Description Logics of Context with Rigid Roles Revisited

Technische Universität Dresden

Institute for Theoretical Computer Science Chair for Automata Theory

LTCS-Report

Description Logics of Context with Rigid Roles Revisited

Stephan Böhme Marcel Lippmann

LTCS-Report 15-04

Postal Address: http://lat.inf.tu-dresden.de

Description Logics of Context with Rigid Roles Revisited

Stephan Böhme and Marcel Lippmann Theoretical Computer Science

TU Dresden

{stephan.boehme,marcel.lippmann}@tu-dresden.de

08.05.2015

To represent and reason about contextualized knowledge often two-dimensional De- scription Logics (DLs) are employed, where one DL is used to describe contexts (or possible worlds) and the other DL is used to describe the objects, i.e. the relational structure of the specific contexts. Previous approaches for DLs of context that com- bined pairs of DLs resulted in undecidability in those cases where so-called rigid roles are admitted, i.e. if parts of the relational structure are the same in all contexts. In this paper, we present a novel combination of pairs of DLs and show that reasoning stays decidable even in the presence of rigid roles. We give complexity results for various combinations of DLs involvingALC,SHOQ, andEL.

1 Introduction

Description logics (DLs) of context can be employed to represent and reason about contextualized knowledge [BS03; BVS+09; KG10; KG11b; KG11a]. Such contextualized knowledge naturally occurs in practice. Consider, for instance, the rôles played by a person in different contexts. The person Bob, who works for the company Siemens, plays the rôle of an employee of Siemens while at work, i.e. in the work context, whereas he might play the rôle of a customer of Siemens in the context of private life. In this example, access restrictions to the data of Siemens might critically depend on the rôle played by Bob. Moreover, DLs capable of representing contexts are vital to integrate distributed knowledge as argued in [BS03; BVS+09].

In DLs, we useconcept names(unary predicates) andcomplex concepts(using certain constructors) to describe subsets of an interpretation domain androles(binary predicates) that are interpreted as binary relations over the interpretation domain. Thus, DLs are well-suited to describe contexts as formal objects with formal properties that are organized in relational structures, which are funda- mental requirements for modeling contexts [McC87; McC93].

However, classical DLs lack expressive power to formalize furthermore that some individuals satisfy certain concepts and relate to other individuals depending on a specific context. Therefore, often two-dimensional DLs are employed [KG10; KG11b; KG11a]. The approach is to have one DL L_M as themetaorouter logic to represent the contexts and their relationships to each other. This logic is combined with theobject or inner logic LO that captures the relational structure within each of the contexts. Moreover, while some pieces of information depend on the context, other pieces of information are shared throughout all contexts. For instance, the name of a person typically stays the same independent of the actual context. To be able to express that, some concepts and roles a designated to berigid, i.e. they are required to be interpreted the same in all contexts. Unfortunately,

if rigid roles are admitted, reasoning in the above mentioned two-dimensional DLs of context turns out to be undecidable; see [KG10].

We propose and investigate a family of two-dimensional context DLsL_MJL_OKthat meet the above requirements, but are a restricted form of the ones defined in [KG10] in the sense that we limit the interaction ofLM andLO. More precisely, in our family of context DLs the outer logic can refer to the internal structure of each context, but not vice versa. That means that information is viewed in a top-down manner, i.e. information of different contexts is strictly capsuled and can only be accessed from the meta level. This means that we cannot express, for instance, that everybody who is employed by Siemens has a certain property in the context of private life. Interestingly, reasoning in LMJLOK stays decidable with such a restriction, even in the presence of rigid roles. In some sense this restriction is similar to what was done in [BGL08; BGL12; Lip14] to obtain a decidable temporalized DL with rigid roles. Even though, our techniques to show complexity results are very similar to the ones employed for those temporalized DLs, we cannot simply reuse these results to reason in our context DLs, and more effort is needed to obtain tight complexity bounds.

For providing better intuition on how our formalism works, we examine the above mentioned example a bit further. Consider the following axioms:

> v J∃worksFor.{Siemens} v ∃hasAccessRights.{Siemens}K (1)

Work v JworksFor(Bob,Siemens)K (2)

J(∃worksFor.>)(Bob)K v ∃related.(PrivateuJHasMoney(Bob)K) (3)

> v J∃isCustomerOf.> vHasMoneyK (4) Private v JisCustomerOf(Bob,Siemens)K (5)

PrivateuWork v ⊥ (6)

¬Work v J∃worksFor.> v ⊥K (7)

The first axiom states that it holds true in all contexts that somebody who works for Siemens also has access rights to certain data. The second axiom states that Bob is an employee of Siemens in any work context. Furthermore, Axioms 3 and 4 say intuitively that if Bob has a job, he will earn money, which he can spend as a customer. Axiom 5 formalises that Bob is a customer of Siemens in any private context. Moreover, Axiom 6 ensures that the private contexts are disjoint from the work contexts. Finally, Axiom 7 states that theworksFor relation only exists in work contexts.

A fundamental reasoning task is to decide whether the above mentioned axioms are consistent altogether, i.e. whether there is a common model. In our example, this is the case; Figure 1 depicts a model. In this model, we also have Bob’s social security number linked to him using a rigid role hasSSN. We require this role to be rigid since Bob’s social security number does not change over the contexts. Furthermore the axioms entail more knowledge such as for example that in any private context nobody will have access rights to work data of Siemens, i.e.

Private v J∃hasAccessRights.{Siemens} v ⊥K

The remainder of the technical report is structured as follows. In the next section, we introduce the syntax and semantics of our family of context DLs LMJLOK. For this, we repeat some basic notions of DLs. In Section 3, we show decidability of the consistency problem inL_MJL_OKfor L_M andL_Obeing DLs betweenALCandSHOQ. There we consider the cases without rigid names, with rigid concepts and roles, and with rigid concepts only, and analyze the computational complexity of the consistency problem in these cases. Thereafter, in Section 4 we investigate the complexity of deciding consistency in DLs of contextLMJLOKwhereLM orLO is the sub-Boolean DLEL. Again

PersonBob,

hasSSN SSN

Siemens, Company worksFor

hasAccessRights

Person hasCEO

. .. Work

Person,Bob, HasMoney

hasSSN SSN

Siemens, Company

isCustomerOf

Person Private

related

Figure 1: Model of Axioms 1–7

we consider the cases with rigid names, with rigid concepts and roles, and with rigid concepts only.

