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An ExpSpace Tableau-based Algorithm for SHOIQ
Chan Le Duc, Myriam Lamolle, Olivier Curé
To cite this version:
Chan Le Duc, Myriam Lamolle, Olivier Curé. An ExpSpace Tableau-based Algorithm for SHOIQ.
Description Logic 2012, Jun 2012, Rome, Italy. pp.11. �hal-00799028�
An E XP S PACE Tableaux-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 EXPSPACEtableaux-based algorithm for SHOIQ. The construction of this algorithm is founded on the standard tableaux- 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 [1] 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 [2]. An interesting feature of logics with nominals (denoted byOinSHOIQ) is that they allow for expressing relationships, represented as role instances, between two sets of individuals which are represented as nominals or standard concepts. Such sets of individuals can be finitely enumerable or infinite.
There were several works on the consistency problem of aSHOIQ knowledge 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. A result in [2] has shown that the consistency problem of aSHOIQknowl- edge base is NEXPTIME-complete. Moreover, tableaux-based algorithms presented in [3] forSHOIQhave been exploited to implement reasoners such as Pellet [4], which inherit from the success of early Description Logic reasoners such as FaCT [5].
It has been shown that when nominals are added to these DLs the consistency prob- lem is harder. In fact, the complexity jumps from EXPTIME-complete for SHIQto NEXPTIME-complete forSHOIQ[2]. The work in [6] has indicated that when nom- inals are allowed inSHIQ, the resolution-based approach yields a triple exponential decision procedure for the consistency problem. The authors have also identified that the interaction between nominals, inverse roles and number restrictions makes termina- tion more difficult to be achieved, and thus, is responsible for this hardness.
Our approach is inspired from a technique that was employed by Ian Pratt-Hartmann in [9] to construct a NEXPTIMEalgorithm for the logicC2includingSHOIQ. Unlike the existing tableaux-based algorithms, this technique does not explicitly build a graph for representing a model but it builds a structure, called aframe, fromstar-typeseach of which represents a set of individuals. A result from [9] shows that a model of aC2 knowledge 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 2EXPSPACEtableaux-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 a result in [9] fromC2 toSHOIQ.
Based on the these results, we propose an EXPSPACE tableaux-based algorithm for SHOIQ. Finally, we discuss the results and future work.
For the lack of place, we refer the reader to [10] 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 define a functionR⊖which returnsR−ifR∈R, and returnsRifR∈RI. Arole hierarchyRis a finite set ofrole inclusion axiomsR⊑SwhereRandSare twoSHOIQ-roles. A relation⊑∗ is defined as the transitive-reflexive closure of⊑onR ∪ {R⊖ ⊑S⊖|R⊑S∈ R}. We define a functionTrans(R)which returnstrueiff there is someQ∈R+∪{P⊖|P ∈R+}such thatQ⊑R. A role∗ Ris 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×∆Isuch thatR−I={hx, yi ∈∆I×∆I | hy, xi ∈RI} for allR∈R, andhx, zi ∈SI,hz, yi ∈SIimplieshx, yi ∈SIfor eachS∈R+. An interpretationIsatisfies a role hierarchyRifRI⊆SIfor eachR⊑S∈ R. Such an interpretation is called amodelofR, denoted byI |=R.
Notice that the simplicity of roles which relies on the functionTrans(·)plays a crucial role in guaranteeing decidability ofSHIQ[11]. The underlying idea is that if a roleRis simple then it is sufficient to count “direct”R-neighborstof an individual s, i.e.hs, ti ∈RIfor some interpretationI, in order to satisfy a restriction that bounds the number ofR-neighbour ofs.
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,⊤,C⊓D,C⊔D,¬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, (C⊓D)I=CI∩DI, (C⊔D)I =CI∪DI, (¬C)I = ∆I\CI,card{oI} = 1for allo ∈ Co,(∃R.C)I = {x ∈ ∆I | ∃y ∈
∆I,hx, yi ∈ RI ∧y ∈ CI},(∀R.C)I = {x ∈ ∆I | ∀y ∈ ∆I,hx, yi ∈ RI ⇒ y ∈ CI}, (≥ n S.C)I = {x ∈ ∆I | card{y ∈ CI | hx, yi ∈ SI} ≥ n}, (≤n S.C)I={x∈∆I|card{y∈CI | hx, yi ∈SI} ≤n}
wherecard{S}is denoted for the cardinality of a setS.
∗ C ⊑ 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 GCIC⊑DifCI ⊆DIandIsatisfies 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 andCI 6= ∅. Such an interpretation is called a modelofCw.r.t.(T,R). A conceptDsubsumesa conceptCw.r.t.(T,R), denoted by C⊑D, ifCI⊆DIholds 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 [11]. 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 [7] 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 [7], we add a new property, namelyP15. This new property imposes an exact number ofS-neighbour individualstof sif(≤ nS.C) ∈ L(s). This property makes explicit nondeterminism implied from the semantics of(≤
nS.C) and requires an extra expansion rule, namely⊲⊳-rule, introduced in Figure 1 (Appendix). The presence of this rule may have an impact on the so-called “pay-as- you-go” behaviour of the tableaux-based algorithm presented in this paper.
