Interpolants and Explicit Definitions in Extensions of the Description Logic $\mathcal{EL}$

The projective Beth definability property (PBDP) of a description logic (DL) $\mathcal{L}$ states that a concept or individual name is explicitly definable under an $\mathcal{L}$-ontology $\mathcal{O}$ by an $\mathcal{L}$-concept using symbols from a signature $\Sigma$ of concept, role, and individual names if, and only if, it is implicitly definable using $\Sigma$ under $\mathcal{O}$. The importance of the PBDP for DL research stems from the fact that it provides a polynomial time reduction of the problem to decide the existence of an explicit definition to the well understood problem of subsumption checking. The existence of explicit definitions is important for numerous knowledge engineering tasks and applications of description logic ontologies, for example, the extraction of equivalent acyclic TBoxes from ontologies (?; ?), the computation of referring expressions (or definite descriptions) for individuals (?), the equivalent rewriting of ontology-mediated queries into concepts (?; ?; ?), the construction of alignments between ontologies (?), and the decomposition of ontologies (?).

The PBDP is often investigated in tandem with the Craig interpolation property (CIP) which states that if an $\mathcal{L}$-concept is subsumed by another $\mathcal{L}$-concept under some $\mathcal{L}$-ontology then one finds an interpolating $\mathcal{L}$-concept using the shared symbols of the two input concepts only. In fact, the CIP implies the PBDP and the interpolants obtained using the CIP can serve as explicit definitions.

Many standard Boolean DLs such as $\mathcal{ALC}$, $\mathcal{ALCI}$, and $\mathcal{ALCQI}$ enjoy the CIP and PBDP and sophisticated algorithms for computing interpolants and explicit definitions have been developed (?). Important exceptions are the extensions of any of the above DLs with nominals and/or role hierarchies. In fact, it has recently been shown that the problem of deciding the existence of an interpolant/explicit definition becomes 2ExpTime-complete for $\mathcal{ALCO}$ ($\mathcal{ALC}$ with nominals) and for $\mathcal{ALCH}$ ($\mathcal{ALC}$ with role hierarchies). This result is in sharp contrast to the ExpTime-completeness of the same problem for $\mathcal{ALC}$ itself inherited from the ExpTime-completeness of subsumption under $\mathcal{ALC}$-ontologies (?).

Our aim in this article is threefold: (1) determine which members of the $\mathcal{EL}$-family of DLs enjoy the CIP/PBDP; (2) investigate the complexity of deciding the existence of interpolants/explicit definitions for those that do not enjoy it; and (3) establish tight bounds on the size of interpolants/explicit definitions and the complexity of computing them.

In what follows we discuss our main results. It has been shown in (?; ?) already that $\mathcal{EL}$ and $\mathcal{EL}$ with role hierarchies enjoy the CIP and PBDP. Rather surprisingly, it turns out that none of the remaining standard DLs in the $\mathcal{EL}$-family enjoy the CIP nor the PBDP.

Theorem 1 also has interesting consequences that are not explicitly stated. For instance, it follows that neither the DL $\mathcal{EL}^{++}$ introduced in (?) nor the extension of $\mathcal{ELI}$ with any combination of nominals, role hierarchies, or transitive roles enjoy the CIP/PBDP. With the exception of the failure of the CIP/PBDP for $\mathcal{EL}$ with nominals (without the universal role in interpolants/explicit definitions) (?), our results are new.

It follows from Theorem 1 that the behaviour of extensions of $\mathcal{EL}$ is fundamentally different from extensions of $\mathcal{ALC}$: adding role hierarchies to $\mathcal{ALC}$ does not preserve the CIP/PBDP (?) but it does for $\mathcal{EL}$; on the other hand, adding the universal role or inverse roles to $\mathcal{ALC}$ preserves the CIP/PBDP (?) but it does not for $\mathcal{EL}$.

Theorem 1 leaves open the behaviour of a few natural DLs between $\mathcal{EL}$ and its extension with arbitrary role inclusions. For instance, what happens if one only adds transitive roles or, more generally, role inclusions using a single role name only? To cover these cases we show a general result that implies that these DLs enjoy the CIP and PBDP. In particular, it follows that in Point 4 of Theorem 1 the combination of role hierarchies with a transitive role is necessary for failure of the CIP/PBDP.

We next discuss our main result about tractable extensions of $\mathcal{EL}$.

It follows that for tractable extensions of $\mathcal{EL}$ the complexity of deciding the existence of interpolants and explicit definitions does not depend on the CIP/PBDP, in sharp contrast to the behaviour of $\mathcal{ALCO}$ and $\mathcal{ALCH}$. Moreover, the proof shows how interpolants and explicit definitions can be computed from the canonical models introduced in (?), if they exist. It applies derivation trees (first introduced in (?) for DLs without nominals and role hierarchies) to estimate the size of interpolants and provide an exponential time algorithm for computing them.

