Living without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article, we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as 𝒜ℒ𝒞𝒪, 𝒜ℒ𝒞ℋ, and 𝒜ℒ𝒞ℋ𝒪ℐ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case, the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.


Introduction
The Craig Interpolation Property (CIP) for a logic L states that an implication ϕ ⇒ ψ is valid in L iff there exists a formula χ in L using only the common symbols of ϕ and ψ such that ϕ ⇒ χ and χ ⇒ ψ are both valid in L. The intermediate formula χ is then called an L-interpolant for ϕ ⇒ ψ [27]. The CIP is generally regarded as one of the most important and useful properties in formal logic [95], with numerous applications ranging from formal verification [74] and software specification [29] to theory combinations [25,39,21,22] and query reformulation and rewriting in databases [91,13]. A particularly important consequence of the CIP is the projective Beth definability property (PBDP), which states that a relation is implicitly definable using a signature Σ of symbols iff it is explicitly definable using Σ. If Σ is the set of all symbols distinct from that relation, then we speak of the (non-projective) Beth definability property (BDP) [16].
In this paper, we investigate interpolants and explicit definitions in description logics (DLs), and we also highlight consequences in modal logic. In DLs, one distinguishes essentially two forms of interpolation, both of which are relevant and have their applications. Given an entailment O |= C ⊑ D, that is, C is subsumed by D w.r.t. some background knowledge in the form of a DL ontology O, one might either be interested in an interpolant between the concepts C and D or in an interpolant between O and the concept inclusion (CI) C ⊑ D. In the first case, the interpolant is a concept, whereas in the second case, the interpolant is an ontology. We refer with CI-interpolation to the latter form and call the interpolant a CI-interpolant. The CIP for CI-interpolation has been shown to be the most important logical property that ensures the robust behaviour of ontology modules and decompositions [55,54].
In this article, we mostly focus on interpolation (in the former sense of an interpolating concept) and only derive some corollaries for CIinterpolation. Hence, unless stated otherwise, here and in what follows we speak about interpolating concepts and the corresponding CIP. For explicit definability, one asks for definitions of concepts, possibly with respect to an ontology; these explicit definitions are strongly related to interpolants and as stated above BDP and PBDP follow from the CIP. In DLs, the BDP and PBDP have been used in ontology engineering to extract explicit definitions of concepts and obtain equivalent acyclic terminologies from ontologies [89,90], they have been investigated in ontology-based data management to equivalently rewrite ontology-mediated queries [85,92,35,34,93], and they have been proposed to support the construction of alignments between ontologies [47]. Interpolants have been used to study P/NP dichotomies in ontology-based query answering [69].
The CIP, PBDP, and BDP are so powerful because potentially very hard existence questions are reduced to straightforward entailment ques-tions: an interpolant exists iff an implication is valid and an explicit definition exists iff a straightforward formula stating implicit definability is valid. The existence problems are thus not harder than validity. Many basic DLs such as ALC, ALCI, and ALCIQ enjoy the CIP and PBDP [90], and consequently the existence of an interpolant or an explicit definition can be decided in ExpTime simply because entailment checking in these DLs is in ExpTime (and without ontology even in PSpace). Unfortunately, the CIP and the PBDP fail to hold for some important DLs. The most basic examples are the extension ALCO of ALC with nominals (concepts of the form {a} with a an individual name), the extension ALCH of ALC with role inclusions (inclusions r ⊑ s between binary relations/role names r and s), and all standard DLs containing either ALCO or ALCH [55,90]. It follows that for these DLs the existence of interpolants and explicit definitions cannot be reduced (directly) to entailment checking.
The aim of this article is to explore the consequences of the failure of the CIP and PBDP for interpolant and explicit definition existence. To this end, we investigate the complexity of deciding the existence of interpolants and explicit definitions for the set DL nr of DLs containing ALCO, ALCH, and their extensions by inverse roles and/or the universal role. We discuss next two more applications of interpolants and explicit definitions for ALCO and its extensions.  1 Concept learning has received significant interest over the past 15 years, where the focus has been on developing and analyzing refinement based algorithms for finding separating concepts [62,63,65,84,30,64,82]. Prominent concept learning systems include the DL Learner [20,19], DL-Foil [31] and its extension DL-Focl [83], SPaCEL [94], YinYang [45]. The existence problem for separating concepts has been investigated in [36,48,49,50]. For DLs extending ALCO, we establish a one-to-one correspondence between interpolants and separating concepts, modulo a rather straightforward polynomial time translation. Hence the existence of separating concepts reduces to the existence of interpolants and finding small such concepts or concepts of a certain syntactic shape, as is often useful in supervised learning, also reduces to the same task for interpolants. We emphasize that the presence of nominals in the DL is critical as they are required to encode the individuals used in D into concepts.
Referring Expressions. The computation of explicit definitions of concept names has been explored in detail since at least [89], see also [7]. Only recently, the focus on defining concept names has been extended to defining individual names, also called referring expression generation in computational linguistics and data management [60,3,17]. In fact, it has been convincingly argued that very often in applications the individual names used in ontologies or data sets are insufficient "to allow humans to figure out what real-world objects they refer to" [18]. A natural way to address this problem is to check for such an individual name a whether there exists a concept C over a set of relevant symbols Σ that provides an explicit definition of {a} and present such a concept C to the human user. Observe that one has to work with DLs extending ALCO to formulate this problem as an explicit definition existence problem.
To conclude, data separation, concept learning, and referring expresssion generation are challenging research problems which directly benefit from a better understanding of interpolant and explicit definition existence in extensions of ALCO. We now discuss the main results of this article, formulated in an informal way. Precise formulations are given later. Recall that DL nr is the set of DLs ALCO, ALCH, and their extensions with inverse roles and the universal role, and that we assume the presence of a background DL ontology. Our first main result is as follows.
Theorem 1. Let L ∈ DL nr . Then L-interpolant existence and L-definition existence are 2ExpTime-complete.
Theorem 1 confirms the suspicion that interpolant and definition existence are much harder problems than entailment if one has to live without Beth and Craig. On the positive side, these problems are still decidable. Interestingly, for DLs in DL nr with nominals, the 2ExpTime lower bound for definition existence already holds if one asks for an explicit definition of an individual over the signature containing all symbols distinct from that individual. In contrast, the same problem for concept names is shown to be ExpTime-complete and thus not harder than entailment. Hence, in contrast to concept name definitions, referring expression existence does not become less complex in the non-projective case when all symbols are allowed in definitions.
We next consider the same problems if the background ontology is empty, or, in the case of DLs in DL nr without nominals, if the ontology contains only role inclusions. Observe that if the DL admits the universal role or both nominals and inverse roles, then the ontology can be encoded as a concept using spy points [1], so nothing changes compared to the case with ontologies covered in Theorem 1. For the remaining cases we show the following. (2) If L ∈ {ALCH, ALCHI}, then for ontologies containing role inclusions only L-interpolant existence and L-definition existence are both coNExpTime-complete.
It follows that without ontology and ontologies containing role inclusions only interpolant existence and explicit definition existence are still harder than entailment which is PSpace-complete.
The proofs of Theorems 1 and 2 can be adapted to also obtain results about CI-interpolation and interpolation in modal logic. Regarding the former we show that for the DLs L which extend ALCO with the universal role or with the universal role and inverse roles the problem of deciding the existence of a CI-interpolant for O |= C ⊑ D is 2ExpTime-complete. It follows that again failure of the CIP leads to an exponentially harder interpolant existence problem than entailment. We conjecture that the same can be proved for all DLs in DL nr , but leave a proof for future work.
In modal logic, the CIP and PBDP have been investigated for many years. In fact, the CIP and PBDP of DLs such as ALC and ALCI follows rather directly from earlier results on the CIP and PBDP in modal logic [71,72,81]. Also the fact that nominals lead to failure of the CIP and PBDP, and how this could be repaired by adding logical connectives, was first analyzed in depth in the literature on hybrid modal logic, in particular [2,86]. In our investigation of interpolant existence in modal logic, we first consider basic modal logic with nominals and show that as a direct consequence of Theorem 2 the problem of deciding interpolant existence is coNExpTime-complete for the standard local consequence relation. We also show using Theorem 1 that if one adds the universal modality, then interpolant existence becomes 2ExpTime-complete. In modal logic, nominals are often considered in tandem with the @-operator, where @ a ϕ states that formula ϕ holds at the world denoted by nominal a. The resulting language is more expressive than modal logic with nominals and less expressive than modal logic with nominals and the universal modality. We show that for the modal logic with both nominals and the @-operator interpolant existence is still coNExpTime-complete. Our complexity results also hold for the modal language with a single modal operator (and the universal modality, if present).
While the focus in this article is on the decision problem, we also make initial observations regarding the problem of actually computing interpolants or explicit definitions if they exist. More specifically, for DLs in DL nr that do not admit nominals, we present a modification of the decision procedure from the proof of Theorem 1 that returns in double exponential time the DAG representation of an interpolant (if it exists). This corresponds to interpolants of worst case triple exponential size which we conjecture to be optimal.
Overview of the Paper. In the following Section 2, we discuss further related work. In Sections 3 and 4, we introduce the preliminaries on description logics and Craig interpolation and Beth definability, respectively. In Section 5, we provide model-theoretic characterizations of the definition and interpolation existence problems and formulate our main results in detail. The subsequent four sections are devoted to the proofs of these main results. In more detail, Section 6 provides the upper bound proof for the case with ontologies and Section 7 provides the matching lower bounds. Sections 8 and 9 cover the ontology-free case and the case of ontologies containing only role inclusions. In Section 10, we investigate the problem of actually computing interpolants and explicit definitions in case they exist, and in Section 11 we draw the connections of our results on DLs to modal logic. Finally, we conclude and point out directions for future work in Section 12.
An appendix available as supplementary material provides a few proofs that were left out of the paper. Here we prove, in particular, our main result about the computational complexity of non-projective definition existence of concept names. This paper is an extended version of [5,6,51]. We include detailed proofs and additionally discuss the link to concept learning, interpolants between ontologies and concept inclusions, and applications to modal logic.
Related work on Craig interpolation and the Beth definability property has been discussed already in the introduction. We therefore focus on work on deciding interpolant and explicit definition existence. These decision problems have only very recently been investigated. A notable exception is linear temporal logic, LTL, for which the CIP fails and for which decidability of interpolant existence has been shown both over finite linear orderings [41,42] and over the natural numbers [79]. Note that these results are formulated as separability results for formal languages of finite and, respectively, infinite words: given two regular languages R 1 and R 2 , does there exist a first-order definable language L separating R 1 and R 2 in the sense that R 1 ⊆ L and L ∩ R 2 = ∅. Neither LTL nor Craig interpolation are mentioned in [79,41,42]. Using the fact that regular languages are projectively LTL definable and that LTL and first-order logic are equivalent over the natural numbers, it is, however, easy to see that interpolant existence is the same problem as separability of regular languages in firstorder logic, modulo the representation of the inputs. We note that this result is just one instance of an ongoing exploration of separation between languages in automata theory. The problem of deciding separation is interesting in this context because obtaining an algorithm for separation yields a far deeper understanding of the class under consideration than just membership [78,80]. We conjecture that deciding interpolant existence could well play a similar role for understanding fragments of first-order logic.
Indeed, interpolant existence has recently also been studied for the guarded fragment (GF), the two-variable fragment (FO 2 ) of FO [52], for Horn description logics extending EL [33], and for first-order modal logics [61]. While GF is a good generalization of modal and description logic in many respects, it neither enjoys the CIP [44] nor the PBDP [10]. Failure of the CIP for FO 2 was shown using algebraic [26,76] and modeltheoretic techniques [73]. Using techniques that are similar to those introduced in this article it is shown in [52] that, in GF, explicit definability and interpolant existence are both 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant c ≥ 3. In FO 2 , explicit definability and interpolant existence are in coN2ExpTime and 2ExpTime-hard [52]. Failure of the CIP and PBDP for first-order modal logics with constant domain is shown in [32,73]. Both properties also fail for their otherwise well-behaved onevariable and monodic fragments [38]. In [61], the complexity of interpolant existence is investigated for first-order S5 with one variable (and some monodic fragments) and for first-order K with one variable. For S5, explicit definability and interpolant existence turn out to be in coN2ExpTime and 2ExpTime-hard while for K only a non-elementary upper bound is shown. These results confirm that for many logics not enjoying the CIP and PBDP, interpolant and explicit definition existence are harder than entailment.
It turns out that this is not always the case. It is shown in [33] that extensions of the description logic EL with any combination of the universal role, nominals, or inverse roles do not enjoy the CIP nor PBDP, but that interpolant existence and explicit definition existence still have the same complexity as entailment (in PTime for those that do not admit inverse roles and ExpTime-complete for those that admit inverse roles). The proofs are rather different from those given in this paper, as they make use of the universal/canonical model that only exists for Horn logics.
We note that for logics that do not enjoy the CIP nor PBDP it is also of interest to look for "small" extensions that enjoy the CIP and PBDP and are decidable. For example, the guarded negation fragment of FO is a decidable extension of GF that enjoys the CIP and the PBDP [11,15,14,12]. Also the two-variable fragment of GF is a decidable extension of ALCH enjoying both properties [44,43]. In both cases the complexity of entailment does not increase for the extension (2ExpTime-complete for the guarded negation fragment and ExpTime-complete for the two-variable fragment of FO). On the other hand, under mild conditions there is no decidable extension of ALCO with the universal role nor of modal logic with nominals and the @-operator enjoying the CIP [86].