P15 If(≤nS.C)∈ L(s)and there ist∈Ssuch thatC∈ L(t)andhs, ti ∈ E(S) then there is some1≤m≤nsuch that{(≤mS.C),(≥mS.C)} ⊆ L(s).
It is not hard to prove that there is a tableau with the new property P15 for a SHOIQknowledge base (T,R)iff(T,R)is consistent. A proof of a similar result forSHIQtableaux can be found in [12].
3 A 2E
XPS
PACEdecision 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 (SHOIQ-tree). Let (T,R) be a SHOIQ knowledge base. For each o∈Co, aSHOIQ-tree for(T,R), denoted byTo = (Vo, Eo,Lo,xbo,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(xbo). A nodex ∈ Vo is callednominalifo′ ∈ Lo(x)for someo′∈Co. In addition, the inequality relation6=·o
is a symmetric binary relation overVo.
∗Eois a set of edges. Each edgehx, yi ∈Eois labelled with a functionLosuch that Lo(hx, yi) ⊆ R(T,R). If hx, yi ∈ Eo 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=xbo, and (ii) each nodex∈ Vowithx6= bxohas a unique predecessor. A nodey is called anR-successor ofx, denoted by y ∈ succ1R(x)(resp.y is called theR-predecessor ofx, denoted byy = pred1R(x)) if there is some role R′ such thatR′ ∈ Lo(hx, yi) (resp. R′ ∈ Lo(hy, xi)) and R′⊑R. 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 ∈ succnR(x)(resp.y = prednR(x)) thenz ∈succ(n+1)R (x) (resp.z=pred(n+1)R (x)) forn≥0withsucc0R(x) ={x}andx=pred0R(x).
∗For a nodex, a roleSando∈Co, we define the setSTo(x, C)ofx’sS-neighbours as follows: :STo(x, C) ={y∈Vo|yis aS-neighbour ofxandC∈ Lo(x)}.
∗A nodexis callediteratedbyyw.r.t. a nodexoifxhas no nominal ancestor except forbxoand there are integersn, m >0and nodesx′, y′ such that : (i)xo=predn(y), y = predm(x), (ii)x′ = pred1(x),y′ = pred1(y), (iii)Lo(x) = Lo(y),Lo(x′) = Lo(y′), (iv)Lo(hx′, xi) = Lo(hy′, yi), and (v) if there arez, z′ andi > 0 such that z′ = pred1(z),predi(z′) =xo,Lo(z) = Lo(y),Lo(z′) =Lo(y′)andLo(hz′, zi) = Lo(hy′, 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),ST(x, C)and6=· instead ofLo(x), Lo(hx, yi),STo(x, C)and 6=·o, respectively. This does not cause any confusion since Vo∩Vo′ =∅andEo∩Eo′ =∅ifo6=o′. In addition,x6=·oyis never defined forx∈Vo
andy∈Vo′ witho6=o′.
We can 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 [11]). Moreover, if we consider the subtree rooted at a 1-iterated node as a SHIQcompletion 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 (SHOIQ-forest).Let(T,R)be aSHOIQknowledge base. ASHOIQ- forest for (T,R) is a pair G = hT, ϕi, whereT = {To | o ∈ Co} is a set of SHOIQ-trees for(T,R) withTo = (Vo, Eo,Lo,xbo,6=·o), and ϕis a partitioning function ϕ : V → 2V withV = S
o∈CoVo. We denote L′(hx, yi) = Lo(hx, yi) if hx, yi ∈Eo, andL′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, x′ ∈ V, ifx′ ∈ ϕ(x)thenϕ(x) = ϕ(x′)andL(x) = L(x′). We de- noteΛ(ϕ(x)) = L(x). In addition, an inequality relation over partitions can be described as follows : forx, x′ ∈ V we defineϕ(x)6=ϕ(x· ′)if there are two nodes y∈ϕ(x)andy′ ∈ϕ(x′)such thaty6=·oy′for someo∈Co;
3. For allϕ(x)andϕ(x′), if there are two edgeshy, y′i ∈Eoandhw, w′i ∈Eo′with o, o′∈Cosuch thaty, w∈ϕ(x),y′, w′ ∈ϕ(x′)andL′(hy, y′i)6=∅,L′(hw, w′i)6=
∅thenL′(hy, y′i) =L′(hw, w′i).
We define a functionΛ(h·,·i)for labelling edges ended by two partitions as follows:
Λ(hϕ(x), ϕ(x′)i) =L′(hz, z′i)wherez∈ϕ(x),z′ ∈ϕ(x′),L′(hz, z′i)6=∅, and {hz, z′i,hz′, zi} ∩Eo′ 6= ∅ for someo′ ∈ Co. We say ϕ(x′)is aS-neighbour partition ofϕ(x)ifS∈Λ(hϕ(x), ϕ(x′)i).
4. For allx, x′∈ V, ifo∈ L(x)∩ L(x′)for someo∈Coandϕ(x)6=ϕ(x· ′)does not hold thenϕ(x) =ϕ(x′); and
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.
