LTCS-Report

To represent and reason about contextualized knowledge often two-dimensional Description 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


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 use concept names (unary predicates) and complex concepts (using certain constructors) to describe subsets of an interpretation domain and roles (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 fundamental 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 the meta or outer logic to represent the contexts and their relationships to each other.This logic is combined with the object or inner logic L O 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 be rigid, 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 DLs L M L O that meet the above requirements, but are a restricted form of the ones defined in [KG10] in the sense that we limit the interaction of L M and L O .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 L M L O 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: 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 the worksFor 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 ∃ hasAccessRights.{Siemens} ⊥
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 L M L O .For this, we repeat some basic notions of DLs.In Section 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.

Basic Notions
As argued in the introduction, our family of two-dimensional context DLs L M L O consists of combinations of two DLs: L M and L O .We focus on the case where L M and L O are the lightweight DL EL or DLs between ALC and SHOQ.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]
Let N C , N R , and N I be non-empty, pairwise disjoint sets of concept names, role names, and individual names, respectively.Furthermore, let N := (N C , N R , N I ).The set of concepts over N is inductively defined starting from concept names A ∈ N C using the constructors in the upper part of Table 1, where r, s ∈ N R , a, b ∈ N I , n ∈ , and C, D are concepts over N. The lower part of Table 1 shows how axioms over N are defined.
Moreover, an RBox R over N is a finite set of role inclusions over N and transitivity axioms over N. A Boolean axiom formula over N is defined inductively as follows: • every GCI over N is a Boolean axiom formula over N, • every concept and role assertion over N is a Boolean axiom formula over N, • if B 1 , B 2 are Boolean axiom formulas over N, then so are ¬B 1 and B 1 ∧ B 2 , and • nothing else is a Boolean axiom formula over N.
Finally, a Boolean knowledge base (BKB) over N is a pair B = (B, R), where B is a Boolean axiom formula over N and R is an RBox over N.
Note that in this definition we refer to the triple N 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 in L M and L O .Sometimes we omit N, however, if it is clear from the context.As usual, we use the following abbreviations: 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 syntactic restriction as follows: if n r.C occurs in B, r must be simple w.r.t.R. In the following, we make this restriction to the syntax of SHQ and all its extensions.This restriction is also the reason why there are no Boolean combinations of role inclusions and transitivity axioms allowed in the RBox R over N in the above definition.Otherwise the notion of a simple role w.r.t.R involves reasoning.Consider, for instance, the Boolean combination of axioms (trans(r) ∨ trans(s)) ∧ r 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 L M L O .Throughout the paper, let O C , O R , and O I be respectively sets of concept names, role names, and individual names for the object logic L O .Analogously, we define the sets M C , M R , and M I for the meta logic L M .Without loss of generality, we assume that all those sets are pairwise disjoint.Moreover, let The set of concepts of the meta logic L M (m-concepts) is the smallest set such that The notion of an m-axiom is defined analogously.
A Boolean m-axiom formula is defined inductively as follows: • every m-axiom is a Boolean m-axiom formula, • if B 1 , B 2 are Boolean m-axiom formulas, then so are ¬B 1 and B 1 ∧ B 2 , and and B is 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 RBox R O over O that contains such o-axioms and holds in all contexts.The same applies to role inclusions over M and transitivity axioms over M, which are only allowed to occur in a RBox R M over M.
Again, we use the usual abbreviations (for disjunctions etc.) for m-concepts and Boolean m-axiom formulas.
The semantics of L M L O is defined by the notion of nested interpretations.These consist of Ointerpretations for the specific contexts and an M-interpretation for the relational structure between Description Logics of Context with Rigid Roles Revisited, 08.05.2015 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 O Crig ⊆ O C denote the set of rigid concepts, and let O Rrig ⊆ O R denote the set of rigid roles.We call concept names and role names in O C \ O Crig and O R \ O Rrig 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]
Moreover, for every c ∈ C, We are now ready to define the semantics of This is extended to Boolean m-axiom formulas inductively as follows: Note that besides the consistency problem there are several other reasoning tasks for L M L O .The entailment problem, for instance, is the problem of deciding, given a BKB B and an m-axiom β, whether B entails β, i.e. whether all models of B 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 the inconsistency problem: is inconsistent.Hence, we focus in the present paper only on the consistency problem.

