Towards Defeasible SROIQ

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Katarina Britz¹and Ivan Varzinczak^2,1

1 CSIR-SU CAIR, Stellenbosch University, South Africa

2 CRIL, Univ. Artois & CNRS, France abritz@sun.ac.za, varzinczak@cril.fr

Abstract. We present a decidable extension of the Description LogicSROIQ that supports defeasible reasoning in the KLM tradition, and extends it through the introduction of defeasible roles. The semantics of the resulting DLdSROIQ extends the classical semantics with a parameterised preference order on binary relations in a domain of interpretation. This allows for the use of defeasible roles in complex concepts, as well as in defeasible concept and role subsumption, and in defeasible role assertions. Reasoning overdSROIQontologies is made possible by a translation of entailment to concept satisfiability relative to an RBox only. A tableau algorithm then decides on consistency ofdSROIQ-concepts in the preferential semantics.

Keywords: SROIQ, non-monotonic reasoning, preferential semantics

1 Introduction

SROIQ[20] is an expressive, yet decidable Description Logic (DL) that serves as semantic foundation for the OWL 2 profile, on which several ontology languages of various expressivity are based. However,SROIQstill allows for meaningful, decidable extension, as new knowledge representation requirements are identified. A case in point is the need to allow for exceptions and defeasibility in reasoning over logic-based ontologies [4, 3, 2, 8, 6, 7, 10, 12–14, 17, 18, 27]. Yet,SROIQdoes not allow for the direct expression of and reasoning with different aspects of defeasibility.

Given the special status of subsumption in DLs in particular, and the historical im- portance of entailment in logic in general, past research efforts in this direction have focused primarily on accounts of defeasible subsumption and the characterisation of defeasible entailment. Semantically, the latter usually take as point of departure or- derings on a class of first-order interpretations, whereas the former usually assume a preference order on objects of the domain.

In this paper, we propose a decidable extension ofSROIQthat supports defeasible knowledge representation and reasoning over defeasible ontologies. Our proposal builds on previous work to resolve two important ontological limitations of the preferential approach to defeasible reasoning in DLs — the assumption of a single preference order on all objects in the domain of interpretation, and the assumption that defeasibility is intrinsically linked to argument form [9, 10].

We achieve this by extendingSROIQwith nonmonotonic reasoning features in the concept language, in subsumption statements and in role assertions via an intuitive

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notion of normality for roles. This parameterises the idea of preference while at the same time introducing the notion of defeasible class membership. Defeasible subsumption allows for the expression of statements of the form “Cis usually subsumed byD”, for example, “Chenin blanc winesare usuallyunwooded”. In our extended language dSROIQ, we can now also refer directly to, for example, “Chenin blanc wines that usually havea wood aroma”. We can also combine these seamlessly, as in: “Chenin blanc wines thatusually havea wood aromaare usuallywooded”. Note that this cannot be expressed in terms of defeasible subsumption alone, nor can it be expressed w.l.o.g. using a typicality operator on concepts. This is because the semantics of the expression is inextricably tied to the two distinct uses of the term ‘usually’. Another defeasible construct that adds to the expressivity ofdSROIQis defeasible role inclusion, e.g. “having a given geographic style usually implies having that region as origin”.

dSROIQalso includes defeasible role assertions, such as defeasible functionality or defeasible disjointness, and defeasible number- andSelf-restrictions.

The remainder of the paper is structured as follows: In Section 2 we introduce the syntax and semantics of the extended languagedSROIQ. Section 3 covers a number of rewriting and elimination results required for effective reasoning with dSROIQ knowledge bases, and which are needed for the tableau algorithm presented in Sec- tion 4. The main results of the paper are Theorem 1, which reduces concept satisfiability indSROIQto concept satisfiability relative to only an RBox, and Theorem 2, which establishes the correctness of the tableau procedure. The latter result is established only for the restriction ofdSROIQwhich excludes role composition in defeasible RIAs.

Space considerations prevent us from providing a summary of the required logi- cal background. We shall therefore assume the reader’s familiarity with DLs in general [1] and withSROIQin particular [20], as well as with the preferential approach to non-monotonic reasoning [23, 24, 28]. Whenever necessary, we refer the reader to the definitions and results in the relevant literature.

2 Defeasible SROIQ

2.1 Defeasibility in RBoxes

LetRbe a set of role names, and letudenote theuniversal role. The set of all roles is given byR := R∪ {r⁻ | r ∈ R} ∪ {u}. We denote roles withr, s, . . ., possibly with subscripts. Moreover, letinv : R −→ Rbe such thatinv : r 7→ r⁻, ifr ∈ R, inv:r7→s, ifr=s⁻, andinv:u7→u.

Letr₁, . . . , r_n, r ∈R\ {u}. Aclassical role inclusion axiomis a statement of the formr1◦ · · · ◦rnvr. Adefeasible role inclusion axiomhas the formr1◦ · · · ◦rn@_∼r, read “usually,r1◦ · · · ◦rnis included inr”. A finite set of role inclusion axioms (RIAs) is called arole hierarchyand is denoted byRh.