∗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∈Co;
3. There is a nodex ∈ V with(≤ nR.C) ∈ Λ(ϕ(x))and there are (n+ 1)nodes x0,· · · , xn ∈ Vsuch thatϕ(xi)∩ϕ(xj) =∅,ϕ(xi)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 tableaux-based algorithm whose goal is to construct from a knowledge base(T,R)aSHOIQ-forestG= hT, ϕi. To do this, the algorithm ap- plies the expansion rules as described in Figure 1 and 2 (Appendix), and terminates when none of the rules is applicable. The obtainedGis calledcomplete, and ifGcon- tains no clash thenGis calledclash-free. In this case, we also sayTois complete and clash-free for all To ∈ T. Before presenting these expansion rules, we introduce an operation, namelyPropagate, which is used in expansion rules.
PropagationPropagate(ϕ(x), ϕ(x′), ϕ(y))is an operation which propagates (i) node labels from a partitionϕ(x)to another partitionϕ(x′), and vice versa, (ii) edge labels
from the edges ended by nodes ofϕ(x)andϕ(y)to the edges ended by nodes ofϕ(x′) andϕ(y), and vice versa. In other terms,Propagate(· · ·)mergesϕ(x)intoϕ(x′), and hϕ(x), ϕ(y)iintohϕ(x′), ϕ(y)i. More precisely, letG=hT, ϕibe aSHOIQ-forest withT={To|o∈Co}andTo= (Vo, Eo,Lo,bxo,6=·o).Propagate(ϕ(x), ϕ(x′), ϕ(y)) updates the label of nodes and edges inTas follows:
1.L(z) =L(x)∪ L(x′)for allz∈ϕ(x)∪ϕ(x′),
2. for all z, z′ ∈ ϕ(x)∪ϕ(x′)andw, w′ ∈ ϕ(y), ifz is a S-neighbour of wand L′(hz′, w′i)6=∅then (i) ifz′is a successor ofw′andS /∈ L(hw′, z′i)thenL(hw′, z′i) = L(hw′, z′i)∪ {S}, (ii) ifw′is a successor ofz′andS /∈ L(hz′, w′i)thenL(hz′, w′i) = L(hz′, w′i)∪ {S⊖}.
The rules in Figure 1 (Appendix) maintain the tree-like structure ofSHOIQ-forest and they are similar to those in [7] except that if a conceptCis added to the label of a nodexdue to application of these rules thenCis propagated to the label of each node y∈ϕ(x). Moreover, all rules in Figure 1 except for∃- and≥-rule update only the label of nodes or edges and do no change on the partitioning functionϕ. Especially, when the≤-rule is applied to a nodexwith twoS-neighbours y, z ofx, it must propagate the label ofhx, yito that of allhx′, z′i(orhz′, x′i) wherex′ ∈ ϕ(x)andz′ ∈ ϕ(z), and set the label ofhx, yito empty set. This may changeϕonly ifϕ(y)is singleton.
By the⊲⊳-rule in Figure 2, each nodexcontaining a term(≤ nS.C)has exactly m S-neighbours containingCwith somem≤n. As a result, this rule and≥-rule ensure that if there are two nodesy, y′ ∈ ϕ(x)theny andy′ 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 tableaux algorithm terminates and builds aSHOIQ-forest whose the size is bounded by a double exponential function in the size of(T,R).
2. If the tableaux algorithm yields a clash-free and complete SHOIQ-forest for (T,R)then there is a tableau for(T,R).
3. If there is a tableau for(T,R)then the tableaux algorithm yields a clash-free and completeSHOIQ-forest for(T,R).
It is straightforward to show that the size of a SHOIQ-forest is bounded by a dou- ble exponential function in the size of (T,R). To prove soundness of the tableaux algorithm, we can devise a model from a clash-free and completeSHOIQ-forest by considering a partition as an individual and unraveling blocked nodes since we can show that each blocking node b(x)has no “core path” from b(x)to every nominal descendant y, i.e., there do not exist terms(≤ miRi.Ci) ∈ predi(y), roles Ri ∈ L(hpredi−1(y),predi(y)i) and concepts Ci ∈ L(predi+1(y)) for k < i ≤ 0 with b(x) =predk(y).
The following theorem is a consequence of Lemma 1.
Theorem 1. Let(T,R)be aSHOIQknowledge base. The tableaux algorithm is a decision procedure for consistency of(T,R)and it runs in2NEXPTIMEin the size of (T,R).
4 An E
XPS
PACEtableaux-based algorithm for SHOIQ
This section starts by translating from results presented in [9] for C2 into those for SHOIQ.
Definition 6 (star-type).Let(T,R)be aSHOIQknowledge base. Astar-typeis a tripletσ = hλσ,ν¯σ,µ¯σi, whereλσ ∈ 2cl(T,R),ν¯σ contains at most a pairhr, li ∈ 2R(T,R) ×2cl(T,R) andµ¯σ = (hr1, l1i,· · ·,hrdσ, ldσi)is ad-tuple over2R(T,R) × 2cl(T,R). A pairhr′, l′iis arayofσifhr′, l′iis a component ofµ¯σorhr′, l′i ∈ν¯σ. We define an inequality relation6=· over the set of rays. A rayhr′, l′iofσisprimaryw.r.t. a term(≤mR.C), denotedhr′, l′ih≤mR.Ci, if(≤mR.C)∈λσ,R∈r′andC∈l′. For a term(≤mR.C)∈λσ, we denoteCh≤mR.Ciσ for the set of all rayshr′, l′iofσsuch thatR∈r′,C∈l′.