While the problem of deciding interpolant and explicit definition existence for logics that neither enjoy the CIP nor the PBDP has only been considered rather recently, the problem of computing and deciding the existence of uniform interpolants for logics that do not enjoy the uniform interpolation property (UIP) has been investigated before. Recall that uniform interpolants generalize Craig interpolants in the sense that a uniform interpolant is an interpolant for a fixed ϕ and all ψ which are entailed by ϕ and share with ϕ a fixed set of symbols. First-order logic enjoys the CIP but not the UIP. Propositional intuitionistic logic, local modal logic, and the modal mu-calculus are examples of expressive logics that enjoy the UIP [77,96,28], see [59,46] for more recent results. In de-scription logic, uniform interpolants of ontologies (extending what we call CI-interpolants in this article) are of particular importance but do not always exist for any standard description logic, including ALC. The complexity of deciding their existence has been investigated in [68,70], their size has been considered in [75,58], and various approaches to computing them have been developed and implemented [56,57,58,97].

Preliminaries
We introduce the syntax and semantics of the relevant description logics, see also [9]. Let N C , N R , and N I be mutually disjoint and countably infinite sets of concept, role, and individual names. A role is a role name s, or an inverse role s − , with s a role name and (s − ) − = s. We use u to denote the universal role. A nominal takes the form {a}, with a an individual name. An ALCOI u -concept is defined according to the syntax rule C, D : where A ranges over concept names, a ranges over individual names, and r over roles and the universal role. We use C ⊔ D as abbreviation for ¬(¬C ⊓ ¬D), C → D for ¬C ⊔ D, C ↔ D for (C → D) ⊓ (D → C), and ∀r.C for ¬∃r.¬C. We use several fragments of ALCOI u , including ALCOI, obtained by dropping the universal role, ALCO u , obtained by dropping inverse roles, ALCO, obtained from ALCO u by dropping the universal role, and ALC, obtained from ALCO by dropping nominals. If L is any of the DLs above, then an L-concept inclusion (L-CI) takes the form C ⊑ D with C and D L-concepts. An L-ontology is a finite set of L-CIs. We also consider DLs with role inclusions (RIs), expressions of the form r ⊑ s, where r and s are roles. As usual, the addition of RIs is indicated by adding the letter H to the name of the DL, where inverse roles occur in RIs only if the DL admits inverse roles. Thus, for example, ALCH-ontologies are finite sets of ALC-CIs and RIs not using inverse roles and ALCHOI u -ontologies are finite sets of ALCOI u -CIs and RIs. In what follows we use DL nr to denote the set of DLs ALCO, ALCOI, ALCH, ALCHI, ALCHO, ALCHOI, and their extensions with the universal role. To simplify notation we do not drop the letter H when speaking about the concepts and CIs of a DL with RIs. Thus, for example, we sometimes use the expressions ALCHO-concept and ALCHO-CI to denote ALCOconcepts and CIs, respectively. An RI-ontology is an ontology containing RIs only.
The semantics is defined in terms of interpretations I = (∆ I , · I ), where ∆ I is a non-empty set, called domain of I, and · I is a function mapping every A ∈ N C to a subset of ∆ I , every s ∈ N R to a subset of ∆ I × ∆ I , the universal role u to ∆ I × ∆ I , and every a ∈ N I to an element in ∆ I . Given a role name s ∈ N R , we set (s − ) I = {(d, e) ∈ ∆ I × ∆ I | (e, d) ∈ s I }. Moreover, the extension C I of an L-concept C in I is defined as follows, where r ranges over roles and the universal role: An interpretation I satisfies an L-CI C ⊑ D if C I ⊆ D I and an RI r ⊑ s if r I ⊆ s I . We say that I is a model of an ontology O if it satisfies all inclusions in it. We say that an inclusion α follows from an ontology O, We use a few well known complexity bounds for reasoning in DLs from DL nr . The L-subsumption problem is the problem to decide for any L-ontology O and L-CI C ⊑ D whether O |= C ⊑ D. The ontology-free L-subsumption problem and the RI-ontology L-subsumption problem are the sub-problems of the Lsubsumption problem in which the ontology is empty or an RI-ontology, respectively. For any L ∈ DL nr , the L-subsumption problem is ExpTimecomplete [8,1]. If L admits the universal role or both inverse roles and nominals, then ontologies can be encoded in concepts and so ontologyfree L-subsumption and RI-ontology L-subsumption are also ExpTimecomplete. In the remaining cases, that is for ALCO, ALCH, ALCHO, and ALCHI, L-subsumption becomes PSpace-complete [8,1].
A signature Σ is a set of concept, role, and individual names, uniformly referred to as symbols. Following standard practice we do not regard the universal role as a symbol but as a logical connective. Thus, the universal role is not contained in any signature. We use sig(X) to denote the set of symbols used in any syntactic object X such as a concept or an ontology. An L(Σ)-concept is an L-concept C with sig(C) ⊆ Σ. A Σ-role r is a role with sig(r) ⊆ Σ. The size of a (finite) syntactic object X, denoted ||X||, is the number of symbols needed to represent it as a word.
[AtomC] for all (d, e) ∈ S: We next recall model-theoretic characterizations when elements in interpretations are indistinguishable by concepts formulated in one of the DLs L introduced above. A pointed interpretation is a pair I, d with I an interpretation and d ∈ ∆ I . For pointed interpretations I, d and J , e and a signature Σ, we write I, d ≡ L,Σ J , e and say that I, d and J , e are L(Σ)-equivalent if d ∈ C I iff e ∈ C J , for all L(Σ)-concepts C.
As for the model-theoretic characterizations, we start with ALC. Let For ALCO, we define ∼ ALCO,Σ analogously, but now demand that, in Figure 1, also condition [AtomI] holds for all individual names a ∈ Σ. For languages L with inverse roles, we demand that, in Figure 1, r additionally ranges over inverse roles. For languages L with the universal role we extend the respective conditions by demanding that the domain dom(S) and range ran(S) of S contain ∆ I and ∆ J , respectively. If a DL L has RIs, then we use I, d ∼ L,Σ J , e to state that I, d ∼ L ′ ,Σ J , e for the fragment L ′ of L without RIs.
The next lemma summarizes the model-theoretic characterizations for all relevant DLs [67,40]. For the definition of ω-saturated structures, we refer the reader to [24]. For the "if "-direction, the ω-saturatednesses condition can be dropped.

Craig Interpolation and Beth Definability
We introduce interpolants and the Craig interpolation property (CIP) as well as implicit and explicit definitions and the (projective) Beth definabil-ity property ((P)BDP). Recall from the introduction that there are two forms of interpolants, one pertaining to concepts and the other pertaining to concept inclusions. We start the discussion here with the former one, and discuss CI-interpolants later. For concept interpolants, we establish a close link between interpolants and separators of positive and negative data examples, show that logics in DL nr do not enjoy the CIP nor PBDP, and determine which DLs in DL nr enjoy the BDP.
Let O be an L-ontology, C 1 , C 2 be L-concepts, and let Σ be a signature. Then, an L-concept D is an L(Σ)-interpolant for If O is empty, then we obtain the standard definition of the Craig interpolation property by demanding that for Σ = sig(C 1 ) ∩ sig(C 2 ) from |= C 1 ⊑ C 2 it follows that there exists an L(Σ)-interpolant for C 1 ⊑ C 2 . The obvious generalization of this definition to non-empty ontologies, however, does not work. Consider, for instance, In fact, to generalize the Craig interpolation property to non-empty ontologies, one has to split the ontology O into two. Hence, we adopt here the following definition of the Craig interpolation property in DLs from [90]. We set sig(O, C) = sig(O) ∪ sig(C), for any ontology O and concept C.
containing RIs only or O 1 = O 2 = ∅, then we say that L enjoys the CIP for RI-ontologies and the CIP for the empty ontology, respectively.
Note that the CIP for the empty ontology coincides with the standard definition of the CIP mentioned before. It is shown in [90] that the DLs ALC and ALCI and their extensions with qualified number restrictions and the universal role all enjoy the CIP. In contrast, no DL in DL nr enjoys the CIP. This is implicitly proved in [90] and is shown in Theorem 12 below. The following illustrating example is folklore and shows that this holds even for the empty ontology for logics admitting nominals. Theorem 6. Let L ∈ DL nr admit nominals. Then one can construct for any ontology O, labelled data sets P, N , and signature Σ with Σ ∩ {a | (D, a) ∈ P ∪ N } = ∅ in polynomial time L-ontologies O 1 , O 2 and Lconcepts C 1 , C 2 such that Σ = sig(O 1 , C 1 ) ∩ sig(O 2 , C 2 ) and the following conditions are equivalent for all L-concepts C: Conversely, asssume that L-ontologies O 1 , O 2 , and L-concepts C 1 , C 2 are given. Then one can construct in polynomial time an ontology O and labelled data sets P, N such that Conditions (1) and (2) are equivalent for Proof. Assume O, P, N , and Σ are given. Let P = {(D 1 , a 1 ), . . . , (D n , a n )} and N = {(D n+1 , a n+1 ), . . . , (D n+m , a n+m )}. If L ∈ DL nr admits nominals and the universal role, then a pair (D, a) can be represented using the L-concept C D,a = {a} ⊓ ∃u.C D , where C D is the conjunction of all {b} ⊓ A with A(b) ∈ D and {b} ⊓ ∃r.{c} with r(b, c) ∈ D. Pick for any symbol X not in Σ a fresh copy X ′ . Let O 1 = O and obtain O 2 from O by replacing all symbols not in Σ by their copies. Let C 1 = C D 1 ,a 1 ⊔ · · · ⊔ C Dn,an and obtain C 2 from ¬(C D n+1 ,a n+1 ⊔ · · · ⊔ C D n+m ,a n+m ) by replacing all symbols not in Σ by their copies. If L admits the universal role, then O 1 , O 2 and C 1 , C 2 are as required. Otherwise replace the universal role in any C D i ,a i by fresh role names not in Σ. The resulting C 1 , C 2 are still as required.
The following example illustrates the link between explicit definitions of nominals and referring expressions discussed in the introduction and also indicates that often one can single out an individual from a set of individuals using an explicit definition without being able to provide an 'absolute' explicit definition of that individual.
Example 7. Let L be an abbreviation for the ALCO-concept where ML, ML n , and ML u n are modal logics introduced below in Section 11. Let O be the ontology consisting of the following CIs: We next define when a concept is implicitly definable. For a signature Σ, the Σ-reduct I |Σ of an interpretation I coincides with I except that no non-Σ symbol is interpreted in I |Σ . A concept C 0 is called implicitly Σdefinable under O and C if the Σ-reduct of any pointed model I, d with I a model of O and d ∈ C I determines whether d ∈ C I 0 . More formally, C 0 is implicitly Σ-definable under O and C if the following holds for all models I and J of O and d ∈ ∆ I = ∆ J : if I |Σ = J |Σ and d ∈ C I , then d ∈ C I 0 iff d ∈ C J 0 . If C = ⊤, then we drop C and say that C 0 is implicitly Σ-definable under O. To illustrate, observe that in Example 7, {ALC} is not implicitly Σ-definable, for any Σ such that ALC ∈ Σ under O. Implicit definability can be reformulated as a standard reasoning problem as follows: a concept C 0 is implicitly Σ-definable under O and C iff where O Σ , C Σ , and C 0Σ are obtained from O, C and, respectively, C 0 , by replacing every non-Σ symbol uniformly by a fresh symbol. If a concept is explicitly L(Σ)-definable under O and C, then it is implicitly Σ-definable under O and C, for any language L. A logic enjoys the projective Beth definability property if the converse implication holds as well. The DLs ALC, ALCI, and their extensions with qualified number restrictions and the universal role all enjoy the PBDP [90]. The following example shows that, in contrast, ALCH does not. Let Σ = {r 1 , r 2 } and C 0 = ∃r.⊤. Then the concept D = ∃r 1 ∩ r 2 .⊤ is an explicit definition of C 0 under O and C in the extension of ALCH with role intersection (the semantics of r 1 ∩ r 2 is defined in the obvious way). Hence C 0 is implicitly Σ-definable under O and C. There does not exist an explicit ALCH(Σ)-definition of C 0 under O and C, however. Intuitively, the reason is that role intersection cannot be expressed in ALCH (see Example 24 below for a proof). Note that an example without "background concept" C can be obtained by taking the ontology ¬∃r.⊤ ⊓ ∃r 1 .A ⊑ ∀r 2 .¬A, ¬∃r.⊤ ⊓ ∃r 1 .¬A ⊑ ∀r 2 .A } and asking for an explicit ALCH({r 1 , r 2 })-definition of ∃r.⊤ under O ′ . ⊣ It is known that the CIP and PBDP are tightly linked [90]. We state the inclusion for logics in DL nr only, but the proof shows that it holds under rather mild conditions. Lemma 10. If L ∈ DL nr enjoys the CIP, then L enjoys the PBDP.
Proof. Assume that an L-concept C 0 is implicitly Σ-definable under an L-ontology O and L-concept C, for some signature Σ. Then (1) holds.
An important special case of explicit definability is the explicit definability of a concept name A from sig(O, C)\{A} under an ontology O and concept C. For this case, we also consider the following non-projective version of the Beth definability property.
