Quasi-Classical Semantics for Expressive Description Logics

Quasi-Classical Semantics for Expressive Description Logics

Xiaowang Zhang^1,4, Guilin Qi², Yue Ma³and Zuoquan Lin¹

1School of Mathematical Sciences, Peking University, Beijing 100871, China

2Institute AIFB, University of Karlsruhe, Karlsruhe 76131, Germany

3Laboratoire d’Informatique de l’universit Paris-Nord (LIPN) - UMR 7030, France

4School of Mathematical Sciences, Anhui University, Hefei 230039, China {zxw,lzq}@is.pku.edu.cn, GQI@aifb.uni-karlsruhe.de, Yue.Ma@lipn.univ-paris13.fr

Abstract. Inconsistency handling in expressive description logics is an impor- tant problem because inconsistency may naturally occur in an open world. In this paper, we first present the quasi-classical semantics for description logicSHIQ, which is based on quasi-classical logic. We show that this semantics can be used to deal with inconsistency and that it is reduced to the standard semantics when there is no inconsistency. Compared with the existing four-valued semantics for description logics, it strengthens the reasoning capability in the sense that it can satisfy some important inference rules, such asModus Ponens. Finally, we ana- lyze the computational complexity of consistency checking based on the quasi- classical semantics.

1 Introduction

In an open, constantly changing and collaborative environment like the forthcoming Semantic Web, a knowledge base (or an ontology) may well contain inconsistencies because of many reasons, such as modeling errors, migration from other formalisms, merging ontologies, and ontology evolution (see [1]). As the logical foundation of On- tology Web Language (OWL), expressive description logics (for short DLs) fail to tol- erate inconsistent data. Therefore, handling inconsistency in expressive DLs, such as SHIQ, is becoming an important topic in recent years.

There are two fundamentally different approaches to handling inconsistency. One is to repair inconsistency in ontologies to obtain consistent ontologies [2,3,4]. The other, called paraconsistent approach, does not simply repair inconsistencies but applies a non- standard reasoning to obtain meaningful answers from inconsistent ontologies [5,6]. For the latter, having inconsistencies is treated as a natural phenomenon in realistic data and is tolerated during reasoning, whilst inconsistencies are viewed as erroneous data for the first approach. So far, the main idea of paraconsistent methods for handling inconsistent ontologies [5] is based on Belnap’s four-valued semantics [7]. However their reasoning capability is rather weak [6]. For instance, they fail in holding some important inference rules, such asModus Ponens,Modus TollensandDisjunctive Syllogism. In [6], a total negation is introduced to enable the resolution principle for paraconsistent reasoning

?This paper is supported by the EU under the IST project NeOn.

in four-valued DLs. However, four-valued DLs with total negation still don’t satisfy some intuitive equivalences mentioned above. The main problem of four-valued DLs is that they can not overcome those inherent shortcomings of four-valued semantics in reasoning [7].

To avoid shortcomings of four-valued DLs and strengthen paraconsistent reason- ing capability, in this paper, following [8], we study a quasi-classical description logic SHIQ(QCSHIQfor short), which is based on quasi-classical logic [9,10] and de- scription logic (for short DL)SHIQ. We concentrate onSHIQsince it is the core of OWL DL [11,12]. Main contributions of this paper are summarized as follows:

– QC semanticsis introduced toSHIQto handle inconsistent ontologies. It com- poses of two kinds of semantics, namely,QC weak semantics|=wandQC strong semantics|=s. QC weak semantics inherits the characteristics of four-valued se- mantics and QC strong semantics redefines the interpretation for disjunction and conjunction of concepts to make the three important inference rules hold.

– A new notion calledcomplement of concept is introduced. We useC to denote the complement of conceptC. The semantics of aninclusion axiomdefined under the weak semantics and the strong semantics satisfy theintuitive equivalences. For instance,I |=_wCvDiffI |=_wCtD(a); andI |=_sCvDiffI |=_s¬CtD(a) for any individuala.