Section 5 concludes the report and lists some possible future work.

2 Basic Notions

As argued in the introduction, our family of two-dimensional context DLs LMJLOK consists of combinations of two DLs: LM andLO. We focus on the case whereLM andLO are the lightweight DLELor DLs betweenALC andSHOQ. First, we recall the basic definitions of those DLs; for a thorough introduction to DLs, we refer the reader to [BCM+07].

Definition 1[Syntax of DLs]

LetNC,NR, andNIbe non-empty, pairwise disjoint sets ofconcept names,role names, andindividual names, respectively. Furthermore, letN:= (NC,NR,NI). The set of concepts overN is inductively defined starting from concept names A ∈NC using the constructors in the upper part of Table 1, wherer, s∈NR, a, b∈NI, n∈Ž, andC, D are concepts overN. The lower part of Table 1 shows howaxioms overNare defined.

Moreover, anRBoxRoverNis a finite set of role inclusions overNand transitivity axioms overN. ABoolean axiom formula overNis defined inductively as follows:

• every GCI overNis a Boolean axiom formula overN,

• every concept and role assertion overNis a Boolean axiom formula overN,

• ifB1,B2 are Boolean axiom formulas overN, then so are ¬B1 andB1∧ B2, and

• nothing else is a Boolean axiom formula overN.

Finally, aBoolean knowledge base (BKB) overNis a pairB= (B,R), whereB is a Boolean axiom

formula overNandRis an RBox overN.

Note that in this definition we refer to the tripleN explicitly although it is usually left implicit in standard definitions. This turns out to be useful as we need to distinguish between the symbols used inLM andLO. Sometimes we omitN, however, if it is clear from the context. As usual, we use the following abbreviations:

• CtD (disjunction) for¬(¬Cu ¬D),

• >(top concept) forAt ¬A, whereA∈NCis arbitrary but fixed,

• ⊥(bottom concept) for¬>,

• ∀r.C (value restriction) for¬∃r.¬C,

Table 1: Syntax and Semantics of DLs syntax semantics

negation ¬C ∆^I\C^I

conjunction CuD C^I∩D^I

existential restriction ∃r.C {d∈∆^I|there is ane∈C^I with (d, e)∈r^I}

nominal {a} {a^I}

at-most restriction 6ⁿr.C {d∈∆^I|]{e∈C^I |(d, e)∈r^I} ≤n}

general concept inclusion (GCI) CvD C^I ⊆D^I

concept assertion C(a) a^I ∈C^I

role assertion r(a, b) (a^I, b^I)∈r^I

role inclusion rvs r^I⊆s^I

transitivity axiom trans(r) r^I= (r^I)⁺

• >n s.C(at-least restriction) for ¬(6n−1s.C), and

• B1∨ B2(disjunction) for ¬(¬B1∧ ¬B2).

Which concept constructors and types of axioms are available depends on the specific DL used. In the smallest propositionally closed DL ALC, the only allowed concept constructors are negation, conjunction, and existential restriction. Thus disjunction, top and bottom concept, and value re- striction can be used as abbreviations. Moreover, no role inclusions and transitivity axioms are allowed inALC. If additional concept constructors or types of axioms are allowed, this is denoted by concatenating a corresponding letter: Omeans nominals,Qmeans at-most restrictions (qualified number restrictions), andHmeans role inclusions (role hierarchies). For instance, the DLALCHO is the extension of ALC that also allows for nominals and role inclusions. The extension of ALC with transitivity axioms is denoted byS. Hence, the DL allowing for all the concept constructors and types of axioms introduced here is calledSHOQ. The sub-Boolean DL ELis the fragment of ALCwhere only conjunction, existential restriction, and the top concept (which cannot be expressed as an abbreviation anymore due to the lack of negation) are admitted as concept constructors. We sometimes writeL-concept over N(L-BKB over N, . . . ) for some DL Lto make clear which DL is used.

The semantics of DLs are defined in a model-theoretic way through the notion ofinterpretations.

Definition 2[Semantics of DLs]

Let N = (NC,NR,NI). An N-interpretation is a pair I = (∆^I,·^I), where ∆^I is a non-empty set (calleddomain), and ·^I is a mapping assigning a setA^I ⊆∆^I to every A∈N_C, a binary relation r^I ⊆∆^I×∆^I to every r∈N_R, and a domain elementa^I ∈∆^I to every a∈N_I. The function ·^I is extended to concepts overNinductively as shown in the upper part of Table 1, where]S denotes the cardinality of the setS.

Moreover,Iis a model of the axiomαoverNif the condition in the lower part of Table 1 is satisfied, where ·⁺ denotes the transitive closure of a binary relation. This is extended to Boolean axiom formulas overNinductively as follows:

• I is a model of¬B1if it is not a model ofB1, and

• I is a model ofB1∧ B2if it is a model of bothB1 andB2.

We writeI |=BifI is a model of the Boolean axiom formulaB overN. Futhermore,I is a model of an RBoxRoverN(writtenI |=R) if it is a model of each axiom inR.

Finally, I is a model of the BKB B = (B,R) over N(written I |=B) if it is a model of both B

andR. We callB consistent if it has a model.

We call a role name r ∈ N_R transitive (w.r.t. R) if every model of R is a model of trans(r).

Moreover, r is a subrole of a role name s ∈ NR (w.r.t. R) if every model of R is a model of r v s. Finally, r is simple w.r.t. R if it has no transitive subrole. It is not hard to see that r ∈ NR is simple w.r.t. Riff trans(r) ∈ R/ and there do not exist roles s1, . . . , sk ∈ NR such that {trans(s1), s1vs2, . . . , s_k−1 vsk, sk vr} ⊆ R. Thus deciding whether r∈NR is simple can be decided in time polynomial in the size ofRby simple syntactic checks.