– A star-typeσisnominalifo∈λσfor someo∈Co.
– A star-typeσ isisomorph to a star-type σ′ if λσ = λσ′, and for each term(≤
mR.C)∈λσ, there is an injectionπ:Cσh≤mR.Ci→ Ch≤σ′mR.Cisuch thatπ(hr, li) = (hr, li).
– Two star-typesσ, σ′areisomorphifλσ=λσ′, and for each term(≤mR.C)∈λσ, there is a bijectionπ:Ch≤σ mR.Ci→ Ch≤σ′mR.Cisuch thatπ(hr, li) = (hr, li).
– A star-typeσ = hλ,ν,¯ µi¯ withµ¯ = (hr1, l1i,· · · ,hrdσ, ldσi)and λ = l0,ν¯ = {hrdσ+1, ldσ+1i}, isvalidif the following conditions are satisfied:
1. IfC⊑D∈ T thennnf(¬C⊔D)∈lifor all0≤i≤dσ+ 1;
2. {A,¬A} 6⊆lifor every concept nameAwith0≤i≤dσ+ 1;
3. IfC1⊓C2∈lithen{C1, C2} ⊆lifor all0≤i≤dσ+ 1;
4. IfC1⊔C2∈lithen{C1, C2} ∩li6=∅for all0≤i≤dσ+ 1;
5. If∃R.C ∈λthen there is some1≤i≤dσ+ 1such thatC∈liandR∈ri; 6. If(≤nS.C) ∈ λand there is some1 ≤ i ≤ dσ + 1such thatS ∈ ri then
C∈lior¬C˙ ∈li;
7. If(≤nS.C)∈λand there is some1≤i≤dσ+1such thatC∈liandS∈ri
then there is some1≤m≤nsuch that{(≤mS.C),(≥mS.C)} ⊆λ;
8. For each1≤i≤dσ+ 1, ifR∈riandR⊑S∗ thenS ∈ri; 9. If∀R.C ∈λandR∈rifor some1≤i≤dσ+ 1thenC∈li;
10. If∀R.D ∈ λ,S⊑R,∗ Trans(S)andR ∈ ri for some 1 ≤ i ≤ dσ + 1then
∀S.D∈li;
11. If(≥nS.C)∈λthen there are1≤i1<· · ·< in≤dσ+ 1such thatC∈lij
andS∈rij for all1≤j≤n;
12. If(≤nS.C)∈λand there are no1 ≤i1 <· · ·< in+1 ≤dσ+ 1such that C∈lij andS∈rij for all1≤j≤n;
We denoteΣfor the set of all star-types for(T,R).
In the context of aSHOIQ-forest, we can think of a star-typeσas the set of nodes which satisfyλσand haveR-neighbours such thatRis included in their rays. Moreover, we can merge nodes satisfying homomorph and isomorph star-types without violating semantic constraints imposed by node and edge labels. A star-type σ is valid if no expansion rule is applicable to a node whose label isλσ.
Definition 7 (frame).Let(T,R)be aSHOIQknowledge base. Aframefor(T,R) is a tupleF =h(N0,· · ·,NH), δ, Φ,δi, whereb H ∈ Nis thedimensionofF,Ni ⊆ Σ for all 0 ≤ i ≤ H, and all star-types in N0 are nominal, δ is a function δ : S
i∈{1,···,H}Ni → N,Φis a functionΦ : S
i∈{1,···,H}Ni → 2Si∈{1,···,H}Ni, andbδ is a functionδ′:Φ(S
i∈{1,···,H}Ni)→N; 1. Two star-typesσ, σ′ ∈ S
i∈{1,···,H}Ni are mergeable inF, denotedσ ≈ σ′, if, eitherσandσ′ are homomorph to a star-typeσ0; orσandσ′are isomorph. The relation of mergeability≈is an equivalence relation overS
i∈{1,···,H}Ni. We de- noteΦ(σ) = {σ′ | σ′ ≈ σ} andΦ(σ)is called mergeable. We say thatΦ(σ)is homomorphw.r.t. a star-typeσ0ifσ′ is homomorph toσ0 for allσ′ ∈ Φ(σ). We say thatΦ(σ)isisomorphifσ′, σ′′are isomorph for allσ′, σ′′ ∈ Φ(σ). For each Φ(σ), we define a set of rays ofΦ(σ)as follows:
– IfΦ(σ)is homomorph w.r.t. a star-typeσ0∈Φ(σ)andhr′, l′iis a primary ray ofσ0then we define a primary rayhr, liofΦ(σ)withr=r′andl=l′; – IfΦ(σ)is isomorph andhr′, l′iis a primary ray of some fixed star-typeσ0 ∈
Φ(σ)then we define a primary rayhr, liofΦ(σ)withr=r′andl=l′; – Ifhr′, l′iis a non primary ray ofΦ(σ)then there is someσ′∈Φ(σ)that has a
non primary rayhr, lisuch thatr=r′andl=l′.