– AQC modelis defined to express a possible world under QC strong semantics. A QC entailmentbased on QC model, written by “|=Q”, between an ontology and an axiom is presented which is shown to be able to handle inconsistency. Compared with the four-valued DLs, QC DLs satisfy three important inference rules:Modus Ponens (MP): {C(a), C v D} |=Q D(a), Modus Tollens (MT): {¬D(a), C v D} |=Q ¬C(a), andDisjunctive Syllogism (DS):{¬C(a), CtD} |=QD(a).

– Two basic query entailment problems, namely, instance checking and subsumption checking are defined and discussed. We show that the two basic inference problems can be reduced into the QC consistency problem.

– We prove that the complexity of deciding QC consistency for an ABox is EXPTIME- Complete.

Compared with QCALCstudied in [13], QCSHIQstudied in this paper has the fol- lowing significant improvements: (1) we redefine the QC strong semantics of concept inclusionC1 v C2 so that intuitive logical equivalences w.r.t. classical negation can be satisfied, which is not the case for QCALC; (2) we properly introduce the weak and strong satisfactions in QCSHIQ, which avoids redundant definition of the seman- tics of disjunction and conjunction of concepts in QCALC; (3) we show that instance checking and subsumption checking can be reduced to the QC consistency problem, which has not been achieved in QCALC; and (4) we study the QC semantics for more expressive language constructors than those inALC.

The paper is organized as follows. Section 2 briefly reviewsSHIQ. Section 3 in- troduces QC semantics forSHIQ. Section 4 discusses the reasoning problems in QC SHIQ. Section 5 concludes this paper and discusses the future work. Due to the space limitation, proofs are omitted but are available in a technical report¹.

1http://www.is.pku.edu.cn/˜zxw/publication/TRQCSHIQ.pdf

2 Preliminaries

In this section, we briefly review some basic notations in description logic (DL)SHIQ.

For comprehensive background reading, please refer to [11].

LetLbe the language ofSHIQ, which contains a set of atomic concepts (or con- cept names), denoted by N_C; a set of individuals, denoted byN_I; and a set of role names, denoted byNR.NR+is a set of transitive role names inNRandRis a transi- tive closure set ofNR. We denoteR=R∪ {R⁻ |R ∈R}whereR⁻ is the inverse role ofR. The inverse role ofRcan be also taken as the inverse transformation onR.

In this way, the inverse role ofRcan be denoted byInv(R), i.e.,Inv(R) =R⁻and Inv(R⁻) =R. Arole inclusion axiomis of the formRvSwhere rolesR, Scan be inverse. Arole hierarchy, denotedR, is a set of role inclusion axioms. A roleR is a sub-roleofS, denoted byRvS, where “∗ v” is the transitive-reflexive closure of∗ vover R ∪ {Inv(R) vInv(S) | R v S ∈ R}. A simple roleS is a role which is neither transitive nor has any transitive sub-roles.

Concept descriptions inSHIQare formed according to the following syntax rule:

C, D→A| > | ⊥ | ¬A|CtD|CuD| ∃R.C| ∀R.C |≥nS.C |≤nS.C

where>is the top concept,⊥is the bottom concept,Ais a concept name,C, D are concepts,Ris a role,Sis a simple role andnis nonnegative integer.

LetC, Dbe concepts,a, bindividuals andRa role. InSHIQ,assertionsare of the formC(a)orR(a, b)ora6.

=b. Ageneral concept inclusion axiom (GCI)is of the formC vD. Informally, an axiomC(a)means that the individualais an instance of conceptC, and an axiomR(a, b)means that individualais related with individualb via the propertyR. The inclusion axiomCvDmeans that each individual ofCis an individual ofD.

A knowledge base (or ontology) comprises two components, aTBoxand anABox.

InSHIQ, a TBox includes two parts: a set of concept inclusion axiomsT and a role hierarchy R; an ABox consists of concept assertions, role assertions, and individual inequalities.

The formal definition of the classical (model-theoretic) semantics ofSHIQis given by means of an interpretationI= (∆^I,·^I)which consists of a non-empty domain∆^I and a mapping·^Isatisfying the conditions in Table 1, where the mapping·^Iinterprets concepts as subsets of the domain and roles as binary relations on the domain∆^I.