It follows from a result in [HST00] that the problem of checking whether a given SHQ-BKB B = (B,R) over N is consistent is undecidable in general. One regains decidability with a syn- tactic restriction as follows: if6ⁿ r.C occurs inB, rmust be simple w.r.t.R. In the following, we make this restriction to the syntax ofSHQand all its extensions.

This restriction is also the reason why there are no Boolean combinations of role inclusions and transitivity axioms allowed in the RBoxRoverNin the above definition. Otherwise the notion of a simple role w.r.t.Rinvolves reasoning. Consider, for instance, the Boolean combination of axioms (trans(r)∨trans(s))∧r v s. It should be clear that s is not simple, but this is no longer a pure syntactic check.

We are now ready to define the syntax of LMJLOK. Throughout the paper, let OC, OR, and OI

be respectively sets of concept names, role names, and individual names for the object logic LO. Analogously, we define the sets MC, MR, and MI for the meta logic LM. Without loss of gen- erality, we assume that all those sets are pairwise disjoint. Moreover, let O := (OC,OR,OI) and M:= (MC,MR,MI).

Definition 3[Syntax ofLMJLOK]

Aconcept of the object logic LO (o-concepts) is an LO-concept over O. An o-axiom is an LO-GCI overO, anLO-concept assertion overO, or an LO-role assertion overO.

The set ofconcepts of the meta logicLM (m-concepts)is the smallest set such that

• everyLM-concept overMis an m-concept and

• JαKis an m-concept ifαis an o-axiom.

The notion of anm-axiomis defined analogously.

ABoolean m-axiom formula is defined inductively as follows:

• every m-axiom is a Boolean m-axiom formula,

• ifB1,B2 are Boolean m-axiom formulas, then so are¬B1 andB1∧ B2, and

• nothing else is a Boolean m-axiom formula.

Finally, aBoolean L_MJL_OK-knowledge base (L_MJL_OK-BKB)is a tripleB= (B,R_O,R_M) where R_O is an RBox overO,R_Man RBox over M, andBis a Boolean m-axiom formula.

For the reasons above, role inclusions over O and transitivity axioms over O are not allowed to constitute m-concepts. However, we fix an RBoxRO overOthat contains such o-axioms and holds in all contexts. The same applies to role inclusions over M and transitivity axioms overM, which are only allowed to occur in a RBoxRM overM.

Again, we use the usual abbreviations (for disjunctions etc.) for m-concepts and Boolean m-axiom formulas.

The semantics of LMJLOK is defined by the notion of nested interpretations. These consist of O- interpretations for the specific contexts and anM-interpretation for the relational structure between

them. We assume that all contexts speak about the same non-empty domain (constant domain assumption).

As argued in the introduction, sometimes it is desired that concepts or roles in the object logic are interpreted the same in all contexts. Let OCrig ⊆ OC denote the set of rigid concepts, and let ORrig ⊆OR denote the set of rigid roles. We call concept names and role names inOC\OCrig and OR\ORrig flexible. Moreover, we assume that individuals of the object logic are always interpreted the same in all contexts (rigid individual assumption).

Definition 4[Nested interpretation]

A nested interpretation is a tuple J = (C,·^J,∆,(·^Ic)c∈C), where C is a non-empty set (called contexts) and (C,·^J) is anM-interpretation.

Moreover, for every c∈C,Ic := (∆,·^Ic) is anO-interpretation such that we have for allc, c⁰ ∈C

thatx^Ic =x^Ic0 for every x∈OI∪OCrig∪ORrig.

We are now ready to define the semantics ofLMJLOK.

Definition 5[Semantics ofL_MJL_OK]

LetJ = (C,·^J,∆,(·^Ic)c∈C) be a nested interpretation. The mapping·^J is extended to o-axioms as follows: JαK

J :={c∈C| I_c |=α}.

Moreover,J is a model of the m-axiomβ if (C,·^J) is a model of β. This is extended to Boolean m-axiom formulas inductively as follows:

• J is a model of¬B1 if it is not a model ofB1, and

• J is a model ofB1∧ B2 if it is a model of bothB1andB2.

We writeJ |=B if J is a model of the Boolean m-axiom formula B. Furthermore, J is a model ofR_M(writtenJ |=R_M) if (C,·^J) is a model ofR_M, andJ is a model ofR_O (writtenJ |=R_O) if Ic is a model ofROfor allc∈C.

Finally,J is a model of theLMJLOK-BKBB= (B,RO,RM) (writtenJ |=B) ifJ is a model ofB, RO, and RM. We callBconsistent if it has a model.

Theconsistency problem in LMJLOK is the problem of deciding whether a givenLMJLOK-BKB is

consistent.

Note that besides the consistency problem there are several other reasoning tasks forLMJLOK. The entailment problem, for instance, is the problem of deciding, given a BKB B and an m-axiom β, whether Bentails β, i.e. whether all models ofB are also models of β. The consistency problem, however, is fundamental in the sense that most other standard decision problems (reasoning tasks) can be polynomially reduced to it (in the presence of negation). For the entailment problem, note that it can be reduced to theinconsistency problem: B= (B,RO,RM) entailsβiff (B ∧ ¬β,RO,RM) is inconsistent. Hence, we focus in the present paper only on the consistency problem.

3 Complexity of the Consistency Problem

Our results for the computational complexity of the consistency problem in LMJLOK are listed in Table 2. In this section, we focus on the cases whereLM andLOare DLs betweenALCandSHOQ. In Section 4, we treat the cases whereLM orLO areEL.

Since the lower bounds of context DLs treated in this section already hold for the fragmentELJALCK, they are shown in Section 4.

Table 2: Complexity results for consistency in LMJLOK(where “c.” is short for “complete”)

LM

LO no rigid names only rigid concepts rigid roles

EL ALC SHOQ EL ALC SHOQ EL ALC SHOQ

EL const. Exp-c. Exp-c. const. NExp-c. NExp-c. const. 2Exp-c. 2Exp-c.

ALC Exp-c. Exp-c. Exp-c. NExp-c. NExp-c. NExp-c. NExp-c. 2Exp-c. 2Exp-c.