Definition 11. A DL L enjoys the Beth definability property ( BDP) if for any L-ontology O, concept C, and any concept name A the following If O ranges over L-ontologies containing RIs only or O = ∅, then we say that L enjoys the BDP for RI-ontologies and the BDP for the empty ontology, respectively.
Clearly the PBDP entails the BDP, but we will see below that the converse direction does not always hold. In fact, the following theorem states that no DL in DL nr enjoys the CIP or PBDP, but that quite a few DLs in DL nr enjoy the BDP. Moreover, all DLs in DL nr enjoy the BDP for RI-ontologies and for the empty ontology.
As mentioned before, the theorem is mostly folklore and therefore proved in the appendix.
Theorem 12. The following statements hold.
(1) No L ∈ DL nr enjoys the CIP nor the PBDP. The CIP and PBDP also do not hold for RI-ontologies and, if L admits nominals, the empty ontology. (2) All L ∈ DL nr \{ALCO, ALCHO} enjoy the BDP. ALCO and ALCHO do not enjoy the BDP. (3) All L ∈ DL nr enjoy the BDP for RI-ontologies and the BDP for the empty ontology.
We have seen that all L ∈ DL nr \ {ALCO, ALCHO} enjoy the BDP. One might be tempted to conjecture that this holds as well if concept names are replaced by nominals; that is to say, a nominal {a} that is implicitly definable using symbols distinct from a is explicitly definable using symbols distinct from a. Rather surprisingly, the following example shows that this is not the case for any DL in DL nr with nominals (for DLs without nominals this notion is clearly meaningless).
Example 13. Let L ∈ DL nr admit nominals and assume that Thus, O implies that a is r-reflexive and that no element distinct from a is r-reflexive. Let Σ = {r, A}. Then {a} is implicitly Σ-definable under O since we have the following explicit definition in first-order logic: The CIP defined above is concerned with interpolating concepts. In the context of modular ontologies and forgetting there is also an interest in interpolating concept inclusions [55]. For simplicity, we only consider DLs without RIs. Let L not admit RIs and let O and O ′ be L-ontologies.
If the particular language L is clear from the context or not important we drop it and call L-CI interpolants simply CI-interpolants. It is known that ALC and its extensions with inverse roles, qualified number restrictions, and the universal role enjoy the CIinterpolation property [55]. The following example shows that no DL in DL nr that does not admit RIs enjoys the CI-interpolation property.
Example 15. We modify the ontology given in Example 13. Let We have that O ′ |= {a} ⊑ ¬∃r.{a}, but there does not exist an ALCOI u -CI-interpolant for O ′ and {a} ⊑ ¬∃r.{a}, since one cannot express using an ALCOI u ({r})-CI that ∀x ¬r(x, x). Indeed, assume for a proof by contradiction that there exists an ALCOI u -CI-interpolant O ′′ for O ′ and {{a} ⊑ ¬∃r.{a}}. Consider the interpretations I 1 , I 2 in Figure 3, where In this article we focus on interpolating concepts and not CIs. The main reasons are that the corresponding notion of an explicit definition of an ontology appears to be less useful than definitions of concepts and nominals and that while the CI-interpolation property is crucial for robust decompositions of ontologies and for robust forgetting [55], checking the existence of an interpolant or computing it for concrete ontologies and CIs appears to not have found any applications yet. Regarding the first point, observe that CI-interpolants correspond to the following notion of an explicit CI-definition of an ontology. Let Σ be a signature and O and O ′ ontologies. Then an In particular, if O is empty then one asks for an ontology using symbols in Σ only that is equivalent to O. While the existence of such ontologies is an interesting theoretical question that could well have applications in the future, investigating this problem is beyond the focus of this article. In what follows we only consider aspects of CI-interpolants that are closely related to concept interpolants, leaving their detailed investigation for future work.
The failure of CIP and (P)BDP reported in Theorem 12 imply that interpolant existence and projective and non-projective definition existence cannot be directly polynomially reduced to subsumption checking. This motivates studying the respective decision problems of interpolant existence and projective and non-projective definition existence. In this section we introduce the decision problems, formulate model-theoretic characterizations of the problems that play a fundamental role in our proofs, and we formulate the main results.
We start with interpolant existence for which we take the definition used in the formulation of the CIP.
Definition 16. Let L be a DL. Then L-interpolant existence is the problem to decide for any L-ontologies O 1 , O 2 and L-concepts In our proofs, we actually focus on a more general version of interpolant existence which has been discussed in the previous section and in which we do not split O into two ontologies and in which Σ is arbitrary.
Definition 17. Let L be a DL. Then generalized L-interpolant existence is the problem to decide for any L-ontology O, L-concepts C 1 , C 2 , and signature Σ whether there exists an L(Σ)-interpolant for We also consider (generalized) L-interpolant existence with empty ontologies, called ontology-free (generalized) L-interpolant existence, and with RI-ontologies, called RI-ontology (generalized) L-interpolant existence, both defined in the obvious way. Observe that in the ontology-free case there is no difference between generalized interpolant existence and interpolant existence. In fact, also with ontologies generalized interpolant existence and interpolant existence are interreducible.
Lemma 18. Let L ∈ DL nr . There are mutual polynomial time reductions between generalized L-interpolant existence and L-interpolant existence.
Proof. The reduction from L-interpolant existence to generalized L-interpolant existence is trivial: For the converse reduction from generalized interpolant existence to interpolant existence, assume that an L-ontology O, L-concepts C 1 , C 2 , and Σ are given. Then there exists an L(Σ)-interpolant for where O Σ and C 2Σ are obtained from O and C 2 by replacing every non-Σ symbol uniformly by a fresh symbol. The latter is an instance of L-interpolant existence. ❏ Note that the reduction above works for all standard DLs including ALC.
Recall that interpolant existence reduces to checking O 1 ∪ O 2 |= C 1 ⊑ C 2 for logics with the CIP. Hence, for DLs which enjoy the CIP such as ALC, interpolant existence and generalized interpolant existence are Ex-pTime-complete and ontology-free interpolant existence and generalized ontology-free interpolant existence are PSpace-complete. We next introduce the relevant definition existence problems. We also consider the (projective) L-definition existence problems with empty ontologies, called ontology-free (projective) L-definition existence, and with RI-ontologies, called RI-ontology (projective) L-definition existence, both defined in the obvious way. Similar to the case of interpolant existence, definition existence reduces to checking implicit definability for logics with the PBDP. We provide model-theoretic characterizations for the non-existence of generalized interpolants and explicit definitions in terms of bisimulations.
Definition 20 (Joint consistency). Let L ∈ DL nr . Let O be an Lontology, C 1 , C 2 be L-concepts, and Σ a signature. Then C 1 and C 2 are called jointly consistent under O modulo L(Σ)-bisimulations if there exist pointed interpretations I 1 , d 1 and The associated decision problem, joint consistency modulo L-bisimu-lations, is defined in the expected way. The following result characterizes the existence of interpolants using joint consistency modulo L(Σ)-bisimulations. The proof uses Lemma 3.
Theorem 21. Let L ∈ DL nr . Let O be an L-ontology, C 1 , C 2 be L-concepts, and Σ a signature. Then the following conditions are equivalent: Proof. The proof is standard and we refer the reader to [40] for similar proofs. We only provide a sketch.
Then O |= D ⊑ C 2 , for any D ∈ Γ . As Γ is closed under conjunction and by compactness (recall that ALCHOI u is a fragment of first-order logic), there exists a model J of O and an element d ∈ ∆ J such that Then, using again compactness, there exists a model I of O and an element e ∈ ∆ I such that e ∈ C I 1 and e ∈ D I for all D ∈ t J (d). Thus I, e ≡ L,Σ J , d. For every interpretation I there exists an ω-saturated elementary extension I ′ of I [24]. Thus, it follows from the fact that ALCHOI u is a fragment of first-order logic that we may assume that both I and J are ω-saturated. By Lemma 3, I, e ∼ L,Σ J , d.
Assume that Condition 2 holds, that is, there are models I 1 and I 2 of O and d i ∈ ∆ I i for i = 1, 2 such that d 1 ∈ C I 1 1 and d 2 ∈ C I 2 2 and I 1 , d 1 ∼ L,Σ I 2 , d 2 . Then, by Lemma 3, I 1 , d 1 ≡ L,Σ I 2 , d 2 . But then from d 1 ∈ C I 1 we obtain d 1 ∈ D I 1 and so d 2 ∈ D I 2 which implies d 2 ∈ C I 2 2 , a contradiction.  Figure 4 (where we set a I i = a and b I i = b, for i = 1, 2) show that C 1 and ¬C 2 are jointly consistent modulo ALCO(Σ)-bisimulations. By extending the bisimulation in Figure 4 to a relation S such that (b I 1 , a I 2 ) ∈ S (so that the domain and range of S contain ∆ I 1 and ∆ I 2 , respectively), one can show that C 1 and ¬C 2 are jointly consistent modulo ALCO u (Σ)-bisimulations. Moreover, by introducing an element e in I 2 so that (e, b I 2 ) ∈ r I 2 and (e, e) ∈ r I 2 , and further extending S by adding (a I 1 , e) ∈ S, it can be seen that C 1 and ¬C 2 are jointly consistent modulo ALCOI u (Σ)-bisimulations (and hence ALCHOI u (Σ)-bisimulations). ⊣ The following characterization of the existence of explicit definitions can be proved similarly to Theorem 21. Example 24. Consider O, C, and Σ from Example 9. The interpretations I 1 , I 2 depicted in Figure 5 show that C ⊓ ∃r.⊤ and C ⊓ ¬∃r.⊤ are jointly consistent under O modulo ALCH(Σ)-bisimulations. Note that the ALCH(Σ)-bisimulation in Figure 5 is also an ALCH u (Σ)-bisimula-tion, but it is not an ALCHI(Σ)-bisimulation, since e 1 has both an r 1 -and an r 2 -predecessor, whereas e 2 and e ′ 2 lack an r 2 -and an r 1 -predecessor, respectively. To repair this, we replace I 2 with an interpretation J that is obtained by taking the union of I 2 with a copy I 1 of I 1 , and further adding (d 1 , e 2 ) ∈ r J 2 and (d 1 , e ′ 2 ) ∈ r J 1 (where d 1 is the copy of d 1 in J ). Then, we extend the ALCH(Σ)-bisimulation in Figure 5 to a relation S that also connects the elements of I 1 with the respective copies in J . It can be seen that J is a model of O, Interpolant existence and explicit definition existence are closely linked. We use Theorems 21 and 23 to show the following reductions.
Lemma 25. Let L ∈ DL nr , O be an L-ontology, C, C 0 , C 1 , and C 2 be Lconcepts, and Σ a signature. Then the following conditions are equivalent: Conversely, the following conditions are also equivalent: Proof. We show the second equivalence. Assume that (1) does not hold.
To show that (2) does not hold, assume O |= C 1 ⊑ C 2 (otherwise we are done). By Theorem 21 there exist pointed interpretations I 1 , d 1 and shows that (2) does not hold by Theorem 23. The other direction is shown similarly. ❏ Hence we obtain the following corollary.
Theorem 26. Let L ∈ DL nr . Then there is a polynomial time reduction of projective L-definition existence to L-interpolant existence (and thus to generalized L-interpolant existence). Conversely, there is a polynomial time reduction of generalized L-interpolant existence (and thus Linterpolant existence) to projective L-definition existence if an oracle for L-subsumption is admitted. Both reductions also exist for the ontology-free case and for RI-ontologies.
We now formulate the main complexity results proved in this article.
Theorem 27. Let L ∈ DL nr . Then L-interpolant existence, generalized Linterpolant existence, and projective L-definition existence are all 2ExpTimecomplete.
It follows that interpolant existence and projective definition existence are one exponential harder than subsumption for logics in DL nr . Our lower bound proofs rely on the presence of ontologies. To understand the ontology-free case (and the case with RI-ontologies) we first recall from our introduction of DLs in DL nr above that for DLs with the universal role or with both inverse roles and nominals, the ontology can be encoded in a concept and so interpolant existence and projective definition existence are still 2ExpTime-complete with empty ontologies and RI-ontologies, respectively. For the remaining DLs in DL nr , interpolant existence and projective definition existence become coNExpTime-complete, however. Thus, less complex than with ontologies, but still harder than subsumption (which is PSpace-complete), under standard complexity theoretic assumptions. We have seen that with the exception of ALCO and ALCHO, all DLs in DL nr enjoy the non-projective Beth definability property. Hence checking the existence of a non-projective definition of a concept name is polynomial time reducible to subsumption checking and so ExpTime-complete in the presence of an ontology. The following result states that even for ALCO and ALCHO checking the existence of non-projective definitions of concept names is not harder than subsumption.
Interestingly, Theorem 29 is the only result where the lack of either the CIP of (P)BDP does not lead to an increase in complexity of the interpolant/explicit definition existence problem. We show Theorem 29 in the appendix provided as supplementary material as it uses techniques that are slightly different from our other main results. We next consider the non-projective explicit definability of nominals. We have seen in Example 13 above that for nominals even the nonprojective Beth definability property does not hold for any DL in DL nr .
In fact, the following result states that the non-projective definability of nominals is as hard as their projective definability.