Definition 1 ((Non-)Simple Role).Letr∈Rand letRhbe a role hierarchy. Thenr isnon-simpleinRhiff:

1. There isr1◦ · · · ◦rnvrorr1◦ · · · ◦rn@_∼rinRhsuch thatn >1, or 2. There issvrors@_∼rinRhsuch thatsis non-simple, or

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3. inv(r)is non-simple.

WithRⁿwe denote the set ofnon-simpleroles inR_h.R^s:=R\Rⁿis the set ofsimple roles inRh.

Intuitively, simple roles are those that are not implied by the composition of roles.

They are needed to restrict the type of roles in certain concept constructors (see below), thereby preserving decidability [20].

Definition 2 (Regular Hierarchy).A role hierarchyRhisregularif there is a strict partial order<onRⁿsuch that:

1. s < riffinv(s)< r, for everyr, sinRⁿ, and

2. every role inclusion inRhis of one of the forms: (1a)r◦r vr, (1b)r◦r@_∼r, (2a)inv(r) vr, (2b)inv(r)@_∼r, (3a) s₁◦ · · · ◦s_n v r, (3b)s₁◦ · · · ◦s_n@_∼r, (4a)r◦s₁◦ · · · ◦s_n vr, (4b)r◦s₁◦ · · · ◦s_n@_∼r, (5a)s₁◦ · · · ◦s_n◦r vr, (5b)s₁◦· · ·◦s_n◦r@_∼r, wherer∈R(i.e., a role name), ands_i< r, fori= 1, . . . , n.

(Regularity prevents a role hierarchy from inducing cyclic dependencies, which are known to lead to undecidability.)

Aclassical role assertionis a statement of the formFun(r)(functionality),Ref(r) (reflexivity),Irr(r)(irreflexivity),Sym(r)(symmetry),Asy(r)(asymmetry),Tra(r)(tran- sitivity), andDis(r, s)(role disjointness), where r, s6= u. Adefeasible role assertion is a statement of the formdFun(r)(ris usually functional),dRef(r)(ris usually re- flexive),dIrr(r)(ris usually irreflexive),dSym(r)(ris usually symmetric),dAsy(r) (ris usually asymmetric),dTra(r)(ris usually transitive), anddDis(r, s)(randsare usually disjoint), also withr, s6=u. WithR_awe denote a finite set of role assertions.

Given a role hierarchyR_h, we say thatR_aissimplew.r.t.R_hif all rolesr, sappear- ing in statements of the formIrr(r),dIrr(r),Asy(r),dAsy(r),Dis(r, s)ordDis(r, s)are simple inRh(see Definition 1).

AdSROIQRBox is a setR:=Rh∪ Ra, whereRhis a regular hierarchy and Rais a set of role assertions which is simple w.r.t.Rh.

2.2 Defeasibility in Concepts and in TBoxes

LetCbe a set of (atomic)concept namesdisjoint fromRand of which N, the set of nominals, is a subset. We use A, B, . . ., possibly with subscripts, to denote concept names. A nominal will also be denoted byo, possibly with subscripts.

Definition 3 (dSROIQConcepts). The set of dSROIQ complex concepts is the smallest set such that>,⊥and everyA ∈ Care concepts, and ifC andDare concepts,r ∈ R,s ∈ R^s, andn∈ N, then¬C (concept complement),CuD(concept conjunction),CtD (concept disjunction),∀r.C (value restriction),∃r.C (existential restriction),^∼^Wr.C (defeasible value restriction),⁻^∼₋^|r.C (defeasible existential restriction),∃r.Self (self restriction),⁻^∼₋^|r.Self (defeasible self restriction),≥ ns.C (at-least restriction), ≤ ns.C (at-most restriction), & ns.C (defeasible at-least restriction), .ns.C(defeasible at-most restriction) are also concepts. WithCwe denote the set of all complex concepts.

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Note that everySROIQconcept is adSROIQconcept, too. We shall useC, D . . ., possibly with subscripts, to denote complexdSROIQconcepts.

GivenC, D∈C,CvDis aclassical general concept inclusion, read “Cis subsumed byD”. (C ≡Dis an abbreviation for bothC v DandD vC.)C@_∼Dis a defeasible general concept inclusion, read “Cisusuallysubsumed byD”. AdSROIQ TBoxT is a finite set of general concept inclusions (GCIs), whether classical or defeasible.

LetIbe a set ofindividual namesdisjoint from bothCandR. GivenC∈C,r∈R anda, b ∈ I, an individual assertionis an expression of the forma : C,(a, b) : r, (a, b) :¬r,a=bora6=b. AdSROIQABoxAis a finite set of individual assertions.

LetAbe an ABox,T be a TBox andRan RBox. Aknowledge base(alias ontology) is a tupleKB:=hA,R,T i.

2.3 Preferential Semantics

We shall anchor our semantic constructions in the well-known preferential approach to non-monotonic reasoning [23, 24, 28] and its extensions [5, 9, 11], especially those to the DL case [8, 16, 25].

LetX be a set and let<be a strict partial order onX. Withmin<X :={x∈X | there is noy∈X s.t.y < x}we denote theminimal elementsofX w.r.t.<. With#X we shall denote thecardinalityofX.