We denoteCΦ(σ)for the set of all rays ofΦ(σ), andCh≤mR.CiΦ(σ) ={hr′, l′i ∈ CΦ(σ)| R∈r, C∈l}.
2. A star-typeσ∈ Nk (0≤k≤H) is calledlinkablewith a star-typeσ′ ∈ Nk−1∪ Nk+1 by a rayhr, liofσifσ′ has a rayhr′, l′isuch thatl = λσ′,l′ = λσand r=r′−wherer′−={R⊖ |R∈r′}.
The frame structure, as introduced in Definition 7, allows us to tile star-types to ob- tain a forest structure. Such a structure is crucial to obtain termination when designing a tableaux-based algorithm. An important difference between a frame and aSHOIQ- forest is that a frame does not represent nodes corresponding to individuals but store the number of individuals satisfying a star-type. The functionδ(σ)is used for this pur- pose. According to Lemma 1, the number of aSHOIQ-forest’s nodes may be double exponential in the size of aSHOIQknowledge base(T,R)while the number of dis- tinct star-types is bounded by an exponential function since star-types are built from the signature of(T,R). This implies thatδ(σ)may take a double exponential value. In the context of aSHOIQ-forest, we can think of aΦ(σ)as the set of partitions each of which satisfies allσ′∈Φ(σ). The functionbδ(σ)is used to store the number of partitions satisfying allσ′∈Φ(σ).
Definition 8 (valid frame).Let(T,R)be aSHOIQknowledge base. A frameF = h(N0,· · · ,NH), δ, Φ,bδiisvalidif the following conditions are satisfied:
1. For eachσ∈S
i∈{1,···,H}Ni, ifδ(σ)≥1thenσis valid;
2. For eacho∈Cothere is a uniqueσo∈ N0such thato∈λσoandδ(σo) = 1;
3. For eacho∈Co,Φ(σo) ={σ∈S
i∈{1,···,H}Ni|o∈λσ}andδ(Φ(σb o)) = 1;
4. For each0 ≤k < H andhλ, r, λ′i ∈2cl(T,R)×2R(T,R)×2cl(T,R)withr− = {R⊖ |R∈r},
X
σ∈Nk
δ(σ)|µ¯σ|hλ,r,λ′i= X
σ′∈Nk+1
δ(σ′)|¯νσ′|hλ′,r−,λi (1)
where|¯νω|hλ,r,λ′iand|µ¯ω|hλ,r,λ′iare denoted for the number of componentshr′, l′i of respectiveν¯ω and µ¯ω such thatλω = λ,r′ = rand l′ = λ′ for a star-type ω=hλω,ν¯ω,µ¯ωi;
5. For eachhλ, r, λ′i ∈2cl(T,R)×2R(T,R)×2cl(T,R)withr−={R⊖ |R∈r}, X
Φ(σ)
bδ(Φ(σ))|Φ(σ)|hλ,r,λ′i= X
Φ(σ′)
bδ(Φ(σ′))|Φ(σ′))|hλ′,r−,λi (2)
where|Φ(ω)|hl,s,l′i is denoted for the number of rays hu, hi ofΦ(ω)with some star-typeωsuch thatλω=l,u=sandh=l′.
6. For eachΦ(σ)withσ∈S
i∈{1,···,H}Ni, and for each term(≤mR.C)∈λσ,
card{CΦ(σ)h≤mR.Ci} ≤m (3)
The notion of validity for a frame is crucial to establish a connection with the tableaux-based algorithm presented in Section 3, i.e., how to build aSHOIQ-forest from a valid frame, and inversely. Condition 1 in Definition 8 requires that every star- type counted byδmust be valid. Condition 2 and 3 ensure that each nominal is counted exactly once. In the context of aSHOIQ-forest, these conditions imply that for each nominalothere is exactly one tree whose root containsoand there is exactly one par- tition containso. Condition 4 allows for linking star-types at levelkwith star-types at levelk−1andk+ 1. It ensures that each nodexsatisfying (or counted for) a star-type σat levelkis linked by its rays to neighbours satisfying star-types at levelk−1and k+ 1. The number of these neighbours corresponds exactly to the number ofx’s rays.
Condition 5 guarantees that each partition satisfyingΦ(σ)can be linked exactly with another partition via a ray ofΦ(σ). Finally, Condition 5 ensures that each partition sat- isfyingΦ(σ)with(≤mR.C)∈λσcan be linked at most withmpartitions containing Cvia rays that includeR.
Lemma 2. Let(T,R)be aSHOIQknowledge base.
1. If the tableaux 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,· · ·,NH), δ, Φ,δib for(T,R)then the tableaux 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 double 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 = 22m+k ×2 withm=card{cl(T,R)}andk=card{R(T,R)}.card{Σ} ≤(card{cl(T,R)})2× card{R(T,R)} δ(σ) ≤ M22m+k×2whereM = P
mi +E,mi occurs in a number restriction term (≥ miR.C) appearing inT, andE is the number of distinct terms
∃R.Cappearing inT forσ∈Σ. 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 there are tremendously non-determinisms which must be dealt with when constructing a valid frame. In the sequel, based on the results obtained so far, we try to design an algorithm which has more goal-directed behaviours.