The proof of Theorem 3 shows how an interpolant or explicit definition can be extracted from a (potentially infinite) tree-shaped canonical model. The ExpTime complexity bound is proved using an encoding as an emptiness problem for tree automata that also uses derivation trees. It does not seem possible to obtain tight bounds on the size of interpolants using derivation trees; instead we generalize transfer sequences for this purpose (also first introduced in (?)).

In the final section, we consider expressive Horn-DLs such as Horn-$\mathcal{ALCI}$. We first observe that Theorem 3 also holds for Horn-$\mathcal{ALCI}$ and extensions with nominals and the universal role, provided one asks for interpolants and explicit definitions in $\mathcal{ELI}$ (and extensions with nominals and the universal role, respectively). If one admits expressive Horn-concepts as interpolants or explicit definitions, then sometimes interpolants and explicit definitions exist that previously did not exist. We show that nevertheless the CIP/PBDP also fail in this case for DLs including Horn-$\mathcal{ALC}$, $\mathcal{ELI}$, and Horn-$\mathcal{ALCI}$.

Detailed proofs are given in the arxiv version of this article.

The CIP and PBDP have been investigated extensively in databases, with applications to query rewriting under views and query compilation (?; ?). The computation of explicit definitions under Horn ontologies can be seen as an instance of query reformulation under constraints (?) which has been a major research topic for many years. The Chase and Backchase approach that is central to this research closely resembles our use of canonical models. We do not assume, however, that the chase terminates. In (?; ?), it is shown that the reformulation of CQs into CQs under tgds can be reduced to entailment using Lyndon interpolation of first-order logic. By linking reformulation into CQs and definability using concepts, this approach can potentially be used to obtain alternative proofs of complexity upper bounds for the existence of interpolants and explicit definitions in our languages. Also relevant is the investigation of interpolation in basic modal logic (?) and hybrid modal logic (?; ?).

The main aim of this article is to investigate explicit definability of concept and individual names under ontologies. We have therefore chosen a definition of the CIP and interpolants that generalizes the projective Beth definability property and explicit definability in a natural and useful way, following (?). There are, however, other notions of Craig interpolation that are of interest. Of particular importance for modularity and various other purposes is the following version: if $\mathcal{O}$ is an ontology and $C\sqsubseteq D$ an inclusion such that $\mathcal{O}\models C\sqsubseteq D$, then there exists an ontology $\mathcal{O}^{\prime}$ in the shared signature of $\mathcal{O}$ and $C\sqsubseteq D$ such that $\mathcal{O}\models\mathcal{O}^{\prime}\models C\sqsubseteq D$. This property has been considered for $\mathcal{EL}$ and various extensions in (?; ?). Currently, it is unknown whether there exists any interesting relationship between this version of the CIP and the version we investigate in this article.

Craig interpolants should not be confused with uniform interpolants (or forgetting) (?; ?; ?; ?). Uniform interpolants generalize Craig interpolants in the sense that a uniform interpolant is an interpolant for a fixed antecedent and any formula implied by the antecedent and sharing with it a fixed set of symbols.

Interpolant and explicit definition existence have only recently been investigated for logics that do not enjoy the CIP or PBDP. Extending work on Boolean DLs we discussed already, it is shown that they become harder than validity also in the guarded and two-variable fragment (?). The interpolant existence problem for linear temporal logic LTL is considered in (?). In the context of referring expressions, explicit definition existence is investigated in (?), see also (?).

Let ${\sf N_{C}}$, ${\sf N_{R}}$, and ${\sf N_{I}}$ be disjoint and countably infinite sets of concept, role, and individual names. A role is a role name $r$ or an inverse role $r^{-}$, with $r$ a role name. Nominals take the form $\{a\}$, where $a$ is an individual name. The universal role is denoted by $u$. $\mathcal{ELIO}_{u}$-concepts $C$ are defined by the following syntax rule:

where $A$ ranges over concept names, $a$ over individual names, and $r$ over roles (including the universal role). Fragments of $\mathcal{ELIO}_{u}$ are defined as usual. For example, $\mathcal{ELI}$-concepts are $\mathcal{ELIO}_{u}$-concepts without nominals and the universal role, and $\mathcal{EL}$-concepts are $\mathcal{ELI}$-concepts without inverse roles. Given any of the DLs $\mathcal{L}$ introduced above, an $\mathcal{L}$-concept inclusion ($\mathcal{L}$-CI) takes the form $C\sqsubseteq D$ with $C,D$ $\mathcal{L}$-concepts. An $\mathcal{L}$-ontology $\mathcal{O}$ is a finite set of $\mathcal{L}$-CIs.

