith star-types at levelk−1 andk+ 1. It ensures that each nodexsatisfying (or counted for) a star-typeσat level kis linked by its rays to neighbours satisfying star-types at levelk−1andk+ 1. The number of these neighbours corresponds exactly to the number ofσ’s rays. Finally, the condition 6 deals with partitions. In the context of aSHOIQ-forest whereΦ(σ)rep- resents the image of a set of partitions, the condition 6 points out how star-typesΦ(σ) would be interconnected.

Lemma 2. Let(T,R)be aSHOIQknowledge base.

1. If the tableau algorithm can build a clash-free and completeSHOIQ-forest for (T,R)then there is a valid frame for(T,R).

2. If there is a valid frameF =h(N₀,· · ·,N_H), δ, Φ,δib for(T,R)then the tableau algorithm can build a clash-free and completeSHOIQ-forest for(T,R).

Lemma 2 points out the equivalence between a clash-free and completeSHOIQ- forest and a valid frame for(T,R). The following lemma affirms that there is an expo- nential structure, a valid frame, which can represent aSHOIQ-forest whose size may be doubly exponential.

Lemma 3. Let(T,R)be aSHOIQknowledge base. The size of a valid frameF = h(N0,· · · ,NH), δ, Φ,bδiis bounded by an exponential function in the size of(T,R).

We can sketch a proof of the lemma here. We haveH ≤K whereK = 2^2m+k ×2 withm = card{cl(T,R)} andk = card{R_(T,R)}. Moreover, each star-type has at mostM distinct rays whereM =Pm_i+E,m_ioccurs in a number restriction term (≥miR.C)appearing inT, andEis the number of distinct terms∃R.Cappearing in T . If we denoteΣfor the set of all star-types thencard{Σ} ≤((card{cl(T,R)})²× card{R_(T,R)})^M. Sinceδ(σ)is bounded byM δ(σ⁰)whereσ⁰, σ are respective star- types at levelk−1andk, it holds thatδ(σ)≤M^22m+k×2. Ifδ(σ)is represented as a binary number then it takes an exponential number of bits.

Based on Lemma 3 and 2, we can present straightforwardly an optimal worst-case algorithm for checking the consistency of aSHOIQknowledge base. However, such an algorithm cannot be used in practice since the non-determinism is not sufficiently constrained to obtain termination in feasible time. In the sequel, based on the results obtained so far, we try to design an algorithm which has more goal-directed behaviour.

Blocking Condition for a FrameLetF=h(N0,· · · ,NH), δ, Φ,bδibe a frame. A star- typeσk ∈ Nk with0< k ≤H isblockedif there areσi ∈ Niwith0 ≤i≤ksuch thatσiis linkable withσ_i−1for alli∈ {1,· · ·, k}, then there are0< k1< k2< k3<

k4≤ksuch that:

1.λσ_k1 =λσ_k2,ν¯σ_k1 = ¯νσ_k2, and there is no0< j < k2such thatj6=k1,λσ_j =λσ_k2

andν¯σ_j = ¯νσ_k2;

2. λσ_k3 = λσ_k4, ¯νσ_k3 = ¯νσ_k4, and there is no k2 < j < k4 such thatj 6= k3, λσj =λσ_k4 andν¯σj = ¯νσ_k4.

Notice that this blocking condition is looser than the blocking condition introduced in Definition 5 for aSHOIQ-forest. Since we cannot determine the path from root to a node satisfying a star-type over a frame, it is not possible to check blocking condition in the same way as for aSHOIQ-forest. The blocking condition for a frame, as described above, implies that a node satisfying a blocked star-type must have an ancestor which is blocked according to the blocking condition for aSHOIQ-forest.

We are now ready to propose an EXPSPACEtableau-based algorithm forSHOIQ.

It starts by generating nominal star-types at level 0. The goal of the algorithm is to replace progressively non-valid star-types with those which are “nearer” the validity.