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 withσi−1for alli∈ {1,· · ·, k}, then there are0< k1< k2< k3<
k4≤ksuch that:
1.λσk1 =λσk2,ν¯σk1 = ¯νσk2, and there is no0< j < k2such thatj6=k1,λσj =λσk2
andν¯σj = ¯νσk2;
2. λσk3 = λσk4, ¯νσk3 = ¯νσk4, and there is no k2 < j < k4 such thatj 6= k3, λσj =λσk4 andν¯σj = ¯νσk4.
Notice that this blocking condition is looser than the blocking condition introduced in Definition 5 for aSHOIQ-forest. Since we can not determine the path from root to a node satisfying a star-type over a frame, it 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 EXPSPACEtableaux-based algorithm forSHOIQ.
Before applying the frame rules described in Figure 3 (Appendix), we initialise a frame F =h(N0,· · ·,NH), δ, Φ,δib from a(T,R)knowledge base as follows:
N0 := {h{o},∅,∅i |o∈ Co};δ(σo) := 1,Φ(σo) ={σo}andδ′(Φ(σo)) = 1for all σo∈ N0.
If no frame rule is applicable to all star-types ofFthen we say thatFis complete.
If we obtain a valid and completeF by applying the frame rules from a(T,R), then we conclude that(T,R)is consistent. Otherwise,(T,R)is not consistent.
Soundness of the tableaux-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 tableaux algorithm for contructing a frame is a decision procedure for consistency of(T,R)and it runs in EXPSPACEin 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 SHOIQ andC2. First, we have based our approach on a technique that constructs tree-like structures for representing a model without adding nominal nodes with new nominals. This technique is founded on the fact that fusions of nodes triggered by merg- ing nominal nodes can be replaced with governing a partitioning function which would simulate this merging process. This allows us to reuse the blocking technique over these tree-like structures to obtain termination. Second, we have transferred toSHOIQthe method used for constructing a NEXPTIMEalgorithm forC2. This enables us to repre- sent a double exponentialSHOIQ-forest by an exponential structure.
The algorithms proposed in the present paper have used several nondeterministic rules, e.g.,⊲⊳or≤o-rules. We think that these rules should be improved in some way such that, for instance, they would take advantage of information from the part of the frame which has already built.
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Appendix
The rules in Figure 3 for building a frame calls the algorithms described in Figure 1, 2, 3 and 4. Basically, these algorithms update the frame by adding a new star-type or modifying the functionsδσandbδσ.
NotationLetF =h(N0,· · ·,NH), δ, Φ,δib be a frame.
– For eachσ∈ Nk, ifk= 0then we defineN(σ) =∅andNb(σ,he, hi) =∅;
– For eachσ∈ Nkwithk >0, we denoteN(σ)⊆ Nk−1for the non-empty set with N(σ) =X N′(σ)∪ {ω0}such that
ω′∈N′(σ)
δ(ω′)< δ(σ), X
ω′∈N′(σ)
δ(ω′) +δ(ω0)≥δ(σ), and for allσ′ ∈ N(σ)it holds thatλσ′ =l0,σ′has a rayhr′′, l′′i∈/ ν¯σ′ withr′′=r−0 andl′′=λσ. – For each rayhr, hiofσwithhr, hi∈/ ν¯σ, we denoteNb(σ,he, hi)⊆ Nk+1for the
non-empty set withX Nb(σ,he, hi) =Nc′(σ,he, hi)∪ {¯ω0}such that
ω′∈Nb′(σ,he,hi)
δ(ω′)< δ(σ), X
ω′∈Nb′(σ,he,hi)
δ(ω′) +δ(¯ω0)≥δ(σ), and for allσ′ ∈ Nb(σ,he, hi)it holds thatλσ′ =h,σ′ has a rayhr′′, l′′i ∈ ν¯σ′ withr′′ =r−and l′′=λσ.
⊑-rule: if C⊑D∈ T andnnf(¬C⊔D)∈ L(x)/
thenL(x′) =L(x′)∪ {nnf(¬C⊔D)}for allx′∈ϕ(x).
⊓-rule: if C1⊓C2 ∈ L(x)and{C1, C2} 6⊆ L(x) thenL(x′) =L(x′)∪ {C1, C2}for allx′∈ϕ(x).
⊔-rule: if C1⊔C2 ∈(x)and{C1, C2} ∩ L(x) =∅
thenL(x′) =Lo(x′)∪ {C}with someC∈ {C1, C2}for allx′∈ϕ(x).
∃-rule: if 1.∃S.C∈ L(x),xis not blocked,xis not non-root nominal, and 2.xhas noS-neighbourywithC∈ L(y)
then create a new nodeywithL(hx, yi)={S},L(y)={C}andϕ(y) ={y}.
∀-rule: if 1.∀S.C∈ L(x), and
2. there is aS-neighbouryofxsuch thatC /∈ L(y) thenL(y′) =L(y′)∪ {C}for ally′∈ϕ(y).