SHOQ Exp-c. Exp-c. Exp-c. NExp-c. NExp-c. NExp-c. NExp-c. 2Exp-c. 2Exp-c.

For the upper bounds, let in the followingB= (B,RO,RM) be aSHOQJSHOQK-BKB. We proceed similar to what was done forALC-LTL in [BGL08; BGL12] (andSHOQ-LTL in [Lip14]) and reduce the consistency problem to two separate decision problems.

For the first problem, we consider the so-calledouter abstraction, which is theSHOQ-BKB overM obtained by replacing each m-concept of the formJαKoccurring inBby a fresh concept name such that there is a 1–1 relationship between them.

Definition 6[Outer abstraction]

LetB = (B,RO,RM) be aLMJLOK-BKB. Let b be the bijection mapping every m-concept of the formJαK occurring inBto the concept name A

JαK∈MC, where we assume w.l.o.g. thatA

JαK does not occur inB.

1. The BooleanLM-axiom formula B^boverMis obtained fromB by replacing every occurrence of an m-concept of the form JαK by b(JαK). We call theLM-BKBB^b = (B^b,RM) the outer abstraction ofB.

2. Given J = (C,·^J,∆,(·^Ic)c∈C), its outer abstraction is the M-interpretation J^b = (C,·^Jb) where

• for everyx∈MR∪MI∪(MC\Im(b)), we have x^Jb =x^J, and

• for everyA∈Im(b), we haveA^Jb = (b⁻¹(A))^J,

whereIm(b) denotes the image ofb.

For simplicity, forB⁰ = (B⁰,RO,RM) where B⁰ is a subformula ofB, we denote by (B⁰)^b the outer abstraction ofB⁰ that is obtained by restrictingbto the m-concepts occurring inB⁰.

Example 7.Let B_ex = (Bex,∅,∅) with Bex := C v (JA v ⊥K) ∧ (C uJA(a)K)(c) be a SHOQJSHOQK-BKB. Then,bmapsJAv ⊥KtoA

JAv⊥KandJA(a)KtoA

JA(a)K. Thus, we have that B^b_ex:=

Cv(A

JAv⊥K) ∧ (CuA

JA(a)K)(c), ∅

is the outer abstraction ofBex. M

The following lemma makes the relationship betweenBand its outer abstractionB^b explicit. It is proved by induction on the structure ofB.

Lemma 8.Let J be a nested interpretation such that J is a model of RO. Then,J is a model of BiffJ^b is a model ofB^b.

Proof. Since r^J =r^Jb for all r∈M_R, we have that J is a model of RM iff J^b is a model ofRM. Thus, it is only left to show that for any m-axiomγoccurring inB, it holds thatJ |=γiffJ^b|=γ^b. Claim: Let C^b be the m-concept obtained from the m-conceptC by replacing every occurrence of JαKbyb(JαK). Then, for anyx∈Cit holds that x∈C^J iffx∈(C^b)^Jb.

Proof: We prove the claim by induction on the structure ofC:

C=A∈MC\Im(b): x∈A^J iffx∈(A^b)^Jb by definition ofJ^band sinceA=A^b C=JαK: x∈JαK

J iffx∈A^Jb

JαK iffx∈(JαK

b)^Jb

C=¬D: x∈(¬D)^J iffx /∈D^J iff, by induction hypothesis,x /∈(D^b)^Jb iffx∈(¬D^b)^Jb iffx∈((¬D)^b)^Jb

C=DuE: x∈(DuE)^J iffx∈D^J andx∈E^J iff, by induction hypothesis,x∈(D^b)^Jb andx∈(E^b)^Jb iffx∈(D^buE^b)^Jb iffx∈((DuE)^b)^Jb

C=∃r.D: x∈(∃r.D)^J iff there existsy∈C s. t. (x, y)∈r^J andy∈D^J iff there exists y∈C s. t. (x, y)∈r^Jb andy∈(D^b)^Jb iffx∈(∃r.D^b)^Jb iffx∈((∃r.D)^b)^Jb C={a}: x∈ {a}^J iffx∈({a}^b)^Jb by definition ofJ^band since{a}={a}^b

C=6nr.D: x ∈ (6n r.D)^J iff there are at most n elements y ∈ C s.t. (x, y) ∈ r^J and y∈D^J iff there are at mostnelementsy∈Cs.t. (x, y)∈r^Jb andy∈(D^b)^Jb iffx∈(6n r.D^b)^Jb iffx∈((6nr.D)^b)^Jb If γ of the form C v D, we have that J |= C v D iff x ∈ C^J implies x ∈ D^J iff (by claim) x∈(C^b)^Jb impliesx∈(D^b)^Jb iffJ^b|=C^bvD^b.

If γ is of the form C(a), we have that J |= C(a) iff a^J ∈ C^J iff (by claim) a^Jb ∈ (C^b)^Jb iff J^b|=C^b(a).

If γ is of the form r(a, b), we have that J |= r(a, b) iff (a^J, b^J) ∈ r^J iff (a^Jb, b^Jb) ∈ r^Jb iff J^b|=r(a, b).

IfBis of the form¬B1, we have thatJ |=Biff not J |=B1 iff notJ^b|=B^b1iffJ^b|=B^b.

IfBis of the formB1∧ B2, we have thatJ |=BiffJ |=B1 andB2 iffJ^b|=B^b1 andJ^b|=B^b2 iff J^b|=B^b.

SinceJ |=RO,J |=RM iffJ^b|=RMand J |=BiffJ^b|=B^b, we haveJ |=BiffJ^b|=B^b. Note that this lemma yields that consistency ofBimplies consistency ofB^b. However, the converse does not hold as the following example shows.

Example 9.Consider againBexof Example 7. Take anyM-interpretationH= (Γ,·^H) with Γ ={e}, d^H=e, and C^H=A^H

JAv⊥K=A^H

JA(a)K={e}.