An interpretationIsatisfiesa terminologyT (role hierarchyR) iff for any concept inclusionC vD (role inclusionR v S) inT (R),I is an interpretation ofC vD (R v S). In this case,I is named amodelofT (R), denotedI |= T (I |= R). A conceptC is calledsatisfiablew.r.t. a terminologyT and a hierarchyRiff there is a modelI ofT andRwithC^I 6=∅. A conceptDsubsumesa conceptCw.r.tT andR iffC^I ⊆D^I holds for each modelIofT andR. For an interpretationI, an element a^I∈∆^Iis called aninstanceof a conceptCiffa^I ∈C^I. An interpretationIsatisfies an ABoxA(writtenI |=A) if it satisfies every assertion inA. An ABoxAis consistent w.r.t.T andRiff there is a modelIofT andRthat satisfies each assertion inA.

Table 1.Syntax and semantics of DLSHIQ

Constructor Name Syntax Semantics

atomic conceptA A A^I⊆∆^I

abstract roleR R R^I⊆∆^I×∆^I

individualsI o o^I∈∆^I

trans.role R∈NR₊ R^I= (R^I)⁺

inverse role R⁻ {(x, y)|(y, x)∈R^I}

top concept > ∆^I

bottom concept ⊥ ∅

conjunction C1uC2 C₁^I∩C₂^I

disjunction C1tC2 C1^I∪C2^I

negation ¬C ∆^I\C^I

exists restriction ∃R.C {x| ∃y,(x, y)∈R^Iandy∈C^I} value restriction ∀R.C {x| ∀y,(x, y)∈R^Iimpliesy∈C^I} qualifying atleast restriction ≥nR.C {x|]({y.(x, y)∈R^I}andy∈C^I)≥n}

qualifying atmost restriction ≤nR.C {x|]({y.(x, y)∈R^I}andy∈C^I)≤n}

Axiom Name Syntax Semantics

concept assertion C(a) a^I ∈C^I

role assertion R(a, b) (a^I, b^I)∈R^I concept inclusion C1vC2 C^I1 ⊆C2^I

role inclusion RvS R^I ⊆S^I

individual inequality a6.

=b a^I 6.

=b^I

3 Quasi-Classical Description Logic SHIQ

Quasi-classical DLSHIQis based on quasi-classical logic defined in [9] andSHIQ.

In this section, we present the syntax and the semantics of QCSHIQ.

3.1 Syntax of QCSHIQ

The syntax of QCSHIQis almost the same as that ofSHIQ. One difference is that a new concept constructor calledcomplement of a conceptis introduced, which is the syntax form of “total negation” in [14]. The complement of conceptCis denoted byC.

The intuition behind a complement of a concept is that we want to use it to characterize QC inconsistency, i.e., to reverse both the information of being true and of being false.

Its formal interpretation will be given in Section 3.2.

The definition of aQC ABox (resp. QC terminology, QC role hierarchy)is similar to that of an ABox (resp. terminology, role hierarchy). The only difference is that we allow a complement of a concept in a complex concept. AQC ontologycomprises a QC terminology, a QC role hierarchy and a QC ABox.

Given the languageLof SHIQ, the language of QCSHIQ is denotedL^∗ and is defined byL^∗ = L ∪ {C | Cis a concept inL}. In the following, we discuss QC SHIQbased on the languageL^∗.

For an atomic conceptA, we callA,¬A,Aand¬Aasconcept literals. A concept C is inNegation Normal Form (NNF)if negation (¬) occurs only in front of concept names. A concept C is in QC NNF, if conceptC is in NNF and complement only occurs over a concept name or negation of a concept name. Arole-involved literalhas the form∀R.Cor∃R.CwithCa concept in NNF. Arole-involved literalhas the form

∀R.C,∃R.C,≥nR.C or ≤nR.C withCa concept in QC NNF. Aliteralis either a concept literal or a role-involved literal, written byL. Aclauseis the disjunction of finite literals. LetL1t · · · tLnbe a clause, thenLit(L1t · · · tLn)is the set of literals {L1, . . . , Ln}that are in the clause. A clause is theempty clause, denoted by♦, if it has no literals. We define∼be acomplementation operationsuch that∼Ais¬Aand

∼(¬A)isA.