∀+-rule: if 1.∀S.C∈ L(x), and
2. there is someQwithTrans(Q)andQ⊑S, and∗ 3. there is anQ-neighbouryofxsuch that∀Q.C /∈ L(y) thenL(y′) =L(y′)∪ {∀Q.C}for ally′∈ϕ(y).
ch-rule: if 1.(≤n S.C)∈ L(x), and
2. there is anS-neighbouryofxwith{C,¬C} ∩ L(y) =˙ ∅
thenL(y′) =Lo(y′)∪ {E}with someE∈ {C,¬C}˙ for ally′∈ϕ(y).
≥-rule: if 1.(≥n S.C)∈ L(x),xis not blocked,xis not non-root nominal, and 2.xhas non S-neighboursy1, ..., ynsuch thatC∈ L(yi), and
yi·
6=yjfor1≤i < j≤n,
then creatennew nodesy1, ..., ynwithL(hx, yii)={S},L(yi)={C}, ϕ(yi) ={yi}andyi·
6=yjfor1≤i < j≤n.
≤-rule: if 1.(≤n S.C)∈ L(x), and
2.card{ST(x, C)}> nand there are twoS-neighboursy, zofxwith C∈ L(y)∩ L(z),yis not an ancestor ofzand noty6=z·
then 1. for allz′∈ϕ(z),x′∈ϕ(x)such thatL′(hx′, z′i)6=∅,
ifx′is an ancestor ofz′thenL(hx′, z′i) =L(hx′, z′i)∪ L(hx, yi) elseL(hz′, x′i) =L(hz′, xi)∪ {R⊖|R∈ L(hx, yi)}
2.L(z′) =L(z′)∪ L(y)for allz′∈ϕ(z)andL(hx, yi) =∅ 3. addu6=z· for allusuch thatu6=y.·
⊲⊳-rule: if 1.(≤nR.C)∈ L(x),{(≤l R.C),(≥l R.C)}*L(x)for alll≤n, 2.(≤k R.C)∈ L(x)/ for allk < n, and
3.xhas aR-neighbourysuch thatC∈ L(y) then 1. guessmwith1≤m≤n, and
2.L(x′) =L(x′)∪ {≤mR.C,≥mR.C}for allx′∈ϕ(x).
Fig. 1.Expansion rules forSHIQ
oϕ-rule: if 1. there are nodesx, x′witho∈ L(x)∩ L(x′)for someo∈Co, 2.ϕ(x)∩ϕ(x′) =∅andϕ(x)6=ϕ(x· ′)does not hold,
then 1.Propagate(ϕ(x), ϕ(x′), ϕ(y))for eachysuch that
{hx′′, yi,hy, x′′i} ∩Eo6=∅forx′′∈ϕ(x)∪ϕ(x′),o∈Co. 2.ϕ(y′) =ϕ(x)∪ϕ(x′)for ally′∈ϕ(x)∪ϕ(x′).
≤ϕ-rule: if 1.(≤nR.C)∈ L(x),
2. there are nodesy0,· · ·, ynwithϕ(yi)∩ϕ(yj) =∅,0≤i < j≤n, C∈Λ(ϕ(yi)),R∈Λ(hϕ(x), ϕ(yi)i)for all0≤i≤n, and 3. there arex′, x′′∈ϕ(x)withx′6=x′′, andx′has aR-neighboury′,
x′′has aR-neighboury′′s.t.C∈ L(y′)∩ L(y′′),ϕ(y′)∩ϕ(y′′) =∅, and notϕ(y′)6=ϕ(y· ′′)
then 1.Propagate(ϕ(y′), ϕ(y′′), ϕ(x)),
2.ϕ(y) =ϕ(y′)∪ϕ(y′′)for ally∈ϕ(y′)∪ϕ(y′′).
Fig. 2.New expansion rules forSHOIQ
Input :σ=hλσ,µ¯σ,ν¯σi ∈ Nk,r⊆R(T,R),l⊆cl(T,R)and F=h(N0,· · ·,NH), δ, Φ,bδi
Output: the frame obtained by updatingF
Build a star-typeωwithλω=λσ,ν¯ω= ¯νωandµ¯ω= (¯µσ,hr, li);
1
Build a star-typeω′withλω=l,¯νω={hr−, λσi}andµ¯ω=∅;
2
Nk:=Nk∪ {ω};
3
δ(ω) :=δ(σ);
4
δ(σ) := 0;
5
ifω′∈ Nk+1then
6
δ(ω′) :=δ(ω′) +δ(σ);
7
else
8
Nk+1:=Nk+1∪ {ω′};
9
δ(ω′) :=δ(σ);
10
ifω6≈σ′for allσ′∈S
i∈{1,···,H}Nithen
11
Φ(ω) :={ω},bδ(Φ(ω)) := 1andδ(Φ(σ)) :=b δ(Φ(σ))b −1;
12
else
13
ifo /∈λωfor allo∈Cothen
14
bδ(Φ(ω)) :=δ(Φ(ω)) + 1;b
15
ifω′6≈σ′for allσ′∈S
i∈{1,···,H}Nithen
16
Φ(ω′) :={ω′},bδ(Φ(ω′)) := 1;
17
else
18
ifo /∈λω′for allo∈Cothen
19
bδ(Φ(ω′)) :=δ(Φ(ωb ′)) + 1;
20
Algorithm 1:addRay(σ, r, l)updates frame when adding a new ray hr, lito a star-typeσ∈ Nk.