Clearly, His a model of B^b_ex. But there is no nested interpretation J = (C,·^J,∆,(·^Ic)c∈C) with J |=B_ex since this would implyC= Γ, and thatI_e is a model of both Av ⊥ andA(a), which is

not possible. M

Therefore, we need to ensure that the concept names inIm(b) are not treated independently. For expressing such a restriction on the modelI ofB^b, we adapt a notion of [BGL08; BGL12]. Here it is worth noting that this problem occurs also in much less expressive DLs asALC or EL^⊥ (i.e.EL extended with the bottom concept).

Definition 10 [N-interpretation (weakly) respects(U,Y)]

LetU ⊆NC and letY ⊆ P(U). TheN-interpretationI= (∆^I,·^I)respects (U,Y) ifZ=Y where Z:={Y ⊆ U |there is somed∈∆^I withd∈(CU,Y)^I}

and

C_U,Y :=_A∈Y

u

u

A∈U \Y ¬A.

Itweakly respects (U,Y) ifZ ⊆ Y.

The second decision problem that we use for deciding consistency is needed to make sure that such a set of concept names is admissible in the following sense.

Definition 11 [Admissibility]

Let X = {X1, . . . , Xk} ⊆ P(Im(b)). We call X admissible if there exist O-interpretations I1= (∆,·^I1), . . . ,Ik = (∆,·^Ik) such that

• x^Ii =x^Ij for allx∈OI∪OCrig∪ORrig and alli, j∈ {1, . . . , k}, and

• everyIi, 1≤i≤k, is a model of theLO-BKBBX_i = (BX_i,RO) overOwhere BX_i := ^

b(JαK)∈Xi

α ∧ ^

b(JαK)∈Im(b)\Xi

¬α.

Note that any subsetX⁰ ⊆ X is admissible ifX is admissible.

Intuitively, the setsXiin an admissible setX consist of concept names such that the corresponding o-axioms “fit together”. Consider again Example 9. Clearly, the set {A

JAv⊥K, A

JA(a)K} ∈ P(Im(b)) cannot be contained in any admissible setX.

The next definition captures the above mentioned restriction on the modelI ofB^b. Definition 12 [Outer consistency]

LetX ⊆ P(Im(b)). We call theLM-BKBB^boverMouter consistent w.r.t.X if there exists a model

ofB^bthat weakly respects (Im(b),X).

The next two lemmas show that the consistency problem in LMJLOK can be decided by checking whether there is an admissible setX and the outer abstraction of the givenLMJLOK-BKB is outer consistent w.r.t.X.

Lemma 13.For every M-interpretationH= (Γ,·^H), the following two statements are equivalent:

1. There exists a modelJ ofB withJ^b=H.

2. His a model ofB^band the set{Xd|d∈Γ}is admissible, whereXd:={A∈Im(b)|d∈A^H}. Proof. (1 ⇒ 2): Let J = (C,·^J,∆,(·^Ic)c∈C) be a model ofB with J^b = H. Since J^b =H, we have thatC= Γ. By Lemma 8, we have thatH is a model ofB^b. Moreover, since bis a bijection between m-concepts of the formJαKoccurring in Band concept names ofM_C, we have that Im(b) is finite, and thus also the setX :={X_d|d∈Γ} ⊆ P(Im(b)) is finite. LetX ={Y₁, . . . , Y_k}. Since C= Γ, there exists an index function ν:C→ {1, . . . , k}such thatX_c=Y_ν(c)for every c∈C, i.e.

Y_ν(c)=

b(JαK)|JαKoccurs inB andc∈JαK

H =

b(JαK)|JαKoccurs inBandI_c |=α . Conversely, for every µ ∈ {1, . . . , k}, there is an element c ∈ C such that ν(c) = µ. The O- interpretations for showing admissibility ofX are obtained as follows. Takec1, . . . , ck ∈Csuch that ν(c1) = 1, . . . ,ν(ck) =k. Now, for everyi, 1≤i≤k, we define theO-interpretationGi:= (∆,·^Ici).

Clearly, we have thatGi|=BY_i and sinceJ |=RO, we have thatGi|=BY_i. Moreover, the definition of a nested interpretation yields thatx^Gi=x^Gj for allx∈OI∪OCrig∪ORrig and alli, j∈ {1, . . . , k}. Hence, theO-interpretationsG1, . . . ,Gk attest admissibility ofX.

(2⇒1): Assume thatH= (Γ,·^H) is a model ofB^band that the setX :={Xd|d∈Γ}is admissible.

Again, since Im(b) is finite, we have that X ⊆ P(Im(b)) is finite. Let X = {Y₁, . . . , Y_k}. Since X is admissible, there areO-interpretations G₁ = (∆,·^G1), . . . , G_k = (∆,·^Gk) such thatG_i |=B_Yi and x^Gi =x^Gj for allx∈O_I∪O_Crig∪O_Rrig and all i, j∈ {1, . . . , k}. Furthermore, there exists an index functionν: Γ→ {1, . . . , k}such thatY_ν(d)=Xd for everyd∈Γ. We define a nested interpretation J = (C,·^J,∆,(·^Ic)c∈C) as follows:

• C:= Γ;

• x^J :=x^H for everyx∈M_C∪M_R∪M_I; and

• x^Ic:=x^Gν(c) for every x∈OC∪OR∪OIand everyc∈C.

By construction ofJ, we have thatx^Jb =x^H for everyx∈M_R∪M_I∪(M_C\Im(b)). LetA∈Im(b), and letb⁻¹(A) =JαK. We have for everyd∈Γ =C thatd∈A^Jb iffd∈(b⁻¹(A))^J iffd∈JαK

J iff Id|=αiffGν(d)|=αiffb(JαK) =A∈Yν(d) (since Gν(d)|=BY_ν(d)) iffA∈Xd iffd∈A^H. Hence, we haveJ^b=H. SinceHis a model ofB^band, by construction ofJ,J is a model ofRO, we have by Lemma 8 thatJ is a model ofB.

The following lemma is a consequence of the previous one.

Lemma 14.The LMJLOK-BKB B is consistent iff there is a set X = {X₁, . . . , Xk} ⊆ P(Im(b)) such that

1. X is admissible, and

2. B^b is outer consistent w.r.t.X.

Proof. (=⇒): Let J be a model of B, and let J^b = (C,·^Jb). By Lemma 13, we have that J^b is a model ofB^b, and the setX :={X_c |c∈C} is admissible. By construction,J^b weakly respects (Im(b),X), and henceB^b is outer consistent w.r.t.X.