Definition 4 (Ordered Interpretation). Anordered interpretationis a tuple O :=

h∆Ô,·Ô,≺Ô,Ôiin whichh∆Ô,·Ôiis aSROIQinterpretation withAÔ ⊆ ∆Ô, for everyA ∈C,AÔ a singleton for everyA∈ N,rÔ ⊆ ∆Ô ×∆Ô, for allr ∈R, and aÔ ∈ ∆Ô, for every a ∈ I, ≺Ô is a strict partial order on ∆Ô, and Ô:=

hÔ₁, . . . ,Ô_#Ri, whereÔ_i ⊆rÔ_i ×rÔ_i , fori = 1, . . . ,#R, and such that≺Ô and eachÔ_i satisfy the smoothness condition [23]. Moreover, for anyr, r1, r2∈R\ {u}, Ointerpretsorderings on role inversesandon role compositionsas follows:

Ô_r−:={((y1, x1),(y2, x2))|((x1, y1),(x2, y2))∈ Ô_r}, andÔ_r₁_◦r₂:={((x1, y1), (x2, y2))|for somez1,z2[((x1, z1),(x2, z2)) ∈Ô_r₁ and((z1, y1),(z2, y2)) ∈Ô_r₂], and for noz1,z2[((x2, z2),(x1, z1))∈Ô_r₁and((z2, y2),(z1, y1))∈Ô_r₂]}.

LetrÔ|x_i :=rÔ_i ∩({x} ×∆Ô)(i.e., the restriction of the domain ofrÔ_i to{x}). The interpretation function·ÔinterpretsdSROIQconcepts in the following way (whenever it is clear which component ofÔis used, we shall drop the subscript inÔ_i ):

>Ô :=∆Ô; ⊥Ô:=∅; (¬C)Ô:=∆Ô\CÔ; (CuD)Ô :=CÔ∩DÔ; (CtD)Ô :=CÔ∪DÔ;

(∀r.C)Ô :={x|rÔ(x)⊆CÔ}; (^∼^Wr.C)Ô :={x|minO(rÔ|x)(x)⊆CÔ};

(∃r.C)Ô :={x|rÔ(x)∩CÔ6=∅}; (⁻₋^∼^|r.C)Ô :={x|minÔ(rÔ|x)(x)∩CÔ 6=∅};

(∃r.Self)Ô:={x|(x, x)∈rÔ}; (⁻^∼₋^|r.Self)Ô :={x|(x, x)∈minÔ(rÔ|x)};

(≥nr.C)Ô :={x|#rÔ(x)∩CÔ≥n}; (≤nr.C)Ô :={x|#rÔ(x)∩CÔ ≤n};

(&nr.C)Ô :={x|# minÔ(rÔ|x)(x)∩CÔ} ≥n;

(.nr.C)Ô :={x|# minÔ(rÔ|x)(x)∩CÔ≤n}.

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It is not hard to see that, analogously to the classical case,^∼^Wand⁻^∼₋^|, as well as&

and., are duals to each other.

Definition 5 (Satisfaction).LetO=h∆Ô,·Ô,≺Ô,Ôiand letr₁, . . . , r_n, r, s∈R, C, D∈C, anda, b∈I. Thesatisfaction relationis defined as follows:

OrvsifrÔ⊆sÔ; Or@_∼sifminOrÔ ⊆sÔ;

Or₁◦ · · · ◦r_nvrif(r₁◦ · · · ◦r_n)^O ⊆r^O; Or₁◦ · · · ◦r_n@_∼rifminO(r₁◦

· · · ◦r_n)^O ⊆r^O;

OFun(r)ifr^Ois a function;OdFun(r)if for allx,# minO(r^O|x)(x)≤1;

O Ref(r)if{(x, x)|x∈∆Ô} ⊆rÔ; O dRef(r)if for everyx∈ min_≺O∆Ô, (x, x)∈rÔ;

OIrr(r)ifrÔ∩ {(x, x)|x∈∆Ô}=∅; OdIrr(r)if for everyx∈min_≺O∆Ô, (x, x)∈/ rÔ;

OSym(r)ifinv(r)Ô ⊆rÔ; OdSym(r)ifminO(r⁻)Ô ⊆rÔ;

OAsy(r)ifrÔ∩inv(r)Ô =∅; OdAsy(r)ifminOrÔ∩minO(r⁻)Ô =∅;

OTra(r)if(r◦r)Ô ⊆rÔ; OdTra(r)ifminÔ(r◦r)Ô⊆rÔ;

ODis(r, s)ifrÔ∩sÔ =∅; OdDis(r, s)ifminÔrÔ∩minÔsÔ =∅;

OCvDifCÔ⊆DÔ; OC@_∼Difmin_≺ÔCÔ ⊆DÔ;

O a : C ifaÔ ∈ CÔ; O (a, b) : rif (aÔ, bÔ) ∈ rÔ; O (a, b) : ¬r if O 6(a, b) :r; Oa=bifaÔ=bÔ; Oa6=bifO 6a=b.

IfOα, then we sayOsatisfiesα.Osatisfies a set of statements or assertionsX (denotedOX) ifOαfor everyα∈X, in which case we sayOis amodelofX.

We sayC ∈ C issatisfiable w.r.t.KB = hA,R,T iif there is a modelOofKBs.t.

C^O 6=∅, andunsatisfiableotherwise.

A statementαis (classically)entailedby a knowledge baseKB, denotedKB |=α, if every model ofKBsatisfiesα.