LetL1t· · ·tLnbe a clause that includes a literal disjunctLi. ThefocusofL1t· · ·t LnbyLi, denoted⊗(L1t · · · tLn, Li), is defined as the clause obtained by removing L_ifromLit(L₁t · · · tL_n). In the case of a clause with just one disjunct, we assume

⊗(L, L) =⊥. For instance, given a clauseL₁tL2tL3,⊗(L1tL2tL3, L₂) =L₁tL3.

3.2 Semantics of QCSHIQ

In this subsection, we define the QC semantics for QC SHIQ by introducing two semantics, namely, QC weak semantics and QC strong semantics.

Firstly, we introduce the notion of a weak interpretation and the notion of a strong interpretation over domain ∆^I by assigning to each concept C a pair h+C,−Ciof subsets of∆^I. Intuitively,+Cis the set of elements known to belong to the extension ofC, while−C is the set of elements known to be not contained in the extension of C.+Cand−Care not necessarily disjoint or mutually complemental with respect to the domain. Thecomplemental setof a setSw.r.t. an interpretationI, denoted byS, is S=∆^I\S.

In QCSHIQ, a weak interpretation is a reformulation of a four-valued interpreta- tion in four-valued DLs (see [6]).

Definition 1 LetI be a pair(∆^I,·^I)with∆^I as domain, where·^I is a function as- signing elements of ∆^I to individuals, subsets of∆^I ×∆^I to concepts and subsets of(∆^I×∆^I)²to roles.I is aweak interpretationin QCSHIQif the conditions in Table 2 are satisfied, whereC_i^I = h+Ci,−Ciifor i = 1,2,C^I = h+C,−Ciand R^I=h+R,−Ri.

Definition 2 Let|=w be a satisfiability relation between a set of weak interpretation and a set of axioms, calledweak satisfaction. For a weak interpretationI, we define

|=_was follows, whereC, C₁, C₂are concepts,R, R₁, R₂are roles andais an individ- ual:

(1)I |=_wR₁vR₂iff+R₁⊆+R₂, ifR^I_i =h+Ri,−Rii,i= 1,2;

(2)I |=_wTrans(R)iff+R= (+R)⁺, ifR^I=h+R,−Riand(R⁺)^I =h+R⁺,−R⁺i;

(3)I |=_wC(a)iffa^I∈+C, C^I=h+C,−Ci;

(4)I |=_wR(a, b)iff(a^I, b^I)∈+R, R^I=h+R,−Ri;

(5)I |=wC1vC2iff+C1⊆+C2, fori= 1,2,C_i^I=h+Ci,−Cii;

(6)I |=wa6.

=biffa^I6=b^I.

Table 2. Weak Semantics of QCSHIQ

Syntax Weak Semantics

A A^I =h+A,−Ai,where+A,−A⊆∆^I

R R^I =h+R,−Ri, where+R,−R⊆∆^I×∆^I

o o^I∈∆^I

> h∆^I,∅i

⊥ h∅, ∆^Ii

C1uC2 h+C1∩+C2,−C1∪ −C2i C1tC2 h+C1∪+C2,−C1∩ −C2i

¬C (¬C)^I =h−C,+Ci

C C^I=h∆^I\+C, ∆^I\ −Ci

∃R.C h{x| ∃y,(x, y)∈+Randy∈+C},{x| ∀y,(x, y)∈+Rimpliesy∈ −C}i,

∀R.C h{x| ∀y,(x, y)∈+Rimpliesy∈+C},{x| ∃y,(x, y)∈+Randy∈ −C}i

≥nR.Ch{x|]({y.(x, y)∈+R}andy∈+C)≥n},{x|]({y.(x, y)∈+R}andy6∈ −C)< n}i

≤nR.Ch{x|]({y.(x, y)∈+R}andy6∈ −C)≤n},{x|]({y.(x, y)∈+R}andy∈+C)> n}i

In QCSHIQ, the concept inclusion under weak satisfaction is defined by the in- ternal inclusion in four-valued DLs because other inclusion, namely, material inclusion and strong inclusion can be transformed into internal inclusion (see [6]).

The following property shows that there exists a close relationship between weak models and 4-models in four-valued DLs defined in [6].

Proposition 1 LetC, Dbe concepts,aan individual,Ra role andIan interpretation in QCSHIQ. We have

(1)I |=wC(a)iffI |=4C(a);

(2)I |=wCvDiffI |=4C@D.