Observe that the characterizations given in Theorems 21 and 23 provide mutual polynomial time reductions of generalized interpolant and definition existence to the complement of joint consistency modulo Lbisimulations. Hence, to prove Theorems 27 to 30, it suffices to prove the corresponding complexity bounds for joint consistency.
We finally discuss an interesting consequence for CI-interpolants. Let L be a DL in DL nr that does not admit RIs. The CI-interpolant existence problem in L is the problem to decide for L-ontologies O and O ′ whether there exists an L-CI-interpolant for O and O ′ .
Observe that the 2ExpTime upper bound is an immediate consequence of Point 1 of Theorem 28 as we can give a polynomial time reduction of CI-interpolant existence to ontology-free interpolant existence. Assume The 2ExpTime lower bound is proved in Section 7 (Lemma 44) by adapting the 2ExpTime lower bound proof for interpolant existence in L.

Upper Bound Proofs With Ontology
We show the double exponential upper bound of Theorem 27 (and thus of Theorem 30) using a new mosaic elimination procedure that decides joint consistency modulo L-bisimulations, for all L ∈ DL nr . Theorem 32. Let L ∈ DL nr . Then joint consistency modulo L-bisimulations is in 2ExpTime.
To motivate our approach, reconsider Example 22. Notice that in interpretations I 1 , I 2 witnessing joint consistency of C 1 and ¬C 2 , a I 1 is bisimilar to both b I 2 and d. Moreover, it can be easily verified that there are no witnessing interpretations where a I 1 is bisimilar to a single element in I 2 .
Using an ontology, one can extend this example so that a I 1 is enforced to be bisimilar to exponentially many elements in I 2 in any interpretations I 1 , I 2 witnessing joint consistency of two concepts (in fact, this will be the basis for showing the lower bound in the subsequent section). Thus, we cannot consider (pairs of) elements in isolation, but instead need to consider sets of elements. As usual in DLs, we abstract elements in interpretations by types, which syntactically describe the behavior of these elements by listing the relevant concepts that are satisfied there. Correspondingly, sets of elements are abstracted to sets of types. Since we need to coordinate two interpretations I 1 , I 2 , we thus consider mosaics, which are pairs (T 1 , T 2 ) of sets of types. The intuitive meaning of such a pair is that it describes collections of elements in two interpretations I 1 and I 2 which realize precisely the types in T 1 and T 2 , respectively, and are all mutually bisimilar. Naturally, not all possible mosaics (T 1 , T 2 ) can be realized in this way and the goal is to determine the realizable ones. For this task, we use an elimination procedure. We start with the set of all possible mosaics and drop the 'bad' ones until a fixed point is reached. We will see that the elimination conditions extend the conditions known from standard type elimination procedures in a relatively natural way to mosaics. Then, concepts C 1 , C 2 will be jointly consistent under an ontology O modulo bisimulations if there is a surviving mosaic (T 1 , T 2 ) such that C 1 is contained in some type in T 1 and C 2 is contained in some type in T 2 .
We will now formalize our approach and start by introducing the relevant notions. Assume L ∈ DL nr and consider an L-ontology O, L-concepts C 1 , C 2 , and a signature Σ. Let Ξ = sub(O, C 1 , C 2 ) denote the closure under single negation of the set of subconcepts of concepts in O, C 1 , C 2 . A Ξ-type t is a subset of Ξ such that there exists a model I of O and d ∈ ∆ I with t = tp Ξ (I, d), where is the Ξ-type realized at d in I. Let Tp(Ξ) denote the set of all Ξ-types. We remark that the number of Ξ-types is at most exponential in ||O|| + ||C 1 ||+||C 2 || and, moreover, the set of all Ξ-types can be computed in time exponential in ||O||+ ||C 1 ||+ ||C 2 || for all considered logics [8,1]. A mosaic is a pair (T 1 , T 2 ) of sets of types T 1 , T 2 ⊆ Tp(Ξ). For interpretations I 1 , I 2 and i ∈ {1, 2}, the mosaic defined by d ∈ ∆ I i in I 1 , I 2 is the pair for j = 1, 2. We say that a pair (T 1 , T 2 ) of sets T 1 , T 2 of types is a mosaic defined by I 1 , I 2 if there exists d ∈ ∆ I 1 ∪ ∆ I 2 such that (T 1, T 2 ) = (T 1 (d), T 2 (d)). Clearly, there are at most doubly exponentially many mosaics. The mosaic defined by a I 1 in I 1 , I 2 is (T 1 (a I 1 ), T 2 (a I 1 )), where ⊣ As announced above, the aim of the mosaic elimination procedure is to determine all mosaics (T 1 , T 2 ) such that all t ∈ T 1 ∪ T 2 can be realized in mutually L(Σ)-bisimilar elements of models I 1 , I 2 of O. In order to formulate the elimination conditions, we define several compatibility conditions between types and between mosaics, similar to the compatibility conditions that are used in standard type elimination procedures. Throughout the rest of the section, we treat the universal role u as a role name contained in Σ, in case L admits the universal role. Note that u − is equivalent to u, and that O |= r ⊑ u, for every role r.
Let t 1 , t 2 be Ξ-types. We call t 1 , t 2 u-equivalent if ∃u.C ∈ t 1 iff ∃u.C ∈ t 2 , for every ∃u.C ∈ Ξ. Notice that the condition is trivially satisfied if L does not admit the universal role. For a role r, we call t 1 , t 2 r-coherent for O, in symbols t 1 r t 2 , if t 1 , t 2 are u-equivalent and the following conditions hold for all roles s with O |= r ⊑ s: (1) if ¬∃s.C ∈ t 1 , then C ∈ t 2 and (2) if ¬∃s − .C ∈ t 2 , then C ∈ t 1 . Note that t r t ′ iff t ′ r − t. We lift the definition of r-coherence from types to mosaics (T 1 , for every t ∈ T i there exists a t ′ ∈ T ′ i such that t r t ′ , and if L admits inverse roles, then for every t ′ ∈ T ′ i , there is a t ∈ T i such that t r t ′ . , for every role r. Example 34. Consider again interpretations I 1 , I 2 from Example 22 and the types t 1 = tp Ξ (I 1 , a I 1 ), t 2 = tp Ξ (I 2 , b I 2 ), and t 3 = tp Ξ (I 2 , d). Then, t 1 r t 1 , t 2 r t 3 , and t 3 r t 3 . Moreover, the mosaic (T 1 , T 2 ) defined by a I 1 in I 1 , We are now in the position to formulate the mosaic elimination conditions. Let S ⊆ 2 Tp(Ξ) × 2 Tp(Ξ) be a set of mosaics. We call (T 1 , T 2 ) ∈ S bad if it violates one of the following conditions.
Σ-concept name coherence A ∈ t iff A ∈ t ′ , for every concept name A ∈ Σ and every t, t ′ ∈ T 1 ∪ T 2 ; Existential saturation for i = 1, 2 and ∃r.C ∈ t ∈ T i , there exists . For didactic purposes and because we need it later in Section 10, we first give the mosaic elimination procedure for logics L that do not admit nominals. The procedure starts with the set S 0 of all mosaics. Then obtain, for i ≥ 0, S i+1 from S i by eliminating all mosaics (T 1 , T 2 ) that are bad in S i . Let S * be where the sequence stabilizes. The elimination procedure decides joint consistency in the following sense.
We refrain from giving the proof of Lemma 35 since it will follow from Lemma 36 below. We note, however, that for L as in the lemma, Theorem 32 is an immediate consequence of the procedure: there are only double exponentially many mosaics, so the elimination terminates after at most double exponentially steps. It remains to observe that every elimination step can be executed in double exponential time. This relatively straightforward elimination procedure does not quite work in the presence of nominals. Intuitively, the reason is that in any two interpretations I 1 , I 2 , every nominal a is realized (modulo bisimulation) in exactly one mosaic. Now, if the set S contains several mosaics mentioning a, they possibly witness existential saturation of each other which, however, cannot be reflected in an interpretation. Thus, for the mosaic elimination procedure to work (in the sense of Lemma 35) one has to "guess" for every nominal a exactly one mosaic that describes a.
To formalize this idea, let us call a set S of mosaics good for nominals if for every individual name a ∈ sig(Ξ) and i = 1, 2 there exists exactly one t i a with {a} ∈ t i a ∈ (T 1 ,T 2 )∈S T i and exactly one pair (T 1 , T 2 ) ∈ S with t i a ∈ T i . Moreover, if a ∈ Σ, then that pair takes the form - 2. there exists a set S * of mosaics that is good for nominals and does not contain a bad mosaic, such that there exist (T 1 , T 2 ) ∈ S * and Ξ-types Proof. "1 ⇒ 2". Let I 1 , d 1 ∼ L,Σ I 2 , d 2 for models I 1 and I 2 of O such that d 1 , d 2 realize Ξ-types t 1 , t 2 and C 1 ∈ t 1 , C 2 ∈ t 2 . Let S * be the set of all mosaics defined by I 1 , I 2 . It is routine to show that no (T 1 , T 2 ) in S * is bad and that S * is good for nominals. Now, the mosaic (T 1 , T 2 ) defined by d I 1 in I 1 , I 2 witnesses Condition (2). "2 ⇒ 1". Suppose there exist a good set S * of mosaics and (S 1 , S 2 ) ∈ S * and Ξ-types s 1 ∈ S 1 , s 2 ∈ S 2 with C 1 ∈ s 1 and C 2 ∈ s 2 . Let I i , for i = 1, 2 be interpretations defined by setting: if L admits the universal role, then (S 1 , S 2 ) u (T 1 , T 2 ) and t, s i are u-equivalent} and for all Σ-roles s : Note that the interpretation of nominals is well-defined since S * is good for nominals.
We verify that interpretations I 1 and I 2 witness Condition (1). Claim 1. For i = 1, 2, all C ∈ Ξ, and all (t, p) ∈ ∆ I i , we have (t, p) ∈ C I i iff C ∈ t.
Proof of Claim 1. Let i ∈ {1, 2}. The proof is by induction on the structure of concepts in Ξ.  (2) if O |= r ⊑ s for some Σ-role s, then p s p ′ . Note that t, t ′ are thus also u-equivalent, so (t ′ , p ′ ) ∈ ∆ I i . We distinguish cases: • If r is a role name, then by definition of r I i , we have that (1) and (2) above imply (1') t ′ r 0 t and (2') if O |= r 0 ⊑ s for some Σ-role s, then p s p ′ . As before, we can then conclude that , p), (t ′ , p ′ )) ∈ r I i and (t ′ , p ′ ) ∈ D I i . By induction, the latter implies D ∈ t ′ . We distinguish cases: • If r is a role name, then by definition of r I i , t r t ′ and thus ∃r.D ∈ t. • If r = r − 0 is an inverse role, then by definition of r I i 0 , t ′ r 0 t. Thus, also ∃r − 0 .D ∈ t. This finishes the proof of Claim 1. Claim 1 implies that (s 1 , (S 1 , S 2 )) ∈ C I 1 1 and (s 2 , (S 1 , S 2 )) ∈ C I 2 2 . Claim 1 also implies that the type realized by (t, p) in I i is t, for all (t, p) ∈ ∆ I i . Since types are, by definition, realized in models of O, it follows that both I 1 and I 2 are models of O.
Proof of Claim 2. Clearly, R satisfies Condition [AtomC] due to Σ-concept name coherence. Condition [AtomI] follows from the fact that S * is good for nominals in case L admits nominals.
-If r = r − 0 is an inverse role, then by definition of r I 1 0 , we have (1) t 1 r 0 t and (2) for all Σ-roles s with O |= r 0 ⊑ s, we have p 1 s p. Since t ′ ∈ T 2 and p 1 r 0 p there is some t ′′ ∈ T ′ 2 with t ′′ r 0 t ′ . Thus, in particular, t ′′ is u-equivalent to t ′ (and thus to s 2 ), which implies (t ′′ , p 1 ) ∈ ∆ I 2 . The definition of r I 2 0 then implies that ((t ′′ , p 1 ), (t ′ , p)) ∈ r I 2 0 . It remains to note that the definition of R yields ((t 1 , p 1 ), (t ′′ , p 1 )) ∈ R.
This finishes the proof of Claim 2. By definition of R, we have ((s 1 , (S 1 , S 2 )), (s 2 , (S 1 , S 2 ))) ∈ R, and thus I 1 , (s 1 , (S 1 , S 2 )) ∼ L,Σ I 2 , (s 2 , (S 1 , S 2 )). ❏ It remains to argue that we can find in double exponential time a set S * as in Condition (2) of Lemma 36. We use a suitable variant of the elimination procedure described after Lemma 35.
Proof. Let L ∈ DL nr , and assume O, C 1 , C 2 , and Σ are given. We can enumerate in double exponential time the maximal good sets U ⊆ 2 T (Ξ) × 2 T (Ξ) by picking, for each nominal a ∈ sig(Ξ) and i = 1, 2, a type t i a , and a mosaic (T 1 , T 2 ) with t i a ∈ T i . In doing so, we make sure that ({t 1 a }, {t 2 a }) is selected in case a ∈ Σ. Crucially, there are only double exponentially many possibilities to make this choice. Remove all mosaics that mention a nominal and have not been selected. The resulting set is good for nominals.