3 Reasoning with dSROIQ Knowledge Bases

As for classical SROIQ[20, Lemma 7], it is possible to eliminate an ABox Aby compiling all individual assertions inAas follows:

1. LetN⁰ :=N∪ {o_a|aappears inA}(i.e., extend the signature with new nominals);

2. LetA⁰:={a:C∈ A} ∪ {a:∃r.ob|(a, b) :r∈ A} ∪ {a:∀r.¬ob|(a, b) :¬r∈ A} ∪ {a:¬ob|a6=b∈ A};

3. For everyC∈C, letC⁰ :=Cud

a:D∈A⁰∃u.(oauD).

It is then easy to see thatCis satisfiable w.r.t.hA,R,T iif and only ifC⁰is satisfiable w.r.t.h∅,R,T i, which allows us to assume from now on and w.l.o.g. that ABoxes have been eliminated.

Next, in the same way that most of the classical role assertions can equivalently be replaced by GCIs or RIAs, under our preferential semantics, all of our defeasible role assertions, with the exception ofdAsy(·)anddDis(·), can be reduced to defeasible RIAs in the following way.dFun(r)can be replaced by> v.1r.>— to be ‘usually functional’ means only non-normal arrows can break functionality. (Note that, since

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the number restriction is unqualified,rneed not be simple.)dRef(r)anddIrr(r)can, respectively, be replaced with>@_∼∃r.Selfand>@_∼¬∃r.Self.dSym(r)can be reduced tor⁻@_∼randdTra(r)tor◦r@_∼r. Furthermore, note thatdAsy(r)can be reduced to dDis(r, r⁻)(cf. Definition 5). Hence, from now on we can assume, w.l.o.g., that the set of role assertionsRacontains only statements of the formDis(r, s)anddDis(r, s).

Next, we observe that defeasible concept inclusions can be made classical by introducing a new role name r≺ to encode≺at the object level. This is similar to the SROIQencoding of the typicality operator of Giordano et al. [16, 15].

Finally, we can apply the same procedure for eliminating both the TBox and the universal role u defined for classical SROIQ[20, Lemma 8][26], extended to the case ofdSROIQconcepts. Hence, from now on we can assume TBoxes (as well as occurrences ofutherein) have been eliminated.

The next theorem summarises the reduction outlined in this section:

Theorem 1. Satisfiability of dSROIQ-concepts w.r.t. TBoxes, ABoxes and RBoxes can be polynomially reduced to satisfiability of dSROIQ-concepts w.r.t. RBoxes in which all role assertions are of the formDis(r, s)anddDis(r, s).

It is known that classical RIAs with role composition on the LHS can be eliminated via automata-based procedures [21] or regular expressions [29]. Hence, we can assume w.l.o.g. that all classical RIAs are of the formr v s, with r, s ∈ R\ {u}. Whether analogous procedures for getting rid of role composition on the LHS ofdefeasibleRIAs are devisable and, if so, feasible in practice, is an open question that we leave for future investigation. (Roughly, the automaton used to ‘memorise’ role-pathsr1, . . . , rnin the classical case must be carefully adapted in order to also recognise preferred role-paths so that a normal r1, . . . , rn-path warrants the existence of ans-path, whenever r1◦ . . .◦rn@_∼sfollows fromR). Hence, in the remainder of the paper, we shall make the assumption that all defeasible RIAs are of the formr@_∼s, forr, s ∈ R\ {u} (and thereforeRcontains no assertions of the formdTra(·)— see above).

Furthermore, note that the special role namer_≺used in the internalisation of defeasible concept inclusions does not appear in^∼^W-,⁻^∼₋^|-,&- or.-concepts or in defeasible RIAs, forr≺∈/R.

4 A Tableau Proof Procedure for dSROIQ

We shall now present a tableau-based algorithm for deciding consistency ofdSROIQ- concepts w.r.t. an RBox. Thanks to the results in Section 3, it also allows for checking concept satisfiability w.r.t. knowledge basesKB=hA,R,T i.

The algorithm extends that forSROIQ[20] to deal with defeasible constructs and also works by generating a completion graph, which, if complete and clash-free (see below), can be used to construct a (possibly infinite) model for the input concept and the RBox.

Withnnf(C)we denote thenegation normal form(NNF) ofC∈C, i.e., the result of transformingCinto an equivalent concept by pushing negation inwards and applying De Morgan’s laws as well as the duality between∀and∃,≥and≤,^W^∼and⁻^∼₋^|, and&

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and.. Note that in NNF negation occurs only in front of concept names or in front of

∃r.Selfor⁻₋^∼^|r.Self.

IfC∈C,sub(C)denotes the set of all (syntactic)sub-conceptsofC(includingC itself), andclos(C)is the smallest set containingC that is closed under sub-concepts and negation:clos(C) :={D|D∈sub(C)} ∪ {nnf(¬D)|D∈sub(C)}.

Definition 6 (Completion Graph).LetC ∈Cbe in NNF and such that the universal roleudoes not occur inC, and letRbe an RBox. Acompletion graphforCw.r.t.R is a directed graphG:=hV, E, M,L,N,6.