The following proposition shows that intuitive equivalence w.r.t. the complement of a concept is satisfied under QC weak semantics.

Proposition 2 LetIbe an interpretation and letC, Dbe concepts, we have I |=wCvDiffI |=wCtD(a)for any individuala∈NI.

The QC weak satisfaction has the drawback that it does not satisfy some basic in- ference rules, such as MP, MT and DS. Therefore, we define a QC strong interpretation that redefines interpretation of disjunction of concepts and conjunction of concepts.

Definition 3 LetI be a pair(∆^I,·^I)with∆^I as the domain, where·^I is a function assigning elements of∆^I to individuals, subsets of∆^I×∆^I to concepts and subsets of(∆^I×∆^I)²to roles.I is astrong interpretationin QCSHIQif the conditions in Table 2, except conditions for conjunction of concepts and disjunction of concepts, are satisfied and the following conditions hold, letC_i^I=h+Ci,−Ciifori= 1,2:

conjunction of concepts:

(C₁uC₂)^I=h+C₁∩+C₂,(−C₁∪ −C₂)∩(−C₁∪+C₂)∩(+C₁∪ −C₂)i disjunction of concepts:

(C₁tC₂)^I=h(+C₁∪+C₂)∩(−C₁∪+C₂)∩(+C₁∪ −C₂),−C₁∩ −C₂i Compared with the weak interpretation, the strong interpretation of disjunction of concepts tightens the condition that an individual is known to belong to a concept; and the strong interpretation of conjunction of concepts is defined by relaxing the condition that an individual known to be not contained in the extension of a concept. The strong interpretation for the disjunction and conjunction characterizes the relationship between concepts and individuals with holding resolution rule.

Definition 4 Let|=_sbe a satisfiability relation between a set of weak interpretation and a set of axioms, calledstrong satisf action. For a strong interpretationI, we define|=_sas follows, whereC, C₁, C₂ are concepts,R, R₁, R₂ are roles andais an individual:

(1)I |=sR1vR2iff+R1⊆+R2, ifR^I_i =h+Ri,−Rii,i= 1,2;

(2)I |=sTrans(R)iff+R= (+R)⁺, ifR^I =h+R,−Riand(R⁺)^I=h+R⁺,−R⁺i;

(3)I |=sC(a)iffa^I ∈+CwhereC^I =h+C,−Ci;

(4)I |=sR(a, b)iff(a^I, b^I)∈+RwhereR^I =h+R,−Ri;

(5)I |=sC1vC2iff−C1⊆+C2,+C1⊆+C2and−C2⊆ −C1, fori= 1,2,C_i^I =h+Ci,−Cii;

(6)I |=sa6.

=biffa^I 6=b^I.

In Definition 4, when defining the strong satisfaction of a concept inclusion under a QC strong interpretation, we use conditions for interpreting a concept inclusion by internal inclusion, material inclusion and strong inclusion in four-valued DLs. By doing so, our strong satisfaction satisfies intuitive equivalence that is falsified by four-valued satisfaction relations.

Proposition 3 LetIbe an interpretation and letC, Dbe concepts. We have I |=_sCvDiffI |=_s¬CtD(a)for any individuala∈N_I.

Proposition 2 and Proposition 3 provide the theoretical base of transforming the problem of reasoning with ABoxes and terminologies into the problem of reasoning with ABoxes.

The following proposition provides a slightly different view on the QC semantics of disjunction of concepts.

Proposition 4 LetC1andC2be two concepts andabe an individual. We have I |=_sC₁tC₂(a)iff for someC_i, a^I∈+C_ianda^I 6∈ −Ci;or

for allC_i, a^I∈+C_ianda^I ∈ −Ci; whereC_i^I=h+Ci,−Ciiandi= 1,2.

Proposition 5 LetIbe an interpretation andφbe an axiom in QCSHIQrespectively.

IfI |=sφthenI |=wφ.