⊑-rule: if C⊑D∈ T andnnf(¬C⊔D)∈/λσ
thenupdateLabel(σ, λσ∪ {nnf(¬C⊔D)}).
⊓-rule: if C1⊓C2∈ L(x)and{C1, C2} 6⊆λσ
thenupdateLabel(σ, λσ∪ {C1, C2}).
⊔-rule: if C1⊔C2∈λσand{C1, C2} ∩λσ=∅
thenupdateLabel(σ, λσ∪ {C})with someC∈ {C1, C2}.
∃-rule: if 1.∃S.C∈λσ,λσis not blocked, and 2.σhas no rayhr, liwithS ∈randC∈l thenaddRay(σ, r, l).
∀-rule: if 1.∀S.C∈λσ, and
2.σhas a rayhr, lisuch thatS∈randC /∈l
then ifhr, li ∈ν¯σthenupdatePredRay(σ,hr, li, r, l∪ {C}) elseupdateSuccRay(σ,hr, li, r, l∪ {C})
∀+-rule: if 1.∀S.C∈λσ, and
2. there is someQwithTrans(Q)andQ⊑S, and∗ 3.σhas a rayhr, lisuch thatQ∈rand∀Q.C /∈l
then ifhr, li ∈ν¯σthenupdatePredRay(σ,hr, li, r, l∪ {∀Q.C}).
elseupdateSuccRay(σ,hr, li, r, l∪ {∀Q.C}) ch-rule: if 1.(≤n S.C)∈λσ, and
2.σhas a rayhr, lisuch thatS∈rand{C,¬C} ∩˙ l=∅ then ifhr, li ∈ν¯σthenupdatePredRay(σ,hr, li, r, l∪ {E}).
elseupdateSuccRay(σ,hr, li, r, l∪ {E})with someE∈ {C,¬C}.˙
≥-rule: if 1.(≥n S.C)∈λσ,σis not blocked, and
2.σhas nonrayshr1, l1i, ...,hrn, lnisuch thatR∈ri,C∈li, and hri, lji6=hr· j, ljifor1≤i < j≤n,
then calladdRay(σ,{R},{C})ntimes to createnrayshr1, l1i, ...,hrn, lni withri={R}andli={C}for1≤i≤n, and
hri, lii6=hr· j, ljifor1≤i < j≤n.
≤-rule: if 1.(≤n S.C)∈σ, and
2.σhas(n+ 1)rayshr0, l0i, ...,hrn, lnisuch thatR∈ri, C∈lifor all 0≤i≤nand there are0≤i < j≤n
such thathri, lii6=hr· j, ljidoes not hold
then 1. For eachhr, li ∈ {hri, lii,hrj, lji}, ifhr, li ∈¯νσ, then,updatePredRay(ω,hr, li, r∪r′, l∪l′), else,updateSuccRay(ω,hr, li, r∪r′, l∪l′) wherehr′, l′i ∈ {hri, lii,hrj, lji}withhr′, l′i 6=hr, li.
2. addhr′, l′i6=hr· i, liifor all rayhr′, l′isuch thathr′, l′i6=hr· j, lji.
o-rule: if 1. there are star-typesσ1,· · ·, σksuch thato∈λσifor someo∈Co thenupdateLabel(σ1,· · ·, σk),
≤o-rule: if 1. there are star-typesσ1,· · ·, σk∈Φ(σ)and(≤mR.C)} ∈λσifor all 1≤i≤k, andσ1,· · ·, σkhave(m+ 1)distinct primary rays hr0, l0i,· · · hrm, lmisuch thatR∈riandC∈lifor all0≤i≤m then 1. Choose two rayshrj, lji,hrj′, lj′iof respectiveσi∈ Nhandσi′ ∈ Nh′
with0≤j < j′≤mand1≤i < i′≤ksuch thathrj, lji6=hr· j′, lj′i 2. For eachhr, li ∈ {hrj, lji,hrj′, lj′i}, ifhr, li ∈ν¯ωwithω∈ {σi, σi′},
then,updatePredRay(ω,hr, li, r∪r′, l∪l′), else,updateSuccRay(ω,hr, li, r∪r′, l∪l′)
wherehr′, l′i ∈ {hrj, lji,hrj′, lj′i}withhr′, l′i 6=hr, li.
⊲⊳-rule: if 1.(≤nR.C)∈λσ,{(≤l R.C),(≥l R.C)}*λσfor alll≤n, 2.(≤k R.C)∈/λσfor allk < n, and
3.σhas a rayhr, lisuch thatR∈r, C∈l then 1. guessmwith1≤m≤n, and
2.updateLabel(σ, λσ∪ {≤mR.C,≥mR.C}).
Fig. 3.Expansion rules for constructing a frame.