=iwhereV is a set ofnodes,E⊆V ×V is a set ofedges,M ⊆E×Eis arelation on edges,L(·)is alabelling functiondefined by:

1. for everyv ∈ V,L(v) ⊆ clos(C)∪N∪ {≤ mr.D | ≤ nr.D ∈ clos(C)and m≤n} ∪ {.mr.D|.nr.D∈clos(C)andm≤n};

2. for everye= (v, v⁰)∈E,L(e)⊆R\ {u};

3. for everym= (e, e⁰)∈M,L(m)⊆ L(e)∩ L(e⁰), N ⊆E×R, with(e, r)∈ N only ifr∈ L(e), and6.

=⊆V×V is a symmetric relation.

Intuitively,N tells us whether(v, v⁰)is anormalr-edge among those leavingv. It is used along withM in the model-unravelling phase to construct a preference relation for each role name. (For the sake of readability, we shall henceforth write L(v, v⁰) andL((v, v⁰),(u, u⁰))instead ofL((v, v⁰))andL(((v, v⁰),(u, u⁰))).)M is the explicit construction of the skeleton of the preference relation on the edges, and is used to construct the model resulting from the unravelling of the completion graph.

If(v, v⁰)∈E, thenv⁰is asuccessorofv, andvis apredecessorofv⁰.Ancestoris the transitive closure of predecessor, anddescendantis the transitive closure of successor. We sayv⁰ is anr-successorofvifr ∈ L(v, v⁰).v is anr-predecessorofv⁰ifv⁰ is anr-successor ofv.Neighbour(resp.r-neighbour) is the union of successor (resp.

r-successor) and predecessor (r-predecessor). Ifr∈R\ {u},C ∈Candv∈V inG, then

r^G(v, C) :={v⁰|(v, v⁰)∈E, r∈ L(v, v⁰), andC∈ L(v⁰)}

denotes allr-successors ofvwithCin their label, and

r^G_N(v, C) :=r^G(v, C)∩ {v⁰ |((v, v⁰), r)∈ N }

denotes (intuitively) ther-successors ofvwithCin their label that are accessible via anr-edge which is minimal amongr-edges leavingv.

Definition 7 (Clash).LetG = hV, E, M,L,N,6.

=ibe a completion graph. We sayG contains aclashif there are nodesv, v⁰, v⁰⁰, v1, . . . , vk, v₁⁰, . . . , v_k⁰ ∈V such that:

1. ⊥ ∈ L(v), or for someA∈C,{A,¬A} ⊆ L(v);

2. r∈ L(v, v)and¬∃r.Self∈ L(v);

3. r∈ L(v, v),((v, v), r)∈ N and¬⁻₋^∼^|r.Self∈ L(v);

4. Dis(r, s)∈ Ra,(v, v⁰)∈Eand{r, s} ⊆ L(v, v⁰);

5. dDis(r, s)∈ Ra,(v, v⁰)∈E,{r, s} ⊆ L(v, v⁰)and((v, v⁰), r),((v, v⁰), s)∈ N; 6. ≤nr.C ∈ L(v)and{v0, . . . , vn} ⊆r^G(v, C), wherevi6.

=vjfor0≤i < j ≤n;

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7. .nr.C ∈ L(v)and{v0, . . . , vn} ⊆r^G_N(v, C), wherevi6.

=vjfor0≤i < j≤n;

8. ((v, v⁰), r)∈ N andr∈ L((v, v⁰⁰),(v, v⁰));

9. r∈ L((vi, v_i⁰),(vi+1, v_i+1⁰ )), fori= 1, . . . , k−1, andvk =v1,v_k⁰ =v⁰₁, or 10. for someo∈N,v6.

=v⁰ando∈ L(v)∩ L(v⁰).

In order to ensure termination of the algorithm in the presence of transitive roles, we extend the standard (classical)blockingtechnique [19, 22] to the case of our richer structures as follows:

Definition 8 (Blocking).LetG=hV, E, M,L,N,6.

=ibe a completion graph andv∈ V. IfL(v)∩N 6= ∅, thenv is anominalnode; otherwisevis ablockablenode. We sayvislabel blockedifvhas ancestorsv⁰,uandu⁰such that:

1. (v⁰, v),(u⁰, u)∈Eand there is a pathu, . . . , v⁰, vwithu, . . . v⁰, vblockable;

2. L(v) =L(u),L(v⁰) =L(u⁰)andL(v⁰, v) =L(u⁰, u);

3. for allr∈ L(v⁰, v),((v⁰, v), r)∈ N iff((u⁰, u), r)∈ N;

4. for every(x, y)such that((x, y),(v⁰, v))∈M, there is(x⁰, y⁰)such that

L(x) = L(x⁰),L(y) = L(y⁰),L(x, y) = L(x⁰, y⁰),((x⁰, y⁰),(u⁰, u)) ∈ M and L((x⁰, y⁰),(u⁰, u)) =L((x, y),(v⁰, v)).

5. for every(x, y)such that((v⁰, v),(x, y))∈M, there is(x⁰, y⁰)such that

L(x) = L(x⁰),L(y) = L(y⁰),L(x, y) = L(x⁰, y⁰),((u⁰, u),(x⁰, y⁰)) ∈ M and L((u⁰, u)(x⁰, y⁰)) =L((v⁰, v),(x, y)).