Complexity of the Consistency Problem
Our results for the computational complexity of the consistency problem in L M L O are listed in Table 2.In this section, we focus on the cases where L M and L O are DLs between ALC and SHOQ.In Section 4, we treat the cases where L M or L O are EL.For the upper bounds, let in the following B = (B, R O , R M ) be a SHOQ SHOQ -BKB.We proceed similar to what was done for ALC-LTL in [BGL08; BGL12] (and SHOQ-LTL in [Lip14]) and reduce the consistency problem to two separate decision problems.
For the first problem, we consider the so-called outer abstraction, which is the SHOQ-BKB over M obtained by replacing each m-concept of the form α occurring in B by a fresh concept name such that there is a 1-1 relationship between them.

Given
where . Thus, we have that is the outer abstraction of B ex .
The following lemma makes the relationship between B and its outer abstraction B b explicit.It is proved by induction on the structure of B. Proof : We prove the claim by induction on the structure of C: x ∈ (∃ r.D) J iff there exists y ∈ C s. t. (x, y) ∈ r J and y ∈ D J iff there exists Note that this lemma yields that consistency of B implies consistency of B b .However, the converse does not hold as the following example shows.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.
Intuitively, the sets X i in an admissible set X consist of concept names such that the corresponding o-axioms "fit together".The next two lemmas show that the consistency problem in L M L O can be decided by checking whether there is an admissible set X and the outer abstraction of the given L M L O -BKB is outer consistent w.r.t.X .
Lemma 13.For every M-interpretation H = (Γ, • H ), the following two statements are equivalent: 1.There exists a model J of B with J b = H.

H is a model of
and all i, j ∈ {1, . . ., k}.Furthermore, there exists an index function ν : Γ → {1, . . ., k} such that Y ν(d) = X d for every d ∈ Γ.We define a nested interpretation J = (C, • J , ∆, (• Ic ) c∈C ) as follows: • C := Γ; • x J := x H for every x ∈ M C ∪ M R ∪ M I ; and By construction of J , we have that and let b −1 (A) = α .We have for every The following lemma is a consequence of the previous one.To obtain a decision procedure for SHOQ SHOQ consistency, we have to non-deterministically guess or construct the set X , and then check the two conditions of Lemma 14. Beforehand, we focus on how to decide the second condition.For that, assume that a set X ⊆ 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 of B b and linear in size of X .
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 of B b and linear in the size of X .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 for SHOQ SHOQ consistency.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.
It is not hard to verify (using arguments of [Lip14]) that X is admissible iff B X is consistent.Note that B X is of size at most exponential in B 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 of B.
Together with the lower bounds shown in Section 4, we obtain 2Exp-completeness for the consistency problem in L M L O for L M and L O being DLs between ALC and SHOQ if O Crig = ∅ and O Rrig = ∅.

Consistency in SHOQ SHOQ 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 fix We can decide consistency of B using Lemma 14.We first non-deterministically guess the set  Summing up the results, we obtain the following corollary.
Corollary 19.For all L M , L O between ALC and SHOQ, the consistency problem in

The Context DLs L M EL
In this section, we consider L M EL where L M is between ALC and SHOQ.The lower bounds already hold for ALC EL .