Proposition 5 shows that a strong model is a weak model. However, the converse does not hold. For instance, given an ABoxA={C(a),¬C(a)}. We can construct an interpretationIsuch thata^I ∈+C,a^I ∈ −Canda^I 6∈+DwhereC^I =h+C,−Ci andD^I = h+D,−Di. Then I 6|=s CtD(a)because I 6|=s D(a). Clearly,I |=w

CtD(a).

Proposition 6 LetCbe a concept,Ra role andna natural number. For any QC weak ( or QC strong) interpretationI, we have

(1)(¬(≤nR.C))^I= (> nR.C)^I; (2)(¬(≥nR.C))^I= (< nR.C)^I; (3)(∃R.C)^I= (≥1R.C)^I; (4)(∀R.C)^I= (<1R.¬C)^I.

Proposition 7 LetAbe a concept name,C, Dconcepts,a, bindividuals,Ra role and na natural number in QCSHIQ. IfIis a QC weak or QC strong interpretation, then the follows properties hold.

(1)I |=xA(a)iffI 6|=xA(a) (6)I |=x>(a)iffI |=x⊥(a)

(2)I |=_xA(a)iffI |=_xA(a) (7)I |=_wCtD(a)iffI |=_wCuD(a) (3)I |=x¬A(a)iffI |=x¬A(a) (8)I |=wCuD(a)iffI |=wCtD(a) (4)I |=x(∀R.C)(a)iffI |=x∃R.C(a) (9)I |=x(≥nR.C)(a)iffI |=x≤n-1R.C(a) (5)I |=x(∃R.C)(a)iffI |=x∀R.C(a) (10)I |=x(≤nR.C)(a)iffI |=x≥n+1R.C(a) where|=xis a place-holder for both|=wand|=s.

Definition 5 Given a QC ontology O and an axiom φ inSHIQ, we sayO quasi- classically entails(for shortQC entails)φ, denoted byO |=Q φ, iff for every interpre- tationI, for any axiomψofOifI |=s ψthenI |=w φ. In this case,|=Q is called a quasi-classical entailment (relation)(for shortQC entailment) betweenOandφ.

The following example shows that|=Q is non-trivializable in the sense that when an ontologyOis classically inconsistent.

Example 1 Given an ABoxA={B(a),¬B(a)}and a concept nameAin QCSHIQ.

It is clear thatAis classically inconsistent. HoweverA |=Q A(a)does not hold. This is because there exists an interpretationI such thata^I ∈ +B anda^I ∈ −B where B^I =h+B,−Bi. SoI |=sBu ¬B(a), butI 6|=wA(a)sinceA(a)does not occur in A.

The following proposition shows that|=_Qsatisfies theresolution rule.

Proposition 8 LetC, D, Ebe concepts andabe an individual.

{CtD(a),¬CtE(a)} |=QDtE(a).

By Proposition 8,|=Qsatisfies three important inferences:MP,MTandDS.

Example 2 SupposeOis an empty QC ontology.

(1)Consider the conceptAt ¬A. We haveO 6|=_Q At ¬A(a), sinceOstrongly satisfies every formula inO, butOdoes not weakly satisfyAt ¬A(a).

(2)Consider the conceptAtA. We haveO |=Q AtA(a), sinceOstrongly satisfies every axiom inOandO |=wA(a)orO |=wA(a), i.e.,O |=wAtA(a).

Example 2 shows that, for some concept nameAin QCSHIQ, the conceptAtA can be considered as the top concept>under QC entailment. However, this is not the case for conceptAt ¬A.

In the following, we show that some properties of QC entailment satisfied in QC propositional logic (see [9]) still hold in QCSHIQ.

Proposition 9 Given a QC ontologyO, axiomsφ, ψ, two conceptsC, Dand an indi- viduala, the following properties hold.

(1) (Reflexivity)O ∪ {φ} |=_Qφ.

(2) (Monotonicity)ifO |=_QφthenO ∪ {ψ} |=_Qφ.

(3) (And-introduction)ifO |=_QC(a)andO |=_QD(a)thenO |=_Q CuD(a).

(4) (Or-elimination)if O ∪ {C(a)} |=Q φand O ∪ {D(a)} |=Q φthenO ∪ {Ct D(a)} |=Q φ.

The following proposition shows that QC entailment is weaker than classical entail- ment inSHIQ.

Proposition 10 LetObe an ontology andφbe an axiom inSHIQ.