If (1)–(5) hold, we say ublocksv. We sayv ∈ V isblockedif either (a) v is label blocked, or (b)vis blockable and there is(v⁰, v) ∈ Esuch thatv⁰ is blocked. Ifv is blocked but is not label blocked, then we sayvisindirectly blocked.

LetCbe the concept of which the satisfiability w.r.t. an RBoxRone wants to check, and leto₁, . . . , o_kbe the nominals occurring inC. The tableau algorithm is initialised with a completion graphG = h{v0, v₁, . . . , v_k},∅,∅,L,∅,∅i, whereL(v0) := {C}, L(vi) := {oi}, for1 ≤ i ≤ k. We then expand G by decomposing concepts in its nodes through the application of the expansion rules in Figures 1–3. These rules are repeatedly applied until either no more rules are applicable or a clash (Definition 7) is found. In either case, we say the completion graph iscomplete. The algorithm returns

“Cis satisfiable w.r.t.R”, if the result of the application of the expansion rules toC andRis a complete and clash-free graph, and “Cis unsatisfiable w.r.t.R”, otherwise.

Note that the rules in Figure 1 are the same as the corresponding ones forSROIQ modulo the new definitions of blocking (see Definition 8), and of merging and pruning (see below). The rules in Figure 3 deal specifically with our new non-monotonic constructs. The rules in Figure 2 correspond to those classical rules that had to be modified in the light of our richer semantics. We here detail the case of the∃-rule, from which the respective explanations for theSelf-,≥- andNN-rules can be constructed. Unlike in the⁻^∼₋^|-rule, we cannot assume the newly addedr-edge is minimal amongr-successors ofv. We therefore need to consider the additional possibility that the newr-edge is not normal. (This has to be dealt with explicitly in order to ensure soundness of the algorithm.) Therefore, when creating a newr-successor, there are two possibilities: either (i) the new edge is normal among ther-edges leavingv, in which case the result is the

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u-rule:

if C1uC2∈ L(v),vis not indirectly blocked, and{C1, C2} 6⊆ L(v) then L(v) :=L(v)∪ {C1, C2}

t-rule:

if C1tC2∈ L(v),vis not indirectly blocked,{C1, C2} ∩ L(v) =∅ then L(v) :=L(v)∪ {C⁰}, for someC⁰∈ {C1, C2}

∀-rule:

if ∀r.C∈ L(v),vis not indirectly blocked,r∈ L(v, v⁰),C /∈ L(v⁰) then L(v⁰) :=L(v⁰)∪ {C}

ch-rule:

if ≤nr.C∈ L(v),vis not indirectly blocked,r∈ L(v, v⁰), and{C,nnf(¬C)} ∩ L(v⁰) =∅ then L(v⁰) :=L(v⁰)∪ {C⁰}, for someC⁰∈ {C,nnf(¬C)}

≤-rule:

if ≤nr.C∈ L(v),vis not indirectly blocked,#r^G(v, C)> n, and

there arev1, v2s.t.r∈ L(v, v1)∩ L(v, v2),C∈ L(v1)∩ L(v2),but notv16.

=v2

then a. ifv1is a nominal node, thenmerge(v2, v1),

b. else ifv2is a nominal node or an ancestor ofv1, thenmerge(v1, v2) c. elsemerge(v2, v1)

o-rule:

if for someo∈Nthere arev, v⁰s.t.o∈ L(v)∩ L(v⁰)and notv6.

=v⁰ then merge(v, v⁰)

Fig. 1.Classical expansion rules fordSROIQ.

same as that of applying the⁻^∼₋^|-rule, or (ii) it is not normal, in which case there must be a most preferredr-edge, which is also more preferred than the newly created one.

(This splitting is of the same nature as that in thet-rule, fitting the purpose of a proof by cases.) The additional indexkin the≥- andNN-rules serve a similar purpose.

The result ofprune(v)inG =hV, E, M,L,N,6.

=iis a new completion graph constructed fromG as follows: (1) For every successorv⁰ ofv,E := E\ {(v, v⁰)}and ifv⁰ is blockable, thenprune(v⁰); (2)V := V \ {v}. (We assume these changes are propagated toL,M,N and6.

=in the expected way.) The result ofmerge(v⁰, v)inG =hV, E, M,L,N,6.

=iis a new completion graph constructed fromGin the following way (conditions (d)–(f) in both clauses (1) and (2) below are used to preserve the relative normality of the edges):

1. For everyus.t.(u, v⁰)∈E:

(a) if{(v, u),(u, v)} ∩E=∅, thenE:=E∪ {(u, v)}andL(u, v) :=L(u, v⁰);

(b) if(u, v)∈E, thenL(u, v) :=L(u, v)∪ L(u, v⁰);

(c) if(v, u)∈E, thenL(v, u) :=L(v, u)∪ {inv(r)|r∈ L(u, v⁰)};

(d) if(x, y)∈Eand((x, y),(u, v⁰))∈M, thenM :=M \ {((x, y),(u, v⁰))} ∪ {((x, y),(u, v))}andL((x, y),(u, v)) :=L((x, y),(u, v))∪L((x, y),(u, v⁰));

(e) if(x, y)∈Eand((u, v⁰),(x, y))∈M, thenM :=M \ {((u, v⁰),(x, y))} ∪ {((u, v),(x, y))}andL((u, v),(x, y)) :=L((u, v),(x, y))∪L((u, v⁰),(x, y));