Then we eliminate from any set U obtained in that process recursively all bad mosaics. Let S U ⊆ U be the largest fixpoint of that procedure. Then one can easily show that there exists a set S * satisfying Condition (2) of Lemma 36 iff there exists a set U that can be obtained by the process described above such that the largest fixpoint S U satisfies Condition (2) of Lemma 36. Since elimination terminates after double exponential time, and there are only double exponentially many possible choices for U , the lemma follows. ❏ Theorem 32 is a direct consequence of Lemmas 36 and 37.

Lower Bound Proofs With Ontology
The goal of this section is to provide the proofs of the lower bounds in Theorems 27, 30, and 31. We start with the former two. By Lemma 25 and Theorem 23, it suffices to consider joint consistency. We will provide two reductions: in Section 7.1, we provide the reduction for DLs in DL nr that admits nominals and, in Section 7.3, the one for DLs that admits role inclusions. In Section 7.2, we will investigate the shape of the interpolants / explicit definitions that arise in the preceding lower bound proof. In Section 7.4, we then show how to adapt the lower bound proof from Section 7.1 to the case of CI-interpolant existence. In all cases we reduce the word problem for languages recognized by exponentially space bounded alternating Turing machines, which we introduce next. where Q = Q ∃ ⊎ Q ∀ is a finite set of states partitioned into existential states Q ∃ and universal states Q ∀ . Further, Θ is the input alphabet and Γ is the tape alphabet that contains a blank symbol / ∈ Θ, q 0 ∈ Q ∀ is the initial state, and ∆ ⊆ Q × Γ × Q × Γ × {L, R} is the transition relation. We assume without loss of generality that the set ∆(q, a) := {(q ′ , a ′ , M ) | (q, a, q ′ , a ′ , M ) ∈ ∆} contains exactly two or zero elements for every q ∈ Q and a ∈ Γ . Moreover, the state q ′ must be from Q ∀ if q ∈ Q ∃ and from Q ∃ otherwise, that is, existential and universal states alternate. Acceptance of ATMs is defined in a slightly unusual way, without using accepting states. Intuitively, an ATM accepts if it runs forever on all branches and rejects otherwise. More formally, a configuration of an ATM is a word wqw ′ with w, w ′ ∈ Γ * and q ∈ Q. We say that wqw ′ is existential if q is, and likewise for universal. Successor configurations are defined in the usual way. Note that every configuration has exactly zero or two successor configurations. A computation tree of an ATM M on input w is a (possibly infinite) tree whose nodes are labeled with configurations of M such that the root is labeled with the initial configuration q 0 w; -if a node is labeled with an existential configuration wqw ′ , then it has a single successor which is labeled with a successor configuration of wqw ′ ; -if a node is labeled with a universal configuration wqw ′ , then it has two successors which are labeled with the two successor configurations of wqw ′ .
An ATM M accepts an input w if there is a computation tree of M on w. Note that we can convert any ATM M in which acceptance is based on accepting states to our model by assuming that M terminates on any input and then modifying it to enter an infinite loop from the accepting states. It is well-known that there are 2 n -space bounded ATMs which recognize a 2ExpTime-hard language [23], where n is the length of the input w.

DLs with Nominals
We start with DLs supporting nominals. By Theorem 23, it suffices to prove the following result. As announced, we reduce the word problem for 2 n -space bounded ATMs. Let us fix such an ATM M = (Q, Θ, Γ, q 0 , ∆) and an input w = a 0 . . . a n−1 of length n. We first provide the reduction for L = ALCO using an ontology O and a signature Σ such that O contains concept names that are not in Σ and uses two role names r, s, and show later how to adapt this proof to Σ = sig(O) \ {b} and DLs supporting inverses and/or the universal role.
The idea of the reduction is as follows. We aim to construct an ontology O such that M accepts w iff {b} and ¬{b} are jointly consistent under O modulo L(Σ)-bisimulations, where  Figure 6, where the trees T * and T i , i ≥ 0 starting in b I and on the path elements, respectively, are also mutually ALC(±)-bisimilar. These trees shall represent the computation tree of M on input w (using symbols from Σ) as follows, cf. Figure 7 which shows the skeleton of a single tree T i . Configurations of M are represented as paths of length 2 n over a role s in which every element is labeled with a symbol A σ , σ ∈ Γ ∪ (Q × Γ ) that represents the content of a single tape cell (omitted in the figure for the sake of readibility). In Figure 7, the start of a configuration is indicated by • and the s-path between consecutive • has length 2 n . Every configuration is marked as existential or universal using concept names B ∀ , B 1 ∃ , B 2 ∃ ; the superscript · 1 /· 2 indicates which successor is chosen for an existential configuration. Existential configurations have a single successor configuration and universal configurations have two successor configurations.  The structure of this computation tree can easily be enforced in ALC using standard techniques (as we detail below). The difficulty is to achieve synchronization between successor configurations in the tree. That is, if a configuration c in the computation tree is followed by another configuration c ′ , then c ′ is actually a successor configuration of c according to M . To achieve this, we first ensure that in T i , for i ≤ 2 n , the (2 n − i)-th cell of each configuration in the computation tree is synchronized with the (2 n −i)-th cell of the next configuration(s), as indicated by the dotted lines in Figure 7. This can be realized in ALC using a set of concept names not in Σ. Then we exploit the fact that the trees T i are mutually ALC(±)bisimilar which implies that in all T i all cells of all configurations are synchronized. In more detail, we do this by using several counters modulo 2 n as follows. The first counter counts modulo 2 n along the path ρ using concept names not in Σ. As announced, each point of ρ starts an infinite tree along role s that is supposed to mimick the computation tree of M on input w. Along this tree, two more counters are maintained: one counter starting at 0 and counting modulo 2 n , and another counter starting at the value of the counter on ρ and also counting modulo 2 n .
The first counter is used to divide the tree into configurations of length 2 n and the second counter is used to link the (2 n − i)-th cell of successive configurations in T i as described above. We will next provide the concept inclusions in O in more detail. The counter along ρ is realized using concept names A i , 0 ≤ i < n and by including the following (standard) concept inclusions, for every i with 0 ≤ i < n: Using again the concept name I s , we start the s-trees with two counters, realized using concept names U i and V i , 0 ≤ i < n, and initialized to 0 and the value of the A-counter, respectively, by including the following concept inclusions for every j with 0 ≤ j < n: Here, (U = 0) is an abbreviation for the concept n−1 i=0 ¬U i ; we use similar abbreviations below without further notice. The counters U i and V i are incremented along s in the same way as A i is incremented along r, so we omit details. Configurations of M are represented between two consecutive points having U -counter value 0. We next enforce the structure of the computation tree (recall that q 0 ∈ Q ∀ ): These concept inclusions enforce that all points which represent a configuration satisfy one of B ∀ , B 1 ∃ , B 2 ∃ indicating the kind of configuration and, if existential, also a choice of the transition function. The symbol Z ∈ Σ enforces the branching.
We next set the initial configuration, for input w = a 0 , . . . , a n−1 .
To coordinate successor configurations, we associate with M functions f i , i ∈ {1, 2} that map the content of three consecutive cells of a configuration to the content of the middle cell in the i-the successor configuration (assuming an arbitrary order on the set ∆(q, a), for all q, a).
In what follows, we ignore the corner cases that occur at the border of configurations; they can be treated in a similar way. Clearly, for each pos- is true at an element a of the computation tree iff a is labeled with A σ 1 , an s-successors b of a is labeled with A σ 2 , and an s-successors c of b is labeled with A σ 3 . In each configuration, we synchronize elements with V -counter 0 by including for every (σ 1 , σ 2 , σ 3 ) and i ∈ {1, 2} the following concept inclusions: At this point, the importance of the superscript in B * ∃ becomes apparent: since different cells of a configuration are synchronized in different trees T k the superscript makes sure that all trees rely on the same choice for existential configurations. The concept names A i σ are used as markers (not in Σ) and are propagated along s for 2 n steps, exploiting the V -counter. The superscript i ∈ {1, 2} determines the successor configuration that the symbol is referring to. After crossing the end of a configuration, the symbol σ is propagated using concept names A ′ σ (the superscript is not needed anymore because the branching happens at the end of the configuration, based on Z).
For those (q, a) with ∆(q, a) = ∅, we add the concept inclusion A q,a ⊑ ⊥.
The following lemma establishes correctness of the reduction. Proof. "1 ⇒ 2". If M accepts w, there is a computation tree of M on w. We construct a single interpretation I with I, b I ∼ ALCO,± I, d for some d = b I as follows. Let J be the infinite tree-shaped interpretation that represents the computation tree of M on w as described above, that is, configurations are represented by sequences of 2 n elements linked by role s and labeled by B ∀ , B 1 ∃ , B 2 ∃ depending on whether the configuration is universal or existential, and in the latter case the superscript indicates which choice has been made for the existential state. Finally, the first element of the first successor configuration of a universal configuration is labeled with Z. Observe that J interprets only the symbols in Σ as nonempty. Now, we obtain interpretations I k , k < 2 n from J by interpreting non-Σ-symbols as follows: the root of I k satisfies I s ; -the U -counter starts at 0 at the root and counts modulo 2 n along each s-path; -the V -counter starts at k at the root and counts modulo 2 n along each s-path; -the auxiliary concept names of the shape A i σ and A ′ σ are interpreted in a minimal way so as to satisfy the concept inclusions starting from concept inclusion ( †). Note that, by definition of these concept inclusions, there is a unique result. Now obtain I from J and the I k by creating an infinite outgoing r-path ρ from some element d = b I (with the corresponding A-counter) and adding I k , k < 2 n to every element with A-counter value k on the r-path, identifying the roots of the I k with the element on the path. Additionally, include (b I , b I ) ∈ r I and add J to I by identifying b I with the root of J . It should be clear that I is as required. In particular, the reflexive, transitive, and symmetric closure of all pairs (b I , e), with e on ρ, and all pairs (e, e ′ ), with e in J and e ′ a copy of e in some tree I k is an ALCO(±)-bisimulation S on I with (b I , d) ∈ S. "2 ⇒ 1". Assume that I, b I ∼ ALCO,± J , d for some d = b J . As argued above, due to the r-self loop at b I , from d there has to be an outgoing infinite r-path on which all s-trees are ALCO(±)-bisimilar. Since I is a model of O, all these s-trees are additionally labeled with some auxiliary concept names not in Σ, depending on the distance from their roots on ρ. Using the concept inclusions in O and the arguments given in their description, it can be shown that all s-trees contain a computation tree of M on input w (which is solely represented with concept names in Σ). ❏ The same ontology O can be used for the remaining DLs with nominals. For ALCO ⊓ , exactly the same proof works; in particular, note that both the bisimulation S constructed in "1 ⇒ 2" and its inverse are surjective. For the DLs with inverse roles the (one-way) infinite r-path ρ has to replaced by a two-way infinite path in "1 ⇒ 2". Proof. "1 ⇒ 2". We modify the interpretation I defined in the proof of Lemma 39 in such a way that we obtain a model of O ′ and such that the ALCO(±)-bisimulation S on I defined in that proof is, in fact, an ALCO(± ′ )-bisimulation on the new interpretation. Formally, obtain I ′ from I by interpreting every r E , E ∈ sig(O) \ Σ as follows: (i) there is an r E -edge from e to b I , for all e ∈ E I ; (ii) there is an r E -edge from e to all elements on the path ρ, for all (e, e ′ ) ∈ S and e ′ ∈ E I ; (iii) there are no more r E -edges.
Note that, by (i), I ′ is a model of O ′ . By (ii), the relation S defined in the proof of Lemma 39 is an ALCO(± ′ )-bisimulation. In particular, by (i), elements e ′ ∈ E I have now an r E -edge to b I , so any element e bisimilar to e ′ , that is, (e, e ′ ) ∈ S, needs an r E -successor to some element bisimilar to b I . Since all elements on the path ρ are bisimilar to b I , these r E -successors exist due to (ii). "2 ⇒ 1". This direction remains the same as in the proof of Lemma 39. ❏ The extension to DLs with inverse roles and the universal role and the restriction to a single role name are again straightforward.
We conclude the section with an observation that will be relevant for the application of our results to modal logic in Section 11. More specificially, we strengthen the lower bound for the case of L = ALCO ⊓ as follows: replacing every subconcept of the shape ∃r.C with ∃r.(X r ⊓ C) and replacing every subconcept of the shape ∃s.C with ∃r.(X s ⊓ C), for fresh concept names X r , X s , and set Σ ′ = Σ ∪ {X r , X s }. It is routine to verify that Lemma 39 holds for O ′ , Σ ′ instead of O, Σ. In particular, we can obtain an interpretation I ′ from I as constructed in "1 ⇒ 2" as follows.
replace all s-connections by r-connections; -every element that has an s-predecessor in I satisfies X s in I ′ , that is, X I ′ s = (∃s − .⊤) I ; -b I and every element on the infinite r-path ρ in I satisfy X r in I ′ , that is, X I ′ r = (∃r.⊤) I (the root of the infinite path has to satisfy X r since it is bisimilar to b I which satisfies X r ). ❏

Shape of Explicit Definitions in the Lower Bound
The goal of this subsection is to provide some intuition on the shape of the explicit definitions that arise in the proof of Lemma 38. We note first that r(x, x) is an explicit FO(Σ)-definition of {b} under O, regardless of whether the ATM accepts its input or not. This means that interpolant and explicit definition existence is 2ExpTime-hard even under the promise that a fixed FO-definition / FO-interpolant exists. We now analyze the ALCO(±)-definitions that arise in the proof of Lemma 38. Recall that such a definition exists iff the ATM M does not accept its input w. So, for the rest of the dicussion we assume the latter.