Theorem 20. The consistency problem in
Proof.Deciding whether a given conjunction of ALC-axioms B is consistent is Exp-hard [Sch91].Obviously, B is also an ALC EL -BKB.
For the cases of rigid names, the lower bounds of NExp are obtained by a careful reduction of the satisfiability problem in the temporalized DL EL-LTL [BT15b; BT15a], which is a fragment of ALC-LTL introduced in [BGL08; BGL12].For the sake of completeness, we recall the basic definitions of L-LTL here, where L is a DL.
Definition 21 [Syntax of L-LTL] L-LTL-formulas over O are defined by induction: • if α is an L-axiom over O, then α is an L-LTL-formula, and • if φ, ψ are L-LTL-formulas over O, then so are φ ∧ ψ, ¬φ, φ U ψ, Xφ, and The usual abbreviations are used: • true for A(a) ∨ ¬A(a), • φ for true U φ, and The semantics of L-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]
A DL-LTL-structure over O is a sequence We are now ready to define the semantics of L-LTL.

Definition 23 [Semantics of L-LTL]
The validity of an L-LTL-formula φ in a DL-LTL-structure I = (I i ) i≥0 at time i ≥ 0, denoted by I, i |= ϕ, is defined inductively: We call an L-LTL-structure I a model of φ if I, 0 |= φ.The satisfiability problem in L-LTL is the question whether a given L-LTL-formula φ has a model.In [BT15b;BT15a], it is shown that the satisfiability problem in EL-LTL is NExp-hard as soon as rigid concept names are available.We reduce the satisfiability problem in EL-LTL to the consistency problem in ALC EL 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 of EL-LTL.Proof.In fact, the lower bounds hold for EL-LTL-formulas of the form φ where φ is an EL-LTLformula that contains only X as temporal operator [BT15a].

Description Logics of
Let φ be such an EL-LTL-formula over O.We obtain now an m-concept C φ from φ by replacing EL-axioms α by α , ∧ by , and subformulas of the form Xψ by ∀t.ψ ∃ t.ψ, where t ∈ M R is arbitrary but fixed.
Proof : (=⇒): Take any DL-LTL-structure I = (∆, • Ii ) i≥0 with I, 0 |= φ.We define the nested interpretation J = (C, • J , ∆, (• Ic ) c∈C ) as follows: We show now that for every i ≥ 0, we have I, i |= φ iff c i ∈ C J φ by induction on the structure of φ.If φ is an EL-axiom over O, then we have If φ is of the form ¬ψ, then we have If φ is of the form ψ 1 ∧ ψ 2 , the claim follows by similar arguments.
If φ is of the form Xψ, we have that Again we show that for every i ≥ 0, that we have c i ∈ C J P φ iff I, i |= φ by induction on the structure of φ.
If φ is of the form ¬ψ, then we have Proof.We again use Lemma 14.First, we non-deterministically guess a set X ⊆ P(Im(b)) and construct the EL-BKB B X over O as in the proof of Theorem 17, which is actually a conjunction of EL-literals over O, i.e. of (negated) EL-axioms over O.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 of EL with nominals and the bottom concept.
Claim: For every conjunction of EL-literals B over O, there exists an equisatisfiable conjunction B of ELO ⊥ -axioms over O.
Proof : Let B be a conjunction of EL-literals over O, i.e.
where α i , 1 ≤ i ≤ n, β j , 1 ≤ j ≤ m are EL-axioms over O.We define B as follows: where 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 all L M between ALC and SHOQ, the consistency problem in L M EL is Next, we show the lower bounds for the consistency problem in EL ALC .We again distinguish the three cases of which names are allowed to be rigid.
Proof.Deciding whether a given conjunction For the case of rigid role names, we have lower bounds of 2Exp.