We also consider ontologies with role inclusions (RIs), expressions of the form $r_{1}\circ\cdots\circ r_{n}\sqsubseteq r$ with $r_{1},\ldots,r_{n},r$ role names. An $\mathcal{ELO}_{u}$-ontology with RIs is called an $\mathcal{ELRO}_{u}$-ontology. A set of RIs is a role hierarchy if all its RIs are of the form $r\sqsubseteq s$ with $r,s$ role names.

A signature $\Sigma$ is a set of concept, role, and individual names, uniformly referred to as (non-logical) symbols. We follow common practice and do not regard the universal role $u$ as a non-logical symbol as its interpretation is fixed. We use $\text{sig}(X)$ to denote the set of symbols used in any syntactic object $X$ such as a concept or an ontology. If $\mathcal{L}$ is a DL and $\Sigma$ a signature, then an $\mathcal{L}(\Sigma)$-concept $C$ is an $\mathcal{L}$-concept with $\text{sig}(C)\subseteq\Sigma$. The size $||X||$ of a syntactic object $X$ is the number of symbols needed to write it down.

The semantics of DLs is given in terms of interpretations $\mathcal{I}=(\Delta^{\mathcal{I}},\cdot^{\mathcal{I}})$, where $\Delta^{\mathcal{I}}$ is a non-empty set (the domain) and $\cdot^{\mathcal{I}}$ is the interpretation function, assigning to each $A\in{\sf N_{C}}$ a set $A^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}$, to each $r\in{\sf N_{R}}$ a relation $r^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}$, and to each $a\in{\sf N_{I}}$ an element $a^{\mathcal{I}}\in\Delta^{\mathcal{I}}$. The interpretation $C^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}$ of a concept $C$ in $\mathcal{I}$ is defined as usual, see (?). An interpretation $\mathcal{I}$ satisfies a CI $C\sqsubseteq D$ if $C^{\mathcal{I}}\subseteq D^{\mathcal{I}}$ and an RI $r_{1}\circ\cdots\circ r_{n}\sqsubseteq r$ if $r_{1}^{\mathcal{I}}\circ\cdots\circ r_{n}^{\mathcal{I}}\subseteq r^{\mathcal{I}}$. We say that $\mathcal{I}$ is a model of an ontology $\mathcal{O}$ if it satisfies all inclusions in it. If $\alpha$ is a CI or RI, we write $\mathcal{O}\models\alpha$ if all models of $\mathcal{O}$ satisfy $\alpha$. We write $\mathcal{O}\models C\equiv D$ if $\mathcal{O}\models C\sqsubseteq D$ and $\mathcal{O}\models D\sqsubseteq C$.

An ontology is in normal form if its CIs are of the form

and

where $A,A_{1},A_{2},B$ are concept names, $r$ is a role or the universal role, and $a$ is an individual name. It is well known that for any $\mathcal{ELIO}_{u}$-ontology $\mathcal{O}$ with or without RIs one can construct in polynomial time a conservative extension $\mathcal{O}^{\prime}$ using the same constructors as $\mathcal{O}$ that is in normal form.

$\mathcal{L}(\Sigma)$-concepts can be characterized using $\mathcal{L}(\Sigma)$-simulations which we define next. Let $\mathcal{I}$ and $\mathcal{J}$ be interpretations. A relation $S\subseteq\Delta^{\mathcal{I}}\times\Delta^{\mathcal{J}}$ is called an $\mathcal{ELO}(\Sigma)$-simulation between $\mathcal{I}$ and $\mathcal{J}$ if the following conditions hold:

1. 1.

   if $d\in A^{\mathcal{I}}$ and $(d,e)\in S$, then $e\in A^{\mathcal{J}}$, for all $A\in{\sf N_{C}}\cap\Sigma$;

2. 2.

   if $d=a^{\mathcal{I}}$ and $(d,e)\in S$, then $e=a^{\mathcal{J}}$, for all $a\in{\sf N_{I}}\cap\Sigma$;

3. 3.

   if $(d,d^{\prime})\in r^{\mathcal{I}}$ and $(d,e)\in S$, then there exists $e^{\prime}$ with $(e,e^{\prime})\in r^{\mathcal{J}}$ and $(d^{\prime},e^{\prime})\in S$, for all $r\in{\sf N_{R}}\cap\Sigma$.