(f) if((u, v⁰), r)∈ N, thenN :=N ∪ {((u, v), r)};

(g) E:=E\ {(u, v⁰)};

2. For every nominal nodeus.t.(v⁰, u)∈E:

(a) if{(v, u),(u, v)} ∩E=∅, thenE:=E∪ {(v, u)}andL(v, u) :=L(v⁰, u);

(b) if(v, u)∈E, thenL(v, u) :=L(v, u)∪ L(v⁰, u);

(c) if(u, v)∈E, thenL(u, v) :=L(u, v)∪ {inv(r)|r∈ L(v⁰, u)};

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∃-rule:

if ∃r.C∈ L(v),vis not blocked, and there is nov⁰s.t.r∈ L(v, v⁰)andC∈ L(v⁰)

then 1. create a new nodev⁰and edge(v, v⁰)withL(v⁰) :={C},L(v, v⁰) :={r}andN:=N ∪ {((v, v⁰), r)}

or 2. create two new nodesv⁰, v⁰⁰and new edges(v, v⁰),(v, v⁰⁰)withL(v⁰) :={C},L(v, v⁰) :={r}, M:=M∪ {((v, v⁰⁰),(v, v⁰))},L((v, v⁰⁰),(v, v⁰)) :={r}andN:=N ∪ {((v, v⁰⁰), r)}

Self-rule:

if ∃r.Self∈ L(v),vis not blocked, andr /∈ L(v, v)

then 1. add an edge(v, v), if it does not exist,L(v, v) :=L(v, v)∪ {r}, andN:=N ∪ {((v, v), r)}

or 2. create a nodev⁰and edges(v, v),(v, v⁰),L(v, v) :=L(v, v)∪ {r},L(v, v⁰) :={r}, M:=M∪ {((v, v⁰),(v, v))},L((v, v⁰),(v, v)) :={r}, andN:=N ∪ {((v, v⁰), r)}

≥-rule:

if ≥nr.C∈ L(v),vis not blocked, and there are nov1, . . . , vns.t.r∈ L(v, vi),C∈ L(vi), i= 1, . . . , n, andvi6.

=vj, for1≤i < j≤n, and eachviis not blocked ifvis not blockable then a. guessk∈ {0, . . . , n},

b. createknew nodesv1, . . . , vkand edges(v, vi), fori= 1, . . . , k, withL(v, vi) :={r}, L(vi) :={C}andN:=N ∪ {((v, vi), r)},

c. create2(n−k)new nodesvk+1, . . . , vnandvk+1⁰ , . . . , v⁰nand edges(v, vi)and(v, v⁰i), fori=k+ 1, . . . , n, withL(vi) :={C},L(v, vi) :={r},L(v, v⁰i) :={r},

M:=M∪ {((v, vi⁰),(v, vi))},L((v, vi⁰),(v, vi)) :={r}andN:=N ∪ {((v, v⁰i), r)}, and d. setvi6.

=vj, for1≤i < j≤n NN-rule:

if 1. ≤nr.C∈ L(v),vis not blockable,r∈ L(v⁰, v),v⁰is blockable, andC∈ L(v⁰) 2. there is nom∈ {1, . . . , n}s.t.≤mr.C∈ L(v)and s.t. there aremnominalr-successors

v1, . . . , vmofvwithC∈ L(vi)andvi6.

=vjfor all1≤i < j≤m

then a. guessm∈ {1, . . . , n}, setL(v) :=L(v)∪ {.mr.C}and guessk∈ {0, . . . , m}, b. createknew nodesv1, . . . , vkand edges(v, vi), fori= 1, . . . , k, withL(v, vi) :={r},

L(vi) :={C, oi}with eachoi∈Nnew inGandN:=N ∪ {((v, vi), r)},

c. create2(m−k)new nodesvk+1, . . . , vmandv⁰k+1, . . . , vm⁰ and edges(v, vi)and(v, v⁰i), for i=k+ 1, . . . , m, withL(v, vi) :={r},L(vi) :={C, oi}, with eachoi∈Nnew inG,L(v, vi⁰) :={r}, M:=M∪ {((v, vi⁰),(v, vi))},L((v, vi⁰),(v, vi)) :={r}andN:=N ∪ {((v, v⁰i), r)}, and

d. setvi6.

=vj, for1≤i < j≤m

Fig. 2.New classical expansion rules fordSROIQ.

(d) if(x, y)∈Eand((x, y),(v⁰, u))∈M, thenM :=M \ {((x, y),(v⁰, u))} ∪ {((x, y),(v, u))}andL((x, y),(v, u)) :=L((x, y),(v, u))∪L((x, y),(v⁰, u));

(e) if(x, y)∈Eand((v⁰, u),(x, y))∈M, thenM :=M \ {((v⁰, u),(x, y))} ∪ {((v, u),(x, y))}andL((v, u),(x, y)) :=L((v, u),(x, y))∪L((v⁰, u),(x, y));

(f) if((v⁰, u), r)∈ N, thenN :=N ∪ {((v, u), r)};

(g) E:=E\ {(v⁰, u)};

3. L(v) :=L(v)∪ L(v⁰);

4. 6.