Instead of directly providing an explicit ALCO(±)-definition of {b}, we give a definition C ¬b of ¬{b}, since the definition of C ¬b is close to the intuitions provided in the proof of Lemma 38. Obviously, ¬C ¬b will be the desired definition of {b}. Let n be the length of the input word w and let k = |Γ ∪ (Q × Γ )| be the number of possible labelings of a cell in some configuration of the ATM. Moreover, set K = k 2 n + 2 n .
The concept C ¬b takes the shape To understand the structure ∃r.⊤ → C ′ of C ¬b , recall that the proof of Lemma 38 relies on the assumption that an element d = b I has an rsuccessor. The concepts C tree , C stop , C start , C i provide an "approximation" of an accepting computation tree of the ATM M on its input w in the following sense. (Note that the definition of ¬{b} cannot describe the full accepting computation since it is not entailed). The concept C tree enforces an s-tree of depth K that acts as the skeleton for encoding (an initial fragment of) a computation tree. It is labeled with concepts Z, B ∀ , B 1 ∃ , B 2 ∃ in the expected way. Formally, C tree is where Path m s,X is a concept that enforces an s-path of length m with each element labeled with X. We refrain from giving the precise definitions of the remaining concepts, and rather provide the intuitions. C start is a concept that enforces the initial configuration to be true in the computation tree, and C stop is a concept that is true if some element within K s-steps is labeled with a concept name A q,a for which ∆(q, a) = ∅. Moreover, each C i is a concept with O |= I s ⊓ (A = i) ⊑ C i ; recall that we denote with (A = i) that the A-counter has value i. The disjunction over all possible C i in C ¬b is needed since the A-counter can take any value between 0 and 2 n − 1 at a given element in d = b I . More precisely, each C i is a conjunction where ⊕ m denotes addition modulo m, and for each m with 0 ≤ m < 2 n , C m sync is a concept that coordinates the content of the m-th cell in every configuration in the computation tree with the same cell in the successor configuration(s). This can be easily realized using value restrictions ∀s.
Observe that O |= ¬{b} ⊑ C ¬b regardless of whether the ATM accepts w or not. In particular, in every model of O, each element d satisfying ¬{b} ⊓ ∃r.⊤ satisfies the concepts C tree , C start , and ¬C stop . Moreover, d satisfies I s and (A = i) for some i, and thus d also satisfies C i .
For the converse, O |= C ¬b ⊑ ¬{b}, suppose that C ¬b is realizable in a model I of O in an element d with (d, d) ∈ r I . We thus also have d ∈ (C tree ⊓ C start ⊓ ¬C stop ) I , and d ∈ C I i , for some i. Due to the r-self loop, d ∈ (C m sync ) I , for all m with 0 ≤ m < 2 n . But this means that at d starts the initial segment of a computation tree of M which is not labeled with a halting configuration, and all of whose cells are coordinated with the corresponding cell of the successor configuration(s). By the choice of K, on every path there is a configuration that occurs twice. We can thus extend the initial fragment of the computation tree to an infinite computation tree for the word w, in contradiction to the fact that M does not accept w.
We conclude with observing that the size of the definition C ¬b of ¬{b} is double exponential in the length n of the input word, due to the depth K of the enforced tree. This is in stark contrast with the (constant!) size of the FO(Σ)-definition. We conjecture that one can enforce explicit definitions of triple exponential size. For example, when using two roles s 1 , s 2 instead of s for encoding the computation tree, already the concept C tree will be of triple exponential size. We leave a detailed analysis for future work.

DLs with Role Inclusions
By Theorem 23, it suffices to prove the following. As in the proof of Lemma 38, we reduce the word problem for exponentially space bounded ATMs, so let M be a 2 n -space bounded ATM and w = a 0 . . . a n−1 an input of length n. In fact, the only difference to the proof of Lemma 38 is the way in which we enforce that exponentially many elements are L(Σ)-bisimilar. We first provide the reduction for L = ALCH and The symbols s, Z, B ∀ , B 1 ∃ , B 2 ∃ and A σ , σ ∈ Γ ∪ (Q × Γ ), play exactly the same role as above. The main difference is that we replace the nominal b by an r-chain of length n. The ontology O contains the RIs r ⊑ r 1 , r ⊑ r 2 and the CI ¬∃r n .⊤ ⊓ ∃r n 1 .⊤ ⊑ R. As usual ∃r n abbreviates a sequence of n times ∃r.
To see how we use these inclusions, suppose there exist models I and J of O and d ∈ ∆ I , e ∈ ∆ J such that then it follows that e ∈ R J : due to I, d ∼ ALC,Σ J , e and d ∈ (∃r n 1 .⊤) I , we also have e ∈ (∃r n 1 .⊤) I . Let now d ′ be an element reachable from d via an r-path of length n (which exists due to d ∈ (∃r n .⊤) I ). Since r ⊑ r i for i = 1, 2, there are also arbitrary r 1 /r 2 -paths of length n from d to d ′ . Since I, d ∼ ALC,Σ J , e, there are also arbitrary r 1 /r 2 -paths of length n starting in e and whose end points are all ALCH(±)-bisimilar to d ′ and thus also mutually ALCH(±)-bisimilar. The concept name R will enforce that ( * ) the end point of any r 1 /r 2 -path of length n starting in e carries a counter value that describes the path in a canonical way.
We can thus use these 2 n different, but bisimilar end points to start the infinite trees which mimick the computation tree of M as in the proof of Lemma 38. Along these we maintain the same two counters as there: one counter starting at 0 and counting modulo 2 n to divide the tree into configurations of length 2 n ; -another counter starting at the value of the counter on the leaf and also counting modulo 2 n .
Formally, the ontology O is constructed as follows. In order to realize ( * ) above, we use concept names A i , 0 ≤ i < n realizing the counter and the following concept inclusions: Using the concept name L R , we start the s-trees with two counters, realized using concept names U i and V i , 0 ≤ i < n, and initialized to 0 and the value of the A-counter, respectively: The structure of the computation tree, the initial configuration, and the coordination between consecutive configurations is done using the same concept inclusions as in the proof of Lemma 38, starting from inclusion ( †) and replacing I s with L R . We can then prove the following very similarly to Lemma 39. Proof. "1 ⇒ 2". If M accepts w, there is a computation tree of M on w. We construct a single interpretation I with I, d ∼ ALCH,± I, e for some d, e with d ∈ (∃r n .⊤) I and e / ∈ (∃r n .⊤) I as follows. Let J be the infinite tree-shaped interpretation that represents the computation tree of M on w as described above, that is, configurations are represented by sequences of 2 n elements linked by role s and labeled by B ∀ , B 1 ∃ , B 2 ∃ depending on whether the configuration is universal or existential, and in the latter case the superscript indicates which choice has been made for the existential state. Finally, the first element of the first successor configuration of a universal configuration is labeled with Z. Observe that J interprets only the symbols in Σ as non-empty. Now, we obtain interpretations I k , k < 2 n from J by interpreting non-Σ-symbols as follows: the root of I k satisfies L R ; -the U -counter starts at 0 at the root and counts modulo 2 n along each s-path; -the V -counter starts at k at the root and counts modulo 2 n along each s-path; -the auxiliary concept names of the shape A i σ and A ′ σ are interpreted in a minimal way so as to satisfy the concept inclusions that enforce the coordination between consecutive configurations (cf. the concept inclusions in proof of Lemma 38). Now obtain I from J and the I k as follows: First, create a path of length n from some element d so that consecutive elements are connected with r, r 1 , r 2 , and identify the end point of the path with the root of J . Then create a binary tree of depth n, rooted in e, in which left children are always r 1 -successors and right children are always r 2 -successors. Label the nodes of the tree with R i and A j as described above and identify the leaf having A-counter value k with the root of I k , for all k < 2 n . I is as required since, by construction, d ∈ (∃r n .⊤) I , e / ∈ (∃r n .⊤) I , and the reflexive, transitive, and symmetric closure of all pairs (d ′ , e ′ ) such that d ′ has distance ℓ ≤ n from d and e ′ has distance ℓ from e, and all pairs (b, b ′ ), with b in J and b ′ a copy of b in some tree I k is an ALCH(±)-bisimulation S on I with (d, e) ∈ S. "2 ⇒ 1". Assume that I, d ∼ ALCH,± J , e for models I, J of O and some d, e with d ∈ (∃r n .⊤) I and e / ∈ (∃r n .⊤) J . As argued above, there are r 1 /r 2 -paths of length n whose end points carry all possible counters < 2 n and are all ALCH(±)-bisimilar. In addition, all these end points root s-trees which are ALCH(±)-bisimilar. Since J is a model of O, all these s-trees are additionally labeled with some auxiliary concept names not in Σ, depending on the value of the A-counter of the corresponding leaf. Using the concept inclusions in O and the arguments given in their description, it can be shown that all s-trees contain a computation tree of M on input w (which is solely represented with concept names in Σ). ❏ The same proof works as well for ALCH ⊓ as the relation S constructed in the direction "1 ⇒ 2" above is actually an ALCH ⊓ (±)-bisimulation. For L ∈ {ALCHI, ALCHI u }, we have to slightly adapt the model construction in "1 ⇒ 2", following the idea provided in Example 24 (except that we do not need to take the union of I 1 , I 2 here, since we construct a single interpretation I = I 1 = I 2 ). Let d 0 , . . . , d n be the elements on the r-path that starts in d, that is, d 0 = d and d ℓ has distance ℓ from d. Recall that (d ℓ , e ′ ) ∈ S for every element e ′ in level ℓ in the binary tree rooted at e. Observe that S is not an L(Σ)-bisimulation since, for ℓ > 0, d ℓ has both an r 1 and an r 2 -predecessor (both are d ℓ−1 ), but elements in the binary tree lack either an r 1 -or an r 2 -predecessor. To repair this, we add for every element e ′ in level ℓ > 0 in the binary tree the following connections: It can be verified that the modified interpretation is still a model of O, and that S is an L(Σ)-bisimulation as required.
We conclude the section by remarking that one can analyze the structure of the explicit ALCH(±)-definitions that arise in the proof of Lemma 42 along the lines of Section 7.2. In contrast to that section, the size of the FO-definition of ∃r n .⊤ under O is not constant, but depends on n. Lemma 45. Let L ∈ DL nr admit neither the universal role nor both inverse roles and nominals simultaneously. Let O be a set of RIs, C 1 , C 2 L-concepts, and Σ a signature. If C 1 and C 2 are jointly consistent under O modulo L(Σ)-bisimulations, then there exist pointed interpretations I 1 , d 1 and I 2 , d 2 with I 1 , I 2 models of O and of at most exponential size in ||O|| + ||C 1 || + ||C 2 || such that d 1 ∈ C I 1 1 , d 2 ∈ C I 2 2 , and I 1 , d 1 ∼ L,Σ I 2 , d 2 .

CI-interpolant Existence
Before we prove Lemma 45, we introduce some notation. The depth of a concept C is the number of nestings of existential restrictions in C. For instance, a concept name has depth 0 and ∃r.∃r.B has depth 2. Given the ontology O, concepts C 1 , C 2 , and the signature Σ, we use the notation introduced in Section 6. For instance, the set of concepts Ξ, Ξ-types t, and mosaics (T 1 , T 2 ) are defined as in Section 6. While in Section 6 we used the relation r between mosaics to guide the construction of interpretations, here we use a relation between mosaics that is directly induced by interpretations. Assume interpretations I 1 and I 2 are given. Consider mosaics p = (T 1 (d), T 2 (d)) and q = (T 1 (d ′ ), T 2 (d ′ )) such that there exists a role name r ∈ Σ with (d, d ′ ) ∈ r I i , for some i ∈ {1, 2}. Then define, for every role name s and i ∈ {1, 2}, relations R s,i p,q ⊆ T i (d) × T i (d ′ ) by setting (t, t ′ ) ∈ R s,i p,q if there exist e and e ′ realizing t and t ′ , respectively, with (T 1 (e), T 2 (e)) = p and (T 1 (e ′ ), T 2 (e ′ )) = q, such that (e, e ′ ) ∈ s I i .
Now assume that C 1 and C 2 are jointly consistent under O modulo L(Σ)-bisimulations. By definition, there exist pointed models I 1 , d 1 and , and I 1 , d 1 ∼ L,Σ I 2 , d 2 . Let k be the maximum depth of C 1 , C 2 .