(⇐=): Let X = {X1, . . . , Xk} ⊆ P(Im(b)) such that X is admissible and B^b is outer consistent w.r.t. X. Hence there is a model G = (C,·^G) of B^b that weakly respects (Im(b),X). We define X⁰ :={Yc | c ∈ C}, where Yc := {A ∈Im(b)| c ∈ A^G}. Since G weakly respects (Im(b),X) and c∈(C_Im(b),Yc)^G for everyc∈C, we have thatX⁰ ⊆ X. SinceXis admissible, this yields admissibility ofX⁰. Lemma 13 yields now consistency ofB.

To obtain a decision procedure for SHOQJSHOQK consistency, we have to non-deterministically guess or construct the setX, and then check the two conditions of Lemma 14. Beforehand, we focus on how to decide the second condition. For that, assume that a setX ⊆ P(Im(b)) is given.

Lemma 15.Deciding whether B^b is outer consistent w.r.t. X can be done in time exponential in the size ofB^b and linear in size ofX.

Proof. It is enough to show that deciding whether B^b has a model that weakly respects (Im(b),X) can be done in time exponential in the size ofB^b and linear in the size ofX. It is not hard to see that we can adapt the notion of a quasimodel respecting a pair (U,Y) of [Lip14] to a quasimodel weakly respecting (U,Y). Indeed, one just has to drop Condition (i) in Definition 3.25 of [Lip14].

Then, the proof of Lemma 3.26 there can be adapted such that our claim follows. This is done by dropping one check in Step 4 of the algorithm of [Lip14].

Using this lemma, we provide decision procedures forSHOQJSHOQKconsistency. However, these depend also on the first condition of Lemma 14. We take care of this differently depending on which names are allowed to be rigid.

3.1 Consistency in SHOQ J SHOQ K without rigid names

In this section, we consider the case where no rigid concept names or role names are allowed. So we fixOCrig=ORrig=∅.

Theorem 16.The consistency problem in SHOQJSHOQKis in ExpifOCrig =ORrig =∅.

Proof. LetB be aSHOQJSHOQK-BKB and B^b its outer abstraction. We can decide consistency of B using Lemma 14. We define X :={X ⊆ Im(b)| BX is consistent} where BX is defined as in Definition 11. We first show that X ={X1, . . . , Xk} is admissible. Let Ii be a model of BX_i, which exists sinceBX_i is consistent. Due to the Löwenheim-Skolem theorem, we can assume that all modelsIi, 1≤i ≤k, have a countably infinite domain. Thus, w.l.o.g. we can assume that all models have the same domain ∆. Furthermore, we can assume that individual names are interpreted the same. SinceOCrig=ORrig=∅, the setX fulfills all conditions of Definition 11 for admissibility.

Thus, ifB^bis outer consistent w.r.t.X, then we have by Lemma 14 thatBis consistent. Conversely, assume thatBis consistent. Then, by Lemma 14, there is an admissible setX⁰ ⊆ P(Im(b)) andB^b is outer consistent w.r.t.X⁰. SinceX is the maximal admissible subset ofP(Im(b)), we haveX⁰⊆ X. If B^b is outer consistent w.r.t. X⁰, it is also outer consistent w.r.t. X. Hence, B is consistent iff B^b is outer consistent w.r.t. X, which yields a decision procedure for the consistency problem in SHOQJSHOQK.

It remains to analyze the complexity. There are exponentially manyX∈ P(Im(b)), but eachSHOQ- BKBBX can be constructed in time polynomial in the size ofB. We can decide consistency ofBX

in time exponential [Lip14]. Thus, the setX can be constructed in time exponential in the size ofB and it is of exponential size. Due to Lemma 15, deciding whetherB^b is outer consistent w.r.t.X can be done in time exponential in the size ofB^b and linear in the size ofX. Thus, overall we can decide the consistency problem in exponential time.

Together with the lower bounds shown in Section 4, we obtainExp-completeness for the consistency problem inL_MJL_OKforL_M andL_O being DLs betweenALC andSHOQifO_Crig=O_Rrig=∅.

3.2 Consistency in SHOQ J SHOQ K with rigid concept and role names

In this section, we consider the case where rigid concept and role names are present. So we fix OCrig6=∅andORrig 6=∅.

Theorem 17.The consistency problem inSHOQJSHOQK is in2Expif OCrig6=∅andORrig6=∅.

Proof. LetB= (B,RO,RM) be aSHOQJSHOQK-BKB and B^b= (B^b,RM) its outer abstraction.

We can decide consistency of B using Lemma 14. For that, we enumerate all setsX ⊆ P(Im(b)), which can be done in time doubly exponential in B. For each of these sets X = {X₁, . . . , X_k}, we check whether B^b is outer consistent w.r.t.X, which can be done in time exponential in the size of B^b and linear in the size of X. Then, we check X for admissibility using the renaming technique of [BGL08; BGL12]. For every i, 1 ≤ i ≤ k, every flexible concept name A occurring inB^b, and every flexible role nameroccurring inB^b orRO, we introduce copiesA⁽ⁱ⁾ andr⁽ⁱ⁾. The SHOQ-BKBB⁽ⁱ⁾_X

i = (B_X⁽ⁱ⁾

i,RO(i)) overOis obtained fromBX_i (see Definition 11) by replacing every occurrence of a flexible namexbyx⁽ⁱ⁾. We define

B_X :=^

1≤i≤kB_X⁽ⁱ⁾

i,[

1≤i≤kR_O⁽ⁱ⁾ .

It is not hard to verify (using arguments of [Lip14]) thatX is admissible iff B_X is consistent. Note thatB_X is of size at most exponential inB and can be constructed in exponential time. Moreover, consistency of B_X can be decided in time exponential in the size of B_X [Lip14], and thus in time doubly exponential in the size ofB.