Theorem 28. The consistency problem in
Proof.To show the lower bound, we adapt the proof ideas of [BGL08; BGL12], and reduce the word problem for exponentially space-bounded alternating Turing machines (i.e. is a given word w accepted by the machine M ) to the consistency problem in EL ALC with rigid roles, i.e.O Rrig = ∅.In [BGL08; BGL12], a reduction was provided to show 2Exp-hardness for the temporalized DL ALC-LTL in the presence of rigid roles.Here, we mimic the properties of the time dimension that are important for the reduction using a role name t ∈ M R .
Our EL ALC -BKB is the conjunction of the EL ALC -BKBs introduced below.First, we ensure that we never have a "last" time point: Note that in the corresponding model, we do not enforce a t-chain since cycles are not prohibited.This, however, is not important in the reduction.
The ALC-LTL-formula obtained in the reduction of [BGL08; BGL12] is a conjunction of ALC-LTLformulas of the form φ, where φ is an ALC-LTL-formula.This makes sure that φ holds in all (temporal) worlds.For the cases where φ is an ALC-axiom, we can simply express this by: φ This captures all except for two conjuncts of the ALC-LTL-formula of the reduction of [BGL08; BGL12].There, a k-bit binary counter using concept names A 0 , . . ., A k−1 was attached to the individual name a, which is incremented along the temporal dimension.We can express something similar in EL ALC , but instead of incrementing the counter values along a sequence of t-successors, we have to go backwards since EL does allow for branching but does not allow for values restrictions, i.e. we cannot make sure that all t-successors behave the same.More precisely, if the counter value n is attached to a in context c, the value n + 1 (modulo 2 k − 1) must be attached to a in all of c's t-predecessors.
First, we ensure which bits must be flipped: Next, we ensure that all other bits stay the same: Note that due to the first m-axiom above, we enforce that every context has a t-successor.By the other m-axioms, we make sure that we enforce a t-chain of length 2 k .As in [BGL08; BGL12], it is not necessary to initialize the counter.Since we decrement the counter along the t-chain (modulo 2 k − 1), every value between 0 and 2 k − 1 is reached.
The conjunction of all the EL ALC -BKBs above yields an EL ALC -BKB B that is consistent iff w is accepted by M .
Finally, we obtain a lower bound of NExp in the case of rigid concept names only.Our EL ALC -BKB is the conjunction of the EL ALC -BKBs introduced below.We proceed in a similar way as in the proof of Theorem 28.First, we ensure that we never have a "last" time point: Note that in the corresponding model, we do not enforce a t-chain since cycles are not prohibited.As in the reduction in the proof of Theorem 28, this is not important in the reduction here.
Next, note that since the -operator distributes over conjunction, most of the conjuncts of the ALC-LTL-formula of the reduction of [BGL08; BGL12] can be rewritten as conjunctions of ALC-LTLformulas of the form α, where α is an ALC-axiom.As already argued in the proof of Theorem 28, this can equivalently be expressed by α .
This can be expressed in EL ALC as shown in the proof of Theorem 28: Note that due to the first m-axiom above, we enforce that every context has a t-successor.By the other m-axioms, we make sure that we enforce a t-chain of length 2 2n+2 .As in [BGL08; BGL12], it is not necessary to initialize the counter.Since we decrement the counter along the t-chain (modulo 2 2n+1 ), every value between 0 and 2 2n+1 is reached.
In [BGL08; BGL12], an ALC-LTL-formula is used to express that the value of the counter in shared by all domain elements belonging to the current (temporal) world.This is expressed using a disjunction, which we can simulate as follows: Next, there is a concept name N , which is required be non-empty in every (temporal) world.We express this using a role name r ∈ O R : It is only left to express the following ALC-LTL-formula of [BGL08; BGL12]: For readability, let D = {d 1 , . . ., d k }.We use non-convexity of ALC as follows to express this: The conjunction of all the EL ALC -BKBs above yields an EL ALC -BKB B that is consistent iff the exponentially bounded version of the domino problem has a solution.
Summing up the results of this section together with the upper bounds of Section 3, we obtain the following corollary.
and R M .We call B consistent if it has a model.The consistency problem in L M L O is the problem of deciding whether a given L M L O -BKB is consistent.
Since the lower bounds of context DLs treated in this section already hold for the fragment EL ALC , they are shown in Section 4. Description Logics of Context with Rigid Roles Revisited, 08.05.2015
BKB.Let b be the bijection mapping every m-concept of the form α occurring in B to the concept name A α ∈ M C , where we assume w.l.o.g. that A α does not occur in B. 1.The Boolean L M -axiom formula B b over M is obtained from B by replacing every occurrence of an m-concept of the form α by b( α ).We call the L M -BKB B b = (B b , R M ) the outer abstraction of B.
and • for every A ∈ Im(b), we have A J b = (b −1 (A)) J , where Im(b) denotes the image of b.For simplicity, for B = (B , R O , R M ) where B is a subformula of B, we denote by (B ) b the outer abstraction of B that is obtained by restricting b to the m-concepts occurring in B .Example 7. Let B ex = (B ex , ∅, ∅) with is only left to show that for any m-axiom γ occurring in B, it holds that J |= γ iff J b |= γ b .Claim: Let C b be the m-concept obtained from the m-concept C by replacing every occurrence of α by b( α ).Then, for any x ∈ C it holds that x ∈ C J iff x ∈ (C b ) J b .Description Logics of Context with Rigid Roles Revisited, 08.05.2015 definition of J b and since {a} = {a} b C = n r.D: x ∈ ( n 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 most n elements y ∈ C s.t.(x, y) ∈ r J b and y ∈