We start with the case involving nominals and without inverse roles. We construct exponential size J 1 , J 2 with the same properties as I 1 , I 2 above. Intuitively, J i is obtained via a suitable unraveling operation up to the depth k of the concepts C 1 , C 2 ; during the unraveling, we take care of the nominals and, moreover, restrict the outdegree of the produced interpretation by keeping only necessary successors. Formally, let B be some minimal set of mosaics defined by I 1 , I 2 such that -B contains every mosaic generated by some nominal, or formally, (T 1 (d), T 2 (d)) ∈ B for every d ∈ ∆ I i such that d = a I i for some nominal a ∈ sig(C i ); -for every type t realized in I i there exists (T 1 , T 2 ) ∈ B with t ∈ T i . Intuitively, B serves to describe the behavior of the root of the unraveling (first item), of the nominals (second item), and of potential witnesses for existential restrictions for non-Σ-roles (third item). Observe that the size of B is at most exponential in the size of O, C 1 , C 2 . To restrict the outdegree, select, for any mosaic p = (T 1 , T 2 ) defined by I 1 , I 2 and any ∃s.C ∈ t ∈ T i such that there exists r ∈ Σ with O |= s ⊑ r, a mosaic q = (T ′ 1 , T ′ 2 ) such that (t, t ′ ) ∈ R s,i p,q and C ∈ t ′ , and denote the resulting set by S(p). Form the set T of sequences with j ≤ k, p 0 ∈ B and p i+1 ∈ S(p i ) for i < j. Let tail(σ) = p j and tail i (σ) = T j i . We next define the domain of J 1 and J 2 as , |σ| > 1, t contains no nominal} and define the interpretation of individual, concept and role names in J 1 , J 2 in the expected way: for any individual name a and (T 1 , T 2 ) ∈ B with {a} ∈ t ∈ T i , we set a J i = (t, (T 1 , T 2 )); -for any concept name A, (t, σ) ∈ A J i iff A ∈ t; -for any role name r we let for σp ∈ T , • ((t, σ), (t ′ , σp)) ∈ r J i if (t, t ′ ) ∈ R r,i tail(σ),p and t ′ contains no nominal; tail(σ),p and t ′ contains a nominal. Next assume that tail(σ) = (T 1 , T 2 ) and σ has length k. If tail(σ ′ ) = (T 1 , T 2 ) for some |σ ′ | < k, then choose as r-successors of any element of the form (t, σ) exactly the r-successors of (t, σ ′ ) defined above. If no such σ ′ exists, then all elements of the form (t, tail(σ)) have distance exactly k from the roots (since no nominal occurs in any type in any mosaic in σ) and no successors are added. It remains to take care of existential restrictions ∃r.C for the role names r that do not entail any role name in Σ. If σ ∈ T , ∃r.C ∈ t ∈ T i with tail i (σ) = T i and O |= r ⊑ s for any s ∈ Σ, we add ((t, σ), (t ′ , p)) to r J i (and all s J i with O |= r ⊑ s) for some p = (T ′ 1 , T ′ 2 ) ∈ B and t ′ ∈ T ′ i with C ∈ t ′ such that there are e, e ′ realizing t, t ′ in I i and (e, e ′ ) ∈ r I i .
The following example illustrates the construction of J 1 , J 2 using the interpretations I 1 , I 2 introduced in Example 22.
We ignore the types realized by b I 1 in I 1 and by a I 2 in I 2 as they are not relevant for understanding the construction. Then only the mosaic p = (T 1 , T 2 ) with T 1 = {t 0 } and T 2 = {t 1 , t 2 } remains and J 1 and J 2 are depicted in Figure 8. We show that J 1 , J 2 are as required. First, for i ∈ {1, 2}, J i |= O follows from the definition of J i and the fact that I i |= O. Indeed, given r ⊑ s ∈ O, let ((t, σ), (t ′ , σ ′ )) ∈ r J i . This means that (t, t ′ ) ∈ R r,i tail(σ),tail(σ ′ ) , that is, there exist e, e ′ realizing t and t ′ , respectively, with (T 1 (e), T 2 (e)) = tail(σ) and (T 1 (e ′ ), T 2 (e ′ )) = tail(σ ′ ), such that (e, e ′ ) ∈ r I i . Since I i |= O, we obtain that (e, e ′ ) ∈ s I i as well, and thus (t, t ′ ) ∈ R s,i tail(σ),tail(σ ′ ) , meaning that ((t, σ), (t ′ , σ ′ )) ∈ s J i . Hence, We next prove that, for every (t, σ) ∈ ∆ J i and every concept C ∈ Ξ of depth ≤ k − |σ|, The proof is by induction on the structure of C. We consider the case C = ∃r.D, where D has depth < k − |σ|. We can assume that |σ| < k, since for |σ| = k the claim holds trivially.
(⇒) Let (t, σ) ∈ ∃r.D J i . Then ∃r.D ∈ t follows by construction of r J i as we only have ((t, σ), (t ′ , σ ′ ) ∈ r J i if there are e, e ′ realizing t, t ′ in I i such that (e, e ′ ) ∈ r I i .
-There exists s ∈ Σ such that O |= r ⊑ s. Then there exists q = (T ′ 1 , T ′ 2 ) ∈ S(p) and t ′ ∈ T ′ i such that (t, t ′ ) ∈ R r,i p,q and D ∈ t ′ . We distinguish two cases.
as required.
We next consider the case with inverse roles, but without nominals. In this case, we let B be some minimal set of mosaics defined by I 1 , I 2 containing (T 1 (d 1 ), T 2 (d 1 )) and such that for every type t realized in I i there exists (T 1 , T 2 ) ∈ B with t ∈ T i . We extend the relations R s,i p,q defined previously to inverse roles s in the obious way and select for any mosaic p = (T 1 , T 2 ) and any ∃s.C ∈ t ∈ T i such that there exists a Σ-role r with O |= s ⊑ r a mosaic q = (T ′ 1 , T ′ 2 ) such that (t, t ′ ) ∈ R s,i p,q and C ∈ t ′ and denote the resulting set by S(p).
Form again the set T of sequences with j ≤ k, p 0 ∈ B and p i+1 ∈ S(p i ) for i < j. Let tail(σ) = p j and tail i (σ) = T j i . We next define the domain of J 1 and J 2 as We define interpretations J 1 , J 2 in the expected way.
-For any concept name A, (t, σ) ∈ A J i iff A ∈ t; -Let r be a role name. Then we let for σp ∈ T , tail(σ),p . -We still have to take care of existential restrictions ∃r.C with r a role that does not entail any Σ-role. If σ ∈ T , ∃r.C ∈ t ∈ T i with tail i (σ) = T i and O |= r ⊑ s for any Σ-role s, we add ((t, σ), (t ′ , p)) to r J i (and all s J i with O |= r ⊑ s) for some p = (T ′ 1 , T ′ 2 ) ∈ B and t ′ ∈ T ′ i with C ∈ t ′ such that there are e, e ′ realizing t, t ′ in I i and (e, e ′ ) ∈ r I i .
The fact that J i |= O, for i ∈ {1, 2}, is proved similarly to the case with nominals. One can also prove again by induction on the structure of C that for every (t, σ) ∈ ∆ J i and every C ∈ Ξ of depth ≤ k − |σ|, Next we observe that the relation is an ALCHI(Σ)-bisimulation. Indeed, it can be seen, similar to the case with nominals, that S satisfies [AtomC]. We now give a proof of [Forth]. We provide the proof for role names; the proof for inverse roles is similar.
Observe that again the models J i , i = 1, 2, are of at most exponential size in the size of O, C 1 , C 2 . We also have (T 1 (d 1 ), T 2 (d 1 )) ∈ B and so (tp Ξ (I 1 , d 1 ), as required.

Lower Bound Proofs without Ontology
In this section, we first show (the hardness part of) Points 1 and 2 of Theorem 28 by a reduction of the case with ontologies, and then show the nondeterministic exponential time lower bounds for Points 3 and 4 of that Theorem. Points 1 and 2 of Theorem 28 are a direct consequence of the following lemma. for all s ∈ {r, r − } with r ∈ sig(O, C, C 0 ) and b ∈ sig(O, C, C 0 ). Observe that if d ∈ (U ⊓ ∀r − 0 .F ) I for some interpretation I and concept F , then e ∈ F I holds for all elements e in ∆ I that can be reached in I from d or any b I with b ∈ sig(O, C, C 0 ) along roles in sig(O, C, C 0 ). It follows that for any L(Σ)-concept E, we have if τ (i, j) = t 1 and τ (i ⊕ 2 n 1, j) = t 2 , then (t 1 , t 2 ) ∈ H; -if τ (i, j) = t 1 and τ (i, j ⊕ 2 n 1) = t 2 , then (t 1 , t 2 ) ∈ V .
It is well-known that the problem of deciding whether there is a solution for given P and c is NExpTime-hard [9, Section 5.2.2]. For the following constructions, assume a tiling system P and an initial condition c of length n.
For the reduction for ALCO, we give concepts C, C 0 and a signature Σ such that here exist I 1 , d 1 ∼ ALCO,Σ I 2 , d 2 with d 1 ∈ (C ⊓ C 0 ) I 1 and d 2 ∈ (C ⊓ ¬C 0 ) I 2 iff P has a solution given c. We start with setting with a ∈ Σ and r ∈ Σ. In addition to r, Σ contains concept names B 0 , . . . , B 2n−1 that serve as bits in the binary representation of grid positions (i, j) with 0 ≤ i, j ≤ 2 n − 1, where bits B 0 , . . . , B n−1 represent the horizontal position i and B n , . . . , B 2n−1 the vertical position j, and concept names T 0 , . . . , T k representing tile types. We also use the following concept names that are not in Σ: another four sets of concepts names A 0 , . . . , A 2n−1 and V 0 , . . . , V 2n−1 with V ∈ {X, Y, Z} that also serve as bits in the binary representation of grid position (i, j) with 0 ≤ i, j ≤ 2 n − 1, and concept names R 0 , . . . , R 2n , M , M 1 , and M 2 . We now define the concept C as a conjunction of several concepts. The first conjunct is Intuitively, R 0 generates a binary r-tree of depth 2n with R i true at level i for 0 ≤ i ≤ 2 2n and each leaf represents a grid position (i, j) using the concept names A i . To achieve this let C contain the following conjuncts for generating the binary tree: As usual, ∀r i abbreviates a sequence of i times ∀r.
We next express using additional conjuncts of C that any leaf d representing (i, j) using A i has the following properties (A) -(C): (A) d has an r-successor representing (i, j) using B i with a tile type T (i,j) true in it; moreover, no r-successor of d representing (i, j) satisfies a tile type different from T (i,j) . This is achieved using the marker M which holds in exactly those r-successors of d that represent (i, j) using B i . The latter condition is expressed using the counter X i which represents (i, j) on all r-successors of d. In detail, we add the following conjuncts to C: (B) d has an r-successor representing (i ⊕ 2 n 1, j) using B i with a tile type T right (i,j) true in it such that (T (i,j) , T right (i,j) ) ∈ H; moreover, no r-successor of d representing (i ⊕ 2 n 1, j) satisfies a tile type different from T right (i,j) . This is achieved in a similar way as (A) using the marker M 1 which holds in exactly those r-successors of d that represent (i ⊕ 2 n 1, j) using B i . The latter condition is expressed using the counter Y i which represents (i ⊕ 2 n 1, j) on all r-successors of d. The implementation of these conditions is similar to (A) and omitted. (C) d has an r-successor representing (i, j ⊕ 2 n 1) using B i with a tile type T up (i,j) true in it such that (T (i,j) , T up (i,j) ) ∈ V ; moreover, no r-successor of d representing (i, j ⊕ 2 n 1) satisfies a tile type different from T up (i,j) . This is achieved in a similar way as (A) using the marker M 2 which holds in exactly those r-successors of d that represent (i, j ⊕ 2 n 1) using B i . The latter condition is expressed using the counter Z i which represents (i, j ⊕ 2 n 1) on all r-successors of d. The implementation is again similar to (A) and omitted.
Finally, we ensure that the initial condition holds, that is T (i,0) = c i for i < n. To this end we add the conjuncts for i < n, where A = (i, 0) stands for the representation of (i, 0) using A i ; for instance, A = (0, 0) stands for 0≤i<2n ¬A i . This finishes the definition of C, C 0 and we verify next that they are as required.
Claim. There exist I 1 , d 1 ∼ ALCO,Σ I 2 , d 2 with d 1 ∈ (C ⊓ C 0 ) I 1 and d 2 ∈ (C ⊓ ¬C 0 ) I 2 iff P has a solution given c. Proof of the Claim. Observe that if I 1 , d 1 ∼ ALCO,Σ I 2 , d 2 with d 1 ∈ (C ⊓ C 0 ) I 1 and d 2 ∈ (C ⊓¬C 0 ) I 2 , then there are elements e (i,j) , 0 ≤ i, j ≤ 2 n −1 such that I 1 , a I 1 ∼ ALCO,Σ I 2 , e (i,j) and e (i,j) has (at least) three r-successors satisfying Conditions (A) to (C) and the initial condition. By Σ-bisimilarity and since r ∈ Σ, all e (i,j) have r-successors satisfying the same concept names in Σ. Hence, since the concept names B i and T i are in Σ, for every grid position (i, j) every e (i ′ ,j ′ ) has an r-successor representing (i, j) using B i and all r-successors representing (i, j) using B i satisfy the same tile type T (i,j) . Moreover, T (i⊕ 2 n 1,j) = T right (i,j) and T (i,j⊕ 2 n 1) = T up (i,j) . It follows that the mapping τ defined by setting τ (i, j) = T (i,j) is a solution of P given c.
Conversely, assume that P and c have a solution τ . The definition of an interpretation I with elements d 1 and d 2 such that I, d 1 ∼ ALCO,Σ I, d 2 with d 1 ∈ (C ⊓ C 0 ) I and d 2 ∈ (C ⊓ ¬C 0 ) I is rather straightforward. An abstract version is depicted in Figure 10. We omit the counters, and note that a I and all elements at level R 2n have, for all 0 ≤ i, j < 2 n , an r-successor representing (using concept names B i ) grid position (i, j) which satisfies the concept name T τ (i,j) . We show only the three special successors from Conditions (A)-(C). This finishes the proof of the Claim and thus the reduction for ALCO.