Together with the lower bounds shown in Section 4, we obtain2Exp-completeness for the consistency problem inLMJLOKforLM andLObeing DLs betweenALCandSHOQifOCrig6=∅andORrig 6=∅.

3.3 Consistency in SHOQ J SHOQ K with only rigid concept names

In this section, we consider the case where rigid concept are present, but rigid role names are not allowed. So we fixO_Crig6=∅but O_Rrig=∅.

Theorem 18.The consistency problem inSHOQJSHOQKis in NExpifO_Crig6=∅andO_Rrig=∅.

Proof. LetB= (B,RO,RM) be aSHOQJSHOQK-BKB and B^b= (B^b,RM) its outer abstraction.

We can decide consistency of B using Lemma 14. We first non-deterministically guess the set X ={X1, . . . , Xk} ⊆ P(Im(b)), which is of size at most exponential inB. Due to Lemma 15 we can check whetherB^bis outer consistent w.r.t.X in time exponential in the size ofB^band linear in the size ofX. It remains to checkXfor admissibility. For that letO_Crig(B)⊆O_CrigandO_I(B)⊆O_Ibe the sets of all rigid concept names and individual names occurring inB, respectively. As done in [BGL08;

BGL12] we non-deterministically guess a setY ⊆ P(O_Crig(B)) and a mappingκ:O_I(B)→ Y which also can be done in time exponential in the size of B. Using the same arguments as in [BGL08;

BGL12] we can show thatX is admissible iff

BbX_i :=



BX_i∧ ^

a∈OI(B)

u

A∈κ(a)Au

u

A∈OCrig(B)\κ(a)¬A

(a), RO





has a model that respects (O_Crig(B),Y), for all 1≤i≤k. TheSHOQ-BKBBb_Xiis of size polynomial in the size ofB and can be constructed in time exponential in the size ofB. We can check if BbX_i

has a model that respects (OCrig(B),Y) in time exponential in the size ofBbX_i [BGL08; BGL12], and thus exponential in the size ofB.

Together with the lower bounds shown in Section 4, we obtainNExp-completeness for the consistency problem inLMJLOKforLM andLObeing DLs betweenALCandSHOQifOCrig6=∅andORrig =∅.

Summing up the results, we obtain the following corollary.

Corollary 19. For allLM,LO betweenALC andSHOQ, the consistency problem in LMJLOKis

• Exp-complete if OCrig=∅andORrig =∅,

• NExp-complete if OCrig6=∅andORrig=∅, and

• 2Exp-complete ifO_Crig6=∅andO_Rrig6=∅.

4 The Case of EL: L

J EL K and EL J L

K

In this section, we give some complexity results for context DLsLMJLOKwhereLM or LO areEL. In Section 4.1, we considerLMJELKwhere LM is betweenALC andSHOQ. Then, in Section 4.2, we consider the remaining context DLsEL L whereL is eitherELor betweenALCandSHOQ.

4.1 The Context DLs L

J EL K

In this section, we consider LMJELK where LM is between ALC and SHOQ. The lower bounds already hold forALCJELK.

Theorem 20.The consistency problem in ALCJELK isExp-hard if OCrig=ORrig =∅.

Proof. Deciding whether a given conjunction of ALC-axioms B is consistent is Exp-hard [Sch91].

Obviously,Bis also anALCJELK-BKB.

For the cases of rigid names, the lower bounds of NExpare obtained by a careful reduction of the satisfiability problem in the temporalized DLEL-LTL [BT15b; BT15a], which is a fragment ofALC- LTL introduced in [BGL08; BGL12]. For the sake of completeness, we recall the basic definitions of L-LTL here, whereLis a DL.

Definition 21 [Syntax of L-LTL]

L-LTL-formulas over Oare defined by induction:

• ifαis anL-axiom overO, then αis anL-LTL-formula, and

• ifφ, ψ areL-LTL-formulas overO, then so are φ∧ψ,¬φ,φUψ,Xφ, and

• nothing else is anL-LTL-formula.

The usual abbreviations are used:

• φ∨ψfor¬(¬φ∧ ¬ψ),

• trueforA(a)∨ ¬A(a),

• 3φfortrue Uφ, and

• 2φfor¬3¬φ.

The semantics ofL-LTL is based on DL-LTL-structures. These are sequences of O-interpretations over the same non-empty domain that additionally respect rigid names and the rigid individual assumption.

Definition 22 [DL-LTL-structure]

ADL-LTL-structure overOis a sequenceI= (Ii)i≥0ofO-interpretations (∆,·^Ii) such thatx^Ii =x^Ij

holds for allx∈O_Crig∪O_Rrig∪O_I, i, j >0.

We are now ready to define the semantics ofL-LTL.

Definition 23 [Semantics ofL-LTL]

The validity of anL-LTL-formulaφ in a DL-LTL-structureI = (I_i)i≥0 at time i≥0, denoted by I, i|=ϕ, is defined inductively:

I, i|=α iff Ii|=αwhereαis anALC-axiom overO, I, i|=φ∧ψ iff I, i|=φand I, i|=ψ,

I, i|=¬φ iff notI, i|=φ, I, i|=Xφ iff I, i+ 1|=φ,

I, i|=φUψ iff there isk≥isuch that I, k|=ψ andI, j|=φfor allj withi≤j < k.

We call anL-LTL-structure Ia model of φif I,0 |=φ. The satisfiability problem in L-LTL is the

question whether a givenL-LTL-formulaφhas a model.

In [BT15b; BT15a], it is shown that the satisfiability problem inEL-LTL isNExp-hard as soon as rigid concept names are available. We reduce the satisfiability problem inEL-LTL to the consistency problem in ALCJELK to obtain the lower bounds of NExp, where we use the fact that the lower bounds of [BT15b; BT15a] hold already for a syntactically restricted fragment ofEL-LTL.

Theorem 24.The consistency problem in ALCJELK isNExp-hard if O_Crig6=∅andO_Rrig=∅.

Proof. In fact, the lower bounds hold for EL-LTL-formulas of the form2φwhere φis an EL-LTL- formula that contains onlyX as temporal operator [BT15a].