Example 9 .
Consider again B ex of Example 7. Take any M-interpretation H = (Γ, • H ) with Γ = {e}, d H = e, and C H = A H A ⊥ = A H A(a) = {e}.Clearly, H is 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 imply C = Γ, and that I e is a model of both A ⊥ and A(a), which is not possible.Therefore, we need to ensure that the concept names in Im(b) are not treated independently.For expressing such a restriction on the model I of B b , we adapt a notion of [BGL08; BGL12].Here it is worth noting that this problem occurs also in much less expressive DLs as ALC or EL ⊥ (i.e.EL extended with the bottom concept).Definition 10 [N-interpretation (weakly) respects (U , Y)] Let U ⊆ N C and let Y ⊆ P(U ).The N-interpretation I = (∆ I , • I ) respects (U , Y) if Z = Y where Z := {Y ⊆ U | there is some d ∈ ∆ I with d ∈ (C U ,Y ) I } Description Logics of Context with Rigid Roles Revisited, 08.05.2015 and C U ,Y := It weakly respects (U , Y) if Z ⊆ Y.
Consider again Example 9. Clearly, the set {A A ⊥ , A A(a) } ∈ P(Im(b)) cannot be contained in any admissible set X .The next definition captures the above mentioned restriction on the model I of B b .Definition 12 [Outer consistency] Let X ⊆ P(Im(b)).We call the L M -BKB B b over M outer consistent w.r.t.X if there exists a model of B b that weakly respects (Im(b), X ).
B b and the set {X d | d ∈ Γ} is admissible, where X d := {A ∈ Im(b) | d ∈ A H }. Proof.(1 ⇒ 2): Let J = (C, • J , ∆, (• Ic ) c∈C ) be a model of B with J b = H.Since J b = H, we have that C = Γ.By Lemma 8, we have that H is a model of B b .Moreover, since b is a bijection between m-concepts of the form α occurring in B and concept names of M C , we have that Im(b) is finite, and thus also the set X := {X d | d ∈ Γ} ⊆ P(Im(b)) is finite.Let X = {Y 1 , . . ., Y k }.Since C = Γ, there exists an index function ν : C → {1, . . ., k} such that X c = Y ν(c) for every c ∈ C, i.e.Y ν(c) = b( α ) | α occurs in B and c ∈ α H = b( α ) | α occurs in B and I c |= α .Conversely, for every µ ∈ {1, . . ., k}, there is an element c ∈ C such that ν(c) = µ.The Ointerpretations for showing admissibility of X are obtained as follows.Take c 1 , . . ., c k ∈ C such that ν(c 1 ) = 1, . . ., ν(c k ) = k.Now, for every i, 1 ≤ i ≤ k, we define the O-interpretation G i := (∆, 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, • J b ).By Lemma 13, we have that J b is a model of B b , and the set X := {X c | c ∈ C} is admissible.By construction, J b weakly respects (Im(b), X ), and hence B b is outer consistent w.r.t.X .(⇐=): Let X = {X 1 , . . ., X k } ⊆ 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 X := {Y c | c ∈ C}, where Y c := {A ∈ Im(b) | c ∈ A G }. Since G weakly respects (Im(b), X ) and c ∈ (C Im(b),Yc ) G for every c ∈ C, we have that X ⊆ X .Since X is admissible, this yields admissibility of X .Lemma 13 yields now consistency of B.
a), R O   has a model that respects (O Crig (B), Y), for all 1 ≤ i ≤ k.The SHOQ-BKB B Xi is of size polynomial in the size of B and can be constructed in time exponential in the size of B. We can check if B Xi has a model that respects (O Crig (B), Y) in time exponential in the size of B Xi [BGL08; BGL12], and thus exponential in the size of B. Together with the lower bounds shown in Section 4, we obtain NExp-completeness for the consistency problem in L M L O for L M and L O being DLs between ALC and SHOQ if O Crig = ∅ and O Rrig = ∅.