We come to the lower bound for ALCH and ALCHI. Let and Σ contains r 1 , r 2 but not r nor v. In addition to r 1 and r 2 , Σ contains exactly the same concept names as in the ALCO proof and we also use the same concept names not in Σ. We aim to construct concepts C, C 0 such that there exist models I 1 , I 2 of O and d 1 ∈ (C ⊓ C 0 ) I 1 and d 2 ∈ (C ⊓ ¬C 0 ) I 2 with I 1 , d 1 ∼ ALCH,Σ I 2 , d 2 iff P has a solution given c.
We set C 0 = ∃r 2n .⊤. The concept C is again a conjunction of several concepts; we start in a similar way as for ALCO with The concept name R 0 will enforce that ( * * ) the end point of any r 1 /r 2 -path of length 2n starting in an element satisfying R 0 carries a pair of counter values (i, j) represented by concept names A i which describe the path in a canonical way. 2 To achieve this, we include the following conjuncts in C: Note that we can use the role name v to address all elements reachable along r 1 /r 2 -paths of length i via ∀v i . We continue the definition of C in exactly the same way as for ALCO except that we use ∀v 2n to reach the end points of the paths mentioned in ( * * ) and r 1 -successors of the leaves to encode a solution of the tiling problem. One can then easily prove the following.
Proof of the Claim. Observe that if I 1 , d 1 ∼ ALCH,Σ I 2 , d 2 with I 1 , I 2 models of O, d 1 ∈ (C ⊓ ∃r 2n .⊤) I 1 , and d 2 ∈ (C ⊓ ¬∃r 2n .⊤) I 2 , then there exists an element e reachable from d 1 along an r-path of length 2n in I 1 .
Since d 2 ∈ R I 2 0 and e is reachable via arbitrary r 1 /r 2 -paths of length 2n from d 1 , Property ( * * ) implies that there are elements e (i,j) , 0 ≤ i, j ≤ 2 n − 1, reachable from d 2 along a v-path of length 2n in I 2 such that I 1 , e ∼ ALCH,Σ I 2 , e (i,j) and e i,j represents the pair (i, j) using the concept names A i . The remaining proof is now essentially the same as for ALCO.
The converse direction is rather straightforward and similar to the proof for ALCO. The difference is that the binary tree over role r in the right side of interpretation I depicted in Figure 10 is now a binary tree over roles r 1 (left successor) and r 2 (right successor). This finishes the proof of the Claim.
To prove the claim above for ALCHI, we adapt the model construction in a similar way as in the case with ontologies (Section 7.3). More precisely, for each element e at level ℓ > 0 in the binary tree below d 2 , add (d, e) ∈ r I 1 and (d, e) ∈ r I 2 , where d is the element in distance ℓ − 1 from d 1 . One can then verify that I is as required, that is, I, d 1 ∼ ALCHI I, d 2 , d 1 ∈ (C ⊓ C 0 ) I , and d 2 ∈ (C ⊓ ¬C 0 ) I .

Computation Problem
In the previous sections, we have presented algorithms for deciding the existence of interpolants and explicit definitions, but these algorithms (and their correctness proofs) do not give immediately rise to a way of computing interpolants and explicit definitions in case they exist. Intuitively, this is due to the fact that compactness is used in the proof of the model-theoretic characterization of interpolant and explicit definition existence in terms of joint consistency modulo bisimulations which was provided in Theorems 21 and 23, respectively. In this section, we address the computation problem for logics in DL nr that do not admit nominals, by showing that we can actually compute interpolants in case they exist. We use DAG representation for the interpolants; recall that in DAG representation common sub-formulas are stored only once, and that thus DAG representation is more succinct than formula representation. Our approach is inspired by a recent note on a type elimination based computation of interpolants in modal logic [88] which was originally provided for the guarded fragment [15].
Theorem 48. Let L ∈ DL nr not admit nominals, O be an L-ontology, C 1 , C 2 be L-concepts, and Σ be a signature. Then, if there is an L(Σ)interpolant for C 1 ⊑ C 2 under O, we can compute the DAG representation of an L(Σ)-interpolant in time 2 2 p(n) where p is a polynomial and n = ||O|| + ||C 1 || + ||C 2 ||. and so using @ does not make any difference. Note, moreover, that it uses only a single role name r which corresponds to using a single modal operator.
(2) can be proved in the same way as (1) by observing that there is a bijection · m between ALCO u -concepts and ML u n -formulas, that |= C ⊑ D iff C m |= loc D m for any ALCO u -concepts C, D, and then applying Point 1 of Theorem 28. Note that the lower bound holds for a single role, see Lemma 41, which again translates to a single modal operator (and the universal modality). ❏ Description logics with RIs correspond to modal logics determined by Kripke models satisfying inclusions R i ⊆ R j between accessibility relations R i and R j . For any finite set I of pairs (i, j) let M I denote the class of Kripke models satisfying R i ⊆ R j for all (i, j) ∈ I. Define the consequence relation |= I loc in the usual way by setting ϕ |= I loc ψ if for all pointed models M, w with M ∈ M I , if M, w |= ϕ then M, w |= ψ. We then obtain the following complexity result directly from Points 4 and 2 of Theorem 28, respectively.
Theorem 53. For all finite I, the interpolant existence problem for |= I loc in ML is in coNExpTime. There exists a finite I such that the interpolant existence problem for |= I loc in ML is coNExpTime-hard. For all finite I, the interpolant existence problem for |= I loc in ML u is in 2ExpTime. There exists a finite I such that the interpolant existence problem for |= I loc in ML u is 2ExpTime-hard.
We close this section with a brief discussion of interpolant existence for the global consequence relation. We say that ϕ globally entails ψ, in symbols ϕ |= glo ψ, if for all models M from M |= ϕ it follows that M |= ψ. Call a formula χ a global interpolant for ϕ, ψ if sig(χ) ⊆ sig(ϕ) ∩ sig(ψ), ϕ |= glo χ and χ |= glo ψ. The global interpolant existence problem for L is the problem to decide for any ϕ, ψ ∈ L whether there exists a global interpolant for ϕ, ψ in L. It is straightforward to show that global interpolant existence corresponds to CI-interpolant existence in DLs in the same way as interpolant existence for the local consequence relation corresponds to ontology-free interpolant existence in DLs. We therefore obtain 2Exp-Time-completeness of global interpolant existence for the language ML u n from Theorem 31. We conjecture that the same result holds for global interpolant existence for ML n and ML @ but leave the proofs for future work.
We have investigated the problem of deciding the existence of interpolants and explicit definitions for description and modal logics with nominals and role inclusions, and we also presented an algorithm computing them for logics with role inclusions. There are many challenging problems left for future work, for instance, an algorithm computing interpolants for logics with nominals and the design and implementation of practical algorithms that could be applied in supervised concept learning and referring expression generation. From a theoretical viewpoint it would be of interest to gain a better understanding of when the existence of interpolants is computationally harder than entailment, for logics that do not enjoy the CIP. Logics to consider include more expressive DLs with nominals such as those also admitting qualified number restrictions and/or transitive roles and extensions of the two-variable fragment of FO with counting and/or further constraints on relations [53]. Another class of interest are decidable fragments of first-order modal logics and products of modal logics which both often do not enjoy the CIP [32,73]. Here it would be of interest to consider logics such as the one-variable or monodic fragments of K and S4 for which the complexity of interpolant existence was left open [61]. Finally, is it possible to prove general transfer results (for example, for families of normal modal logics) stating that decidable entailment implies decidability of interpolant existence?
roles, the image of S under f i is fully closed in I i . We now define interpretations J 1 and J 2 as the interpretation I except that A J i = f −1 i (A I i ), for i = 1, 2. Then the f i are ALCOI(Σ ∪ {A})-bisimulations between J i and I i , for i = 1, 2. Note, however, that the J i do not necessarily interpret all nominals, as for a nominal {a}, the element a I 1 might be in a different connected component than d 1 (equivalently: a I 2 is in a different connected component than d 2 ). To address this, let I ′ be the restriction of I 1 to ∆ I 1 \ f i (S). Then, obtain J ′ i by taking the disjoint union of J i and I ′ , i = 1, 2. It follows from the construction, the preservation properties of ALCOI-bisimulations, and the fact that the universal role is not used in O nor C that the following conditions hold: But then A is not implicitly Σ-definable under O and C, as required. Finally, to show Point (2), we have to argue that ALCO and ALCHO do not enjoy the BDP. A counterexample to the BDP is given in [89]. It illustrates nicely the way in which the addition of inverse roles or the universal role to ALCHO restores the BDP. As this is relevant in the proof of Point (3) and also in the analysis of non-projective definitions later in Section B, we give the example here. Let  Figure 11 (where a I = a in I and a J = a in J , and similarly for b) show that I, a I ∼ ALCO,Σ J , a J , with a I ∈ A I and a J ∈ A J . By Lemma 3, we have I, a I ≡ ALCO,Σ J , a J , and hence there cannot be any ALCO(Σ)-concept C such that O |= A ≡ C.
It follows that non explicit ALCO(Σ)-definability of A is caused by the fact that one cannot reach b from a along a path following the role name r. One can reach b, however, using the universal role or inverse roles.
For Point (3), it remains to show that ALCHO enjoys the BDP for ontologies containing RIs only. The BDP for ALCO with empty ontology follows immediately. The proof is similar to the proof of Point (2) above for ALCHIO with fully closed subsets replaced by point generated interpretations. Assume O contains RIs only, C is an ALCO-concept, and A is a concept name. Let Σ = sig(O, C)\{A}. Assume A is not explicitly ALCO(Σ)definable under O and C. By Theorem 23, we find pointed models I 1 , d 1 and I 2 , d 2 such that I i is a model of O and d i ∈ C I i for i = 1, 2, d 1 ∈ A I 1 , d 2 ∈ A I 2 , and I 1 , d 1 ∼ ALCHO,Σ I 2 , d 2 . Take a bisimulation S witnessing this. As we do not have the universal role nor inverse roles, we may assume that S is a ALCO(Σ)-bisimulation between the set ∆ I 1 ↓d 1 generated by d 1 in I 1 and the set ∆ I 2 ↓d 2 generated by d 2 in I 2 . Let I be defined as the bisimulation product induced by S. By Lemma 54, the projection functions f i : S → ∆ I i are ALCO(Σ)-bisimulations between I and I i . Let J 1 and J 2 be again defined as the interpretation I except that A J i = f −1 i (A I i ), for i = 1, 2. Then the f i are ALCO(Σ ∪ {A})-bisimulations between J i and I i , for i = 1, 2. As in the proof above, I does not necessarily interpret all nominals in C. As O is an ontology using RIs only, this problem can be addressed in a straightforward manner. Define interpretations J ′ i as the disjoint union of J i and the singleton interpretation I ′ with domain {d} such that a J i = d for all a not interpreted in I and B I ′ = r I ′ = ∅ for all concept and role names B and r. Then J ′ 1 and J ′ 2 satisfy the conditions (a) to (c) above and show that A is not implicitly Σ-definable under O and C. ❏

B Proof of Theorem 29
We show Theorem 29 which states that for L ∈ {ALCO, ALCHO} nonprojective L-definition existence of concept names is ExpTime-complete. The lower bound follows from the corresponding lower bound for subsumption and any upper bound for ALCHO trivially implies the same upper bound for ALCO. We therefore focus on the upper bound for ALCHO and show the following criterion for (the complement of) non-projective explicit definability of concept names. the Σ-reducts of I 1↓d and I 2↓d coincide; -d ∈ A I 1 and d ∈ A I 2 .
Proof. Clearly, if the conditions of Lemma 55 hold, then A is not explicitly ALCHO(Σ)-definable under O and C, by Theorem 23. Conversely, assume A is not explicitly ALCHO(Σ)-definable under O an C. By Theorem 23, we find pointed models I 1 , d 1 and I 2 , d 2 such that I i is a model of O and d i ∈ C I i for i = 1, 2, d 1 ∈ A I 1 , d 2 ∈ A I 2 , and I 1 , d 1 ∼ ALCHO,Σ I 2 , d 2 . Take a bisimulation S witnessing this. As we do not admit the universal role nor inverse roles, we may assume that S is a bisimulation between the set ∆ I 1 ↓d 1 generated by d 1 in I 1 and the set ∆ I 2 ↓d 2 generated by d 2 in I 2 . Let I be the bisimulation product induced by S. By Lemma 54, the projection functions f i : S → ∆ I i are ALCO(Σ)bisimulations between I and I i . However, as in the proof of Theorem 12, I does not necessarily interpret all nominals. We address this in the following. Let J i be the restriction of I i to ∆ I i \ ∆ I i ↓d i , for i = 1, 2. We now define interpretations J ′ 1 and J ′ 2 as follows: J ′ i is the disjoint union of I and J i extended by Observe that we have to treat the individual names in Σ differently from the concept names in Σ as we have to ensure that they are interpreted in D 1 and D 2 respectively. As their interpretation might be different outside D, we have to introduce copies of the individual names and then state that those that are interpreted in D are actually interpreted in the same way. It is now straightforward to show that the conditions of Lemma 55