Let2φ be such anEL-LTL-formula overO. We obtain now an m-conceptCφ from φby replacing EL-axioms α by JαK, ∧ by u, and subformulas of the form Xψ by ∀t.ψu ∃t.ψ, where t ∈ MR is arbitrary but fixed.

Claim: 2φis satisfiable iffB=> vC_φu ∃t.>is consistent.

Proof: (=⇒): Take any DL-LTL-structure I = (∆,·^Ii)i≥0 with I,0 |=2φ. We define the nested interpretationJ = (C,·^J,∆,(·^Ic)c∈C) as follows:

C:={c_i|i≥0},

·^Ici :=·^Ii,

t^J :={(ci, ci+1)}.

We show now that for everyi≥0, we have I, i|=φiffci ∈C_φ^J by induction on the structure ofφ. Ifφis anEL-axiom over O, then we haveI, i|=φiffI_i|=φiffI_ci |=φiffc_i∈JφK

J. Ifφis of the form¬ψ, then we haveI, i|=φiffI, i6|=ψiffci∈/C_ψ^J iffci∈(¬Cψ)^J =C_φ^J. Ifφis of the formψ1∧ψ2, the claim follows by similar arguments.

Ifφis of the formXψ, we have thatI, i|=φiffI, i+ 1|=ψiffci+1∈C_ψ^J iffci ∈(∀t.Cψu ∃t.Cψ)^J iffc_i∈C_φ^J.

It follows thatI,0|=2φiffJ |=> vC_φ.

Furthermore, since (ci, ci+1)∈t^J, we haveci∈(∃t.>)^J. Thus,J |=> v ∃t.>.

(⇐=): Take any nested interpretationJ = (C,·^J,∆,(·^Ic)c∈C) that is a model of> vCφu∃t.>. Let Pbe an infinite pathP =c0c1. . . withci∈Cand (ci, ci+1)∈t^J for everyi≥0. Such a path exists, because J |=> v ∃t.>. We define the nested interpretation JP := ({ci |i ≥0},·^JP,∆,(·^Ici)i≥0) where·^JP is the restriction of·^J to the domain{ci|i≥0}.

By construction we have that JP |= > v ∃t.>. We show by a simple case distinction that JP |=> vCφ.

IfC_φdoes not contain any role namer∈M_R, the restriction on the set of worlds preserves the entail- ment relation. Otherwise,C_φ is of the form ∀t.Cψu ∃t.C_ψ. SinceJP |=> v ∃r.>,JP |=> vC_ψ, and there is only onet-successor, we haveJP |=> vC_φ.

Hence,J_P |=> vC_φu ∃t.>.

We define the DL-LTL-structureI overOasI:= (∆,·^Ii)i≥0 where·^Ii :=·^Ici.

Again we show that for everyi≥0, that we haveci∈C_φ^JP iffI, i|=φby induction on the structure ofφ.

Ifφis anEL-axiom over O, we havec_i∈JφK

JP iffIci|=φiffIi|=φiffI, i|=φ. Ifφis of the form¬ψ, then we havec_i∈C_φ^JP iffc_i∈/C_ψ^JP iffI, i6|=ψiffI, i|=φ. Ifφis of the formψ₁∧ψ₂, the claim follows by similar arguments.

Ifφis of the formXψ, we have thatci∈C_φ^JP iffci∈(∀t.Cψu∃t.Cψ)^JP iffci+1∈C_ψ^JP iffI, i+1|=ψ iffI, i|=φ.

It follows thatJP |=> vC_φ iffI,0|=2φ. This claim yields the lower bound ofNExpfor the consistency problem inALCJELKifO_Crig6=∅.

Next, we prove the upper bound of NExpfor the consistency problem in the case of rigid names.

Theorem 25.The consistency problem in SHOQJELKis in NExpif OCrig6=∅andORrig6=∅.

Proof. We again use Lemma 14. First, we non-deterministically guess a set X ⊆ P(Im(b)) and construct theEL-BKBB_X overOas in the proof of Theorem 17, which is actually a conjunction of EL-literals over O, i.e. of (negated)EL-axioms overO. The following claim shows that consistency ofB_X can be reduced to consistency of a conjunction of ELO^⊥-axioms over O, where ELO^⊥ is the extension ofELwith nominals and the bottom concept.

Claim: For every conjunction of EL-literals B overO, there exists an equisatisfiable conjunction B⁰ ofELO^⊥-axioms overO.

Proof: LetBbe a conjunction ofEL-literals overO, i.e.

B=α₁∧ · · · ∧α_n∧ ¬β1∧ · · · ∧ ¬βm

whereα_i, 1≤i≤n,β_j, 1≤j ≤m areEL-axioms over O. We defineB⁰ as follows:

B⁰=α₁∧ · · · ∧α_n∧γ₁∧ · · · ∧γ_m, where

γi:=







C(ai)∧D⁰(ai)∧DuD⁰v ⊥ ifβi=CvD, A⁰(a)∧AuA⁰v ⊥ ifβi=A(a), and {a} u ∃r.{b} v ⊥ ifβi=r(a, b)

withA⁰, D⁰ being fresh concept names andai being fresh individual names. It is easy to see that if anO-interpretationI is a model of ¬β1∧ · · · ∧ ¬βm, there exists an extension ofI that is a model ofγ1∧ · · · ∧γm. Conversely, if anO-interpretationI⁰ is a model ofγ1∧ · · · ∧γm, it is also a model of¬β1∧ · · · ∧ ¬βm. Hence BandB⁰ are equisatisfiable.

By this claim and the fact that consistency of conjunctions of ELO^⊥-axioms can be decided in polynomial time [BBL05], we obtain our claimed upper bound.

Summming up the results of this section, we obtain the following corollary.

Corollary 26. For allLM between ALC andSHOQ, the consistency problem inLMJELKis

• Exp-complete if OCrig=∅andORrig =∅, and

• NExp-complete otherwise.

Proof. The lower bounds follow from Theorems 20 and 24. The upper bound of Expin the case O_Crig=O_Rrig=∅follows immediately from Theorem 16, whereas the upper bound of NExpfollows from Theorem 25.