4
The Case of EL: L M EL and EL L O In this section, we give some complexity results for context DLs L M L O where L M or L O are EL.In Section 4.1, we consider L M EL where L M is between ALC and SHOQ.Then, in Section 4.2, we consider the remaining context DLs EL L O where L O is either EL or between ALC and SHOQ.Description Logics of Context with Rigid Roles Revisited, 08.05.2015 Context with Rigid Roles Revisited, 08.05.2015Theorem 24.The consistency problem in ALC EL is NExp-hard if O Crig = ∅ and O Rrig = ∅.
, the claim follows by similar arguments.If φ is of the form Xψ, we have thatc i ∈ C J P φ iff c i ∈ (∀t.C ψ ∃ t.C ψ ) J P iff c i+1 ∈ C J P ψ iff I, i+1 |= ψ iff I, i |= φ.Description Logics ofContext with Rigid Roles Revisited, 08.05.2015It follows that J P |= C φ iff I, 0 |= φ.This claim yields the lower bound of NExp for the consistency problem in ALC EL if O Crig = ∅.Next, we prove the upper bound of NExp for the consistency problem in the case of rigid names.Theorem 25.The consistency problem in SHOQ EL is in NExp if O Crig = ∅ and O Rrig = ∅.
a), and {a} ∃ r.{b} ⊥ if β i = r(a, b) with A , D being fresh concept names and a i being fresh individual names.It is easy to see that if an O-interpretation I is a model of ¬β 1 ∧ • • • ∧ ¬β m , there exists an extension of I that is a model of γ 1 ∧ • • • ∧ γ m .Conversely, if an O-interpretation I is a model of γ 1 ∧ • • • ∧ γ m , it is also a model of ¬β 1 ∧ • • • ∧ ¬β m .Hence B and B are equisatisfiable.

Theorem 29 .
The consistency problem in EL ALC is NExp-hard if O Crig = ∅ and O Rrig = ∅.Proof.To show the lower bound, we again adapt the proof ideas of [BGL08; BGL12], and reduce an exponentially bounded version of the domino problem to the consistency problem in EL ALC with rigid concepts, i.e.O Crig = ∅ and O Rrig = ∅.In [BGL08; BGL12], a reduction was provided to show NExp-hardness for the temporalized DL ALC-LTL in the presence of rigid concepts.As in the proof of Theorem 28, we mimic the properties of the time dimension that are important for the reduction using a role name t ∈ M R .