When a non-valid star-typeσat a levelhis replaced with a “better” oneσ⁰at the same level, the algorithm addsδ(σ)toδ(σ⁰)and setsδ(σ) = 0. This may lead to updateδ(ω) whereω is linkable withσ. In addition, the algorithm has to maintain two functions Φ(σ)andδ(Φ(σ))b such that the conditions 5 and 6 in Definition 9 always hold after each update of star-types.

An important difference between the tableau algorithm presented in Section 3 and the tableau algorithm for constructing a valid frame is that the latter adds star-types to a frame and updates functionsδ(σ)andbδ(Φ(σ))instead of adding nodes for representing individuals. Again, we refer the readers to [5] where the rules for building a valid frame, calledframe rules, can be found.

Soundness of the tableau-based algorithm for building a frame can be established thanks to Lemma 2. Since each frame rule has its counterpart in the expansion rules, completeness of the algorithm can be shown by using the same arguments as those employed to prove Lemma 1. From these results and Lemma 3, we obtain the following main result of the section:

Theorem 2. Let(T,R)be aSHOIQknowledge base. The tableau algorithm for con- structing a frame is a decision procedure for consistency of(T,R)and it runs inEX-

PSPACEin the size of(T,R).

5 Conclusion and Discussion

We have presented in this paper a practical EXPSPACEdecision procedure for the logic SHOIQ. The construction of this algorithm is founded on the well-known results for SHOIQandC². First, we have based our approach on a technique that constructs tree- like structures for representing a model. This allows us to reuse the standard blocking technique over these tree-like structures to obtain termination. Second, we have trans- ferred toSHOIQthe method used for constructing a NEXPTIMEalgorithm for C². This enables us to represent a doubly exponentialSHOIQ-forest by an exponential structure.

The tableau algorithms proposed in the present paper have introduced several new non-deterministic rules, e.g.,./- or≤_o-rules. In particular, the most non-deterministic behaviour of the tableau algorithm for building a valid frame is to update the function δfor star-typesωwhen applying frame rules to a star-typeσwhich is linkable withω.

Such an update may lead to choose a subset from an exponential set of linkable star- types. An open issue consists in investigating whether the complexity resulting from this behaviour is comparable to that caused by the≤- and≤o-rules.

Acknowledgements.Thanks to the reviewers for their helpful comments.

References

1. Horrocks, I.: The FaCT system. In: de Swart, H. (ed.) Proc. of the 2nd Int. Conf. on Analytic Tableaux and Related Methods (TABLEAUX’98). Lecture Notes in Artificial Intelligence, vol. 1397, pp. 307–312. Springer (1998), download/1998/Horr98b.pdf

2. Horrocks, I., Sattler, U.: A tableau decision procedure forSHOIQ. Journal Of Automated Reasoning 39(3), 249–276 (2007)

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

Proceedings of the International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR 1999). Springer (1999)

4. Kazakov, Y., Motik, B.: A resolution-based decision procedure forSHOIQ. Journal of Au- tomated Reasoning 40(2–3), 89–116 (2008)

5. Le Duc, C., Lamolle, M., Cur´e, O.: AnEXPSPACEtableau-based algorithm forSHOIQ. In:

Technical Report. http://www.iut.univ-paris8.fr/∼leduc/papers/RR2012a.pdf (2012) 6. Patel-Schneider, P., Hayes, P., Horrocks, I.: OWL web ontology language semantics and ab-

stract syntax. In: W3C Recommendation (2004)

7. Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Jour- nal of Logic, Language and Information 14(3), 369–395 (2005)

8. Sirin, E., Parsia, B., Grau, B.C., Kalyanpur, A., Katz, Y.: Pellet: a pratical OWL-DL reasoner.

Journal of Web Semantics 5(2), 51–53 (2007)

9. Tobies, S.: The complexity of reasoning with cardinality restrictions and nominals in expres- sive description logics. Journal of Artificial Intelligence Research 12, 199–217 (2000)