$S$ is called an $\mathcal{ELO}_{u}(\Sigma)$-simulation if $\Delta^{\mathcal{I}}$ is the domain of $S$ and an $\mathcal{ELIO}(\Sigma)$-simulation if Condition 3 also holds for inverse roles from $\Sigma$. Condition 2 is dropped if $\mathcal{L}$ does not use nominals. We write $(\mathcal{I},d)\preceq_{\mathcal{L},\Sigma}(\mathcal{J},e)$ if there exists an $\mathcal{L}(\Sigma)$-simulation $S$ between $\mathcal{I}$ and $\mathcal{J}$ with $(d,e)\in S$. We write $(\mathcal{I},d)\leq_{\mathcal{L},\Sigma}(\mathcal{J},e)$ if $d\in C^{\mathcal{I}}$ implies $e\in C^{\mathcal{J}}$ for all $\mathcal{L}(\Sigma)$-concepts $C$. The following characterization is well known (?; ?).

We introduce the Craig interpolation property (CIP) as defined in (?) and the projective Beth definability property (PBDP) and prove Theorem 1 from the introduction to this article. We observe that the CIP implies the PBDP, but lack a proof of the converse direction. Nevertheless, all DLs considered in this paper enjoying the PBDP also enjoy the CIP.

Set $\text{sig}(\mathcal{O},C)=\text{sig}(\mathcal{O})\cup\text{sig}(C)$, for any ontology $\mathcal{O}$ and concept $C$. Let $\mathcal{O}_{1},\mathcal{O}_{2}$ be $\mathcal{L}$-ontologies and let $C_{1},C_{2}$ be $\mathcal{L}$-concepts. Then an $\mathcal{L}$-concept $D$ is called an $\mathcal{L}$-interpolant¹¹1Important variations of this definition are to drop $\mathcal{O}_{2}$ in Point 2 and $\mathcal{O}_{1}$ in Point 3, respectively, or to consider only one ontology $\mathcal{O}=\mathcal{O}_{1}=\mathcal{O}_{2}$ and regard the signature $\Sigma$ of the interpolant as an input given independently from $\mathcal{O},C_{1},C_{2}$. This has an effect on the CIP, but our results on interpolant computation and existence are not affected. for $C_{1}\sqsubseteq C_{2}$ under $\mathcal{O}_{1},\mathcal{O}_{2}$ if

• •

  $\text{sig}(D)\subseteq\text{sig}(\mathcal{O}_{1},C_{1})\cap\text{sig}(\mathcal{O}_{2},C_{2})$;

• •

  $\mathcal{O}_{1}\cup\mathcal{O}_{2}\models C_{1}\sqsubseteq D$;

• •

  $\mathcal{O}_{1}\cup\mathcal{O}_{2}\models D\sqsubseteq C_{2}$.

We next define the relevant definability notions. Let $\mathcal{O}$ be an ontology and $A$ a concept name. Let $\Sigma\subseteq\text{sig}(\mathcal{O})$ be a signature. An $\mathcal{L}(\Sigma)$-concept $C$ is an explicit $\mathcal{L}(\Sigma)$-definition of $A$ under $\mathcal{O}$ if $\mathcal{O}\models A\equiv C$. We call $A$ explicitly definable in $\mathcal{L}(\Sigma)$ under $\mathcal{O}$ if there is an explicit $\mathcal{L}(\Sigma)$-definition of $A$ under $\mathcal{O}$. The $\Sigma$-reduct $\mathcal{I}_{|\Sigma}$ of an interpretation $\mathcal{I}$ coincides with $\mathcal{I}$ except that no symbol that is not in $\Sigma$ is interpreted in $\mathcal{I}_{|\Sigma}$. A concept $A$ is called implicitly definable using $\Sigma$ under $\mathcal{O}$ if the $\Sigma$-reduct of any model $\mathcal{I}$ of $\mathcal{O}$ determines the set $A^{\mathcal{I}}$; in other words, if $\mathcal{I}$ and $\mathcal{J}$ are both models of $\mathcal{O}$ such that $\mathcal{I}_{|\Sigma}=\mathcal{J}_{|\Sigma}$, then $A^{\mathcal{I}}=A^{\mathcal{J}}$. It is easy to see that implicit definability can be reformulated as a standard reasoning problem as follows: a concept name $A\not\in\Sigma$ is implicitly definable using $\Sigma$ under $\mathcal{O}$ iff $\mathcal{O}\cup\mathcal{O}_{\Sigma}\models A\equiv A^{\prime}$, where $\mathcal{O}_{\Sigma}$ is obtained from $\mathcal{O}$ by replacing every symbol $X$ not in $\Sigma$ (including $A$) uniformly by a fresh symbol $X^{\prime}$.