IfO |=QφthenO |=φ.

The converse of Proposition 10 does not hold. In Example 1,A |=φfor any axiom becauseAis classically inconsistent whileA 6|=QA(a).

Proposition 11 LetObe an ontology andφbe an axiom inSHIQ.

IfO |=₄φthenO |=_Q φ.

In QCSHIQ, a strong interpretationI satisfiesa terminologyT (role hierarchy R) iffI |=sC vD(R vS) for eachCvDinT (RvSinR). In this case,I is called aQC modelofT, writtenI |=sT (aQC modelofR, writtenI |=sR). A strong interpretationIsatisfiesan ABoxAiffI |=s φfor each assertionφinA, whereφis one of the following forms:C(a),R(a, b),a6.

=b. In this case,Iis called aQC model ofA, writtenI |=sA.

The following proposition shows the close relationship between the QC entailment and the QC model.

Proposition 12 LetObe a QC ontology,C, Dtwo concepts andaan individual.

(1)O |=QC(a)iff there is no any QC model ofO ∪ {C(a)}.

(2) O |=Q C v D iff there is no any QC model ofO ∪ {CuD(t)} for some new individualtnot occurring inO.

4 Reasoning in Quasi-Classical Description Logic SHIQ

4.1 The QC Consistency Problem

In QCSHIQ, a conceptCisQC satisfiablew.r.t. a QC ABoxAif there exists a QC modelIofAsuch thatI |=s C(a)for some individuala, andQC unsatisfiablew.r.t.

Aotherwise. A concept C isQC satisfiable w.r.t. a QC role hierarchyRand a QC terminology T if there exists a QC modelI ofRandT such thatI |=_s C(a)for some individuala, andQC unsatisfiablew.r.t.RandT otherwise. A QC ABoxAis QC consistentif there exists a QC modelIofA, andQC inconsistentotherwise. A QC ABoxAisQC consistentw.r.t. a QC role hierarchyRand a QC terminologyT if there exists a QC modelIofRandT such thatIis a QC model ofA, andQC inconsistent w.r.t.RandT otherwise.

We are able to show that the complexity of QC consistency checking in QCSHIQ is in the same level as that of satisfiability check inSHIQ.

Proposition 13 Checking QC consistency of a QCSHIQABox w.r.t. a QC role hier- archyRand a QC terminologyT is EXPTIME-Complete.

4.2 The Inference Problems

In QCSHIQ, there are two basic inference problems given as follows.

– instance checking:an individualais called aQC instanceof a conceptCw.r.t. a QC ABoxAiff for any QC modelIofA,Iis a QC model ofC(a).

– subsumption:a conceptCQC subsumesa conceptDw.r.t. a QC role hierarchyR and a QC terminologyT iff for any QC modelIofRandT,I is a QC model of CvD.

Proposition 14 Given a QCSHIQontologyO, two conceptsC, Dand an individual a, we have

(1)O |=QC(a)iffO ∪ {C(a)}is QC inconsistent.

(2)O |=QCvDiffO ∪ {CuD(t)}is QC inconsistent for some new individualtnot occurring inO.

Proposition 14 shows that two basic inference problems can be reduced to the prob- lem of QC consistency checking.

5 Conclusions

In this paper, we presented QCSHIQto handle inconsistency inSHIQ. The syntax of SHIQ was extended by introducing the notion of the complement of a concept.

QC semantics was defined by two kinds of semantics: weak semantics and strong se- mantics. We showed that both of them satisfy some important inference rules. A QC entailment relation based on strong interpretations and weak interpretations was intro- duced and we showed that this entailment relation satisfies some desirable properties.

We introduced the notion of QC consistency of a QC ontology and two basic QC en- tailment problems and showed that these two entailment problems can be reduced to the problem of QC consistency checking. Furthermore, we showed that QC consistency problem of ABoxes is EXPTIME-Complete. As a future work, following [8], we will develop an algorithm based on the classical tableau for querying. In the other direc- tion, we will consider employing a classical DL reasoner, such as KAON2 or Pellet, to implement paraconsistent reasoning based on QC semantics. Further, we will also consider introducing QC semantics into more expressive DLs such asSHOIN(D).

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