= :=6.

=∪{(v, w)|v⁰6.

=w}; and 5. prune(v⁰).

As in the classical case, in order to ensure termination of the tableau algorithm, one has to assign higher priorities to certain rules. Here we assume the following strategy is adopted: Theo-rule is applied with the highest priority; theNN- anddNN-rules are applied before the≤- and.-rules; the other rules are applied with a lower priority.

Theorem 2. LetC∈Cand letRbe an RBox.

1. The algorithm terminates if started withnnf(C)andR;

2. When exhaustively applied tonnf(C)andR, the expansion rules yield a complete and clash-free completion graph iffCis satisfiable w.r.t.R.

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Rh-rule:

if r∈ L(v, v⁰),vis not indirectly blocked and eitherrvs∈ Ror bothr@_∼s∈ Rand((v, v⁰), r)∈ N then L(v, v⁰) :=L(v, v⁰)∪ {s}

−∼

−^|-rule:

if ⁻−^∼^|r.C∈ L(v),vis not blocked and there is nov⁰s.t.r∈ L(v, v⁰),C∈ L(v⁰)and((v, v⁰), r)∈ N then create a new nodev⁰and edge(v, v⁰)withL(v⁰) :={C},L(v, v⁰) :={r}andN:=N ∪ {((v, v⁰), r)}

dSelf-rule:

if ⁻−^∼^|r.Self∈ L(v),vis not blocked and eitherr /∈ L(v, v)or((v, v), r)∈ N/

then add a new edge(v, v), if required,L(v, v) :=L(v, v)∪ {r}, andN:=N ∪ {((v, v), r)}

∼W-rule:

if ^W^∼r.C∈ L(v),vis not indirectly blocked,r∈ L(v, v⁰),((v, v⁰), r)∈ NandC /∈ L(v⁰) then L(v⁰) :=L(v⁰)∪ {C}

dch-rule:

if .nr.C∈ L(v),vis not indirectly blocked,r∈ L(v, v⁰),((v, v⁰), r)∈ Nand {C,nnf(¬C)} ∩ L(v⁰) =∅

then L(v⁰) :=L(v⁰)∪ {C⁰}, for someC⁰∈ {C,nnf(¬C)}

&-rule:

if &nr.C∈ L(v),vis not blocked, and there are nov1, . . . , vns.t.r∈ L(v, vi),((v, vi), r)∈ N, C∈ L(vi), fori= 1, . . . , n, and s.t.vi6.

=vj, for1≤i < j≤n, and eachviis not blocked ifvis not blockable

then creatennew nodesv1, . . . , vnwithL(v, vi) ={r},N:=N ∪ {((v, vi), r)},L(vi) ={C}, fori= 1, . . . , n, and setvi6.

=vj,1≤i < j≤n .-rule:

if .nr.C∈ L(v),vis not indirectly blocked,#r^G_N(v, C)> n, and there arev1, v2s.t.

r∈ L(v, v1)∩ L(v, v2),((v, v1), r),((v, v2), r)∈ N,C∈ L(v1)∩ L(v2)but notv16.

=v2

then a. ifv1is a nominal node, thenmerge(v2, v1), else

b. ifv2is a nominal node or an ancestor ofv1, thenmerge(v1, v2) c. elsemerge(v2, v1)

dNN-rule:

if 1. .nr.C∈ L(v),vis not blockable,r∈ L(v⁰, v),v⁰is blockable andC∈ L(v⁰)

2. there is nom∈ {1, . . . , n}s.t..mr.C∈ L(v)and s.t. there aremnominal nodesv1, . . . , vmwith (v, vi)∈E,r∈ L(v, vi),((v, vi), r)∈ N,C∈ L(vi), fori= 1, . . . , m, and withvi6.

=vj, for all1≤i < j≤m

then a. guessm∈ {1, . . . , n}and setL(v) :=L(v)∪ {.mr.C}

b. createmnew nodesv⁰1, . . . , vm⁰ withL(v, vi⁰) :={r},N:=N ∪ {((v, v⁰i), r)|1≤i≤n}, L(v⁰i) :={C, oi}, withoi∈Nnew inG,i= 1, . . . , m, and setv⁰i6.

=v⁰j,1≤i < j≤m

Fig. 3.Defeasible expansion rules fordSROIQ.

5 Summary and Future Work

The main contributions of the present paper are: (i) a meaningful extension ofSROIQ with defeasible reasoning constructs in the concept language, in both concept and role inclusions, and in role assertions, together with an intuitive KLM-style preferential semantics; (ii) a translation of the entailment problem w.r.t.dSROIQknowledge bases to concept satisfiability relative to an RBox only, and (iii) a terminating, sound and complete tableau-based algorithm for checking concept satisfiability w.r.t. dSROIQ RBoxes.

As for the next steps, we have (i) extending the tableau procedure to allow role composition in defeasible RIAs, (ii) an analysis of the computational complexity of concept satisfiability fordSROIQ, (iii) an investigation of the correspondence between dSROIQand an extension of the OWL 2 RDF semantics³, and (iv) the definition of an appropriate notion ofnon-monotonicentailment fordSROIQontologies.

3https://www.w3.org/TR/2012/REC-owl2-rdf-based-semantics-20121211

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