Corollary 30 .
For all L O between ALC and SHOQ, the consistency problem in EL L O is• Exp-complete if O Crig = ∅ and O Rrig = ∅, • NExp-complete if O Crig = ∅ and O Rrig = ∅, and • 2Exp-complete if O Crig = ∅ and O Rrig = ∅.Description Logics of Context with Rigid Roles Revisited, 08.05.2015 worksFor.{Siemens}∃ hasAccessRights.{Siemens}(1) 3, we show decidability of the consistency problem in L M L O for L M and L O being DLs between ALC and SHOQ.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 context L M L O where L M or L O is the sub-Boolean DL EL.Again C , N R , N I ).An N-interpretation is a pair I = (∆ I , • I ), where ∆ I is a non-empty set (called domain), and • I is a mapping assigning a set A I ⊆ ∆ I to every A ∈ N C , a binary relation r I ⊆ ∆ I × ∆ I to every r ∈ N R , and a domain element a I ∈ ∆ I to every a ∈ N I .The function • I is extended to concepts over N inductively as shown in the upper part of Table1, where S denotes the cardinality of the set S.Finally, I is a model of the BKB B = (B, R) over N (written I |= B) if itis a model of both B and R. We call B 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 ∈ N R (w.r.t.R) if every model of R is a model of r s.Finally, r is simple w.r.t.R if it has no transitive subrole.It is not hard to see that r ∈ N R is simple w.r.t.R iff trans(r) / ∈ R and there do not exist roles s 1 , . . ., s k ∈ N R such that {trans(s 1 ), s 1 s 2 , . . ., s k−1 s k , s k r} ⊆ R. Thus deciding whether r ∈ N R is simple can be decided in time polynomial in the size of R by simple syntactic checks.

Table 2 :
Complexity results for consistency in L M L O (where "c." is short for "complete") • Ic i ).Clearly, we have that G i |= B Yi and since J |= R O , we have that G i |= B Yi .Moreover, the definition of a nested interpretation yields that x Gi = x Gj for all x ∈ O I ∪ O Crig ∪ O Rrig and all i, j ∈ {1, . . ., k}.Hence, the O-interpretations G 1 , . . ., G k attest admissibility of X .H ) is a model of B b and that the set X := {X d | d ∈ Γ} is admissible.Again, since Im(b) is finite, we have that X ⊆ P(Im(b)) is finite.Let X = {Y 1 , . . ., Y k }.Since X is admissible, there are O-interpretations G 1 Description Logics of Context with Rigid Roles Revisited, 08.05.2015 (2 ⇒ 1): Assume that H = (Γ, • which is of size at most exponential in B. Due to Lemma 15 we can check whether B b is outer consistent w.r.t.X in time exponential in the size of B b and linear in the size of X .It remains to check X for admissibility.For that let O Crig (B) ⊆ O Crig and O I (B) ⊆ O I be the sets of all rigid concept names and individual names occurring in B, respectively.As done in [BGL08; BGL12] we non-deterministically guess a set Y ⊆ P(O Crig (B)) and a mapping κ : 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 that X is admissible iff we have c i ∈ (∃ t. ) J .Thus, J |= ∃ t. .(⇐=):Takeany nested interpretation J = (C, • J , ∆, (• Ic ) c∈C ) that is a model of C φ ∃ t. .Let P be an infinite path P = c 0 c 1 ...with c i ∈ C and (c i , c i+1 ) ∈ t J for every i ≥ 0. Such a path exists, because J |= ∃ t. .We define the nested interpretationJ P := ({c i | i ≥ 0}, • J P , ∆, (• Ic i ) i≥0 ) where • J P is the restriction of • J to the domain {c i | i ≥ 0}.If C φ does not contain any role name r ∈ M R , the restriction on the set of worlds preserves the entailment relation.Otherwise, C φ is of the form ∀t.C ψ ∃ t.C ψ .Since J P |= ∃ r. , J P |= C ψ , and there is only one t-successor, we have J P |= C φ .
The lower bounds follow from Theorems 20 and 24.The upper bound of Exp in the case O Crig = O Rrig = ∅ follows immediately from Theorem 16, whereas the upper bound of NExp follows from Theorem 25.In this section, we consider EL L O where L M is either EL or between ALC and SHOQ.Instead of considering EL L O -BKBs, we allow only conjunctions of m-axioms.Then the consistency problem becomes trivial in the case of EL EL since all EL EL -BKBs are consistent, as EL lacks to express contradictions.This restriction, however, does not yield a better complexity in the cases of EL L O , where L O is between ALC and SHOQ.