We next prove that the majority of tractable extensions of $\mathcal{EL}$ does not enjoy the CIP nor PBDP.

We next discuss a general positive result on interpolation and explicit definition existence that shows that Theorem 1 is essentially optimal. A set $\mathcal{R}$ of RIs is safe for a signature $\Sigma$ if for each RI $r_{1}\circ\dots\circ r_{n}\sqsubseteq r\in\mathcal{R}$, $n\geq 1$, if $\{r_{1},\dots,r_{n},r\}\cap\Sigma\neq\emptyset$ then $\{r_{1},\dots,r_{n},r\}\subseteq\Sigma$.

The proof technique is based on simulations and similar to (?; ?). Theorem 4 has a few interesting consequences. For instance, $\mathcal{EL}$ with transitive roles enjoys both the CIP and PBDP since transitivity is expressed by the role inclusion $r\circ r\sqsubseteq r$ which is safe for any signature (as it only uses a single role name).

We introduce interpolant and explicit definition existence as decision problems and establish a polynomial time reduction of the latter to the former. We then show that it suffices to consider ontologies in normal form and that the addition of $\bot$ does not affect the complexity of the decision problems.

Observe that interpolant existence reduces to checking $\mathcal{O}_{1}\cup\mathcal{O}_{2}\models C_{1}\sqsubseteq C_{2}$ for logics with the CIP but that this is not the case for logics without the CIP.

We next observe that replacing the original ontologies by a conservative extension preserves interpolants and explicit definitions. Thus, it suffices to consider ontologies in normal form and interpolants for inclusions between concept names.

The aim of this section is to analyse interpolants and explicit definitions for extensions of $\mathcal{EL}$ with any combination of nominals, role inclusions, or the universal role. We show the following result from the introduction.

Before we start with a sketch of the proof we give instructive examples showing that the exponential bound on the size of explicit definitions is optimal.

Observe that using $\mathcal{O}_{b}$ one enforces explicit definitions of exponential size by generating a binary tree of linear depth whereas using $\mathcal{O}_{p}$ this is achieved by generating a path of exponential length. The latter can only happen if role inclusions are used in the ontology. One insight provided by the exponential upper bound on the size of explicit definitions in Theorem 2 is that the two examples cannot be combined to enforce a binary tree of exponential depth.

To continue with the proof we introduce ABoxes as a technical tool that allows us to move from interpretations to (potentially incomplete) sets of facts and concepts. An ABox $\mathcal{A}$ is a (possibly infinite) set of assertions of the form $A(x)$, $r(x,y)$, $\{a\}(x)$, and $\top(x)$ with $A\in{\sf N_{C}}$, $r\in{\sf N_{R}}$, $a\in{\sf N_{I}}$, and $x,y$ individual variables (we call individuals used in ABoxes variables to distinguish them from individual names used in nominals). We denote by $\text{ind}(\mathcal{A})$ the set of individual variables in $\mathcal{A}$. A $\Sigma$-ABox is an ABox using symbols from $\Sigma$ only. Models of ABoxes are defined as usual. We do not make the unique name assumption.

Every interpretation $\mathcal{I}$ defines an ABox $\mathcal{A}_{\mathcal{I}}$ by identifying every $d\in\Delta^{\mathcal{I}}$ with a variable $x_{d}$ and taking $A(x_{d})$ if $d\in A^{\mathcal{I}}$, $r(x_{c},x_{d})$ if $(c,d)\in r^{\mathcal{I}}$, $\{a\}(x_{d})$ if $a^{\mathcal{I}}=d$. Conversely, ABoxes $\mathcal{A}$ define interpretations in the obvious way (by identifying variables $x,y$ if $\{a\}(x),\{a\}(y)\in\mathcal{A}$). We associate with every ABox $\mathcal{A}$ a directed graph $G_{\mathcal{A}}=(\text{ind}(\mathcal{A}),\bigcup_{r\in{\sf N_{R}}}\{(x,y)\mid r(x,y)\in\mathcal{A}\})$. Let $\Gamma$ be a set of individual names. Then $\mathcal{A}$ is ditree-shaped modulo $\Gamma$ if after dropping some facts of the form $r(x,y)$ with $\{a\}(y)\in\mathcal{A}$ for some $a\in\Gamma$, it is ditree-shaped in the sense that $G_{\mathcal{A}}$ is acyclic and $r(x,y)\in\mathcal{A}$ and $s(x,y)\in\mathcal{A}$ imply $r=s$. A pointed ABox is a pair $\mathcal{A},x$ with $x\in\text{ind}(\mathcal{A})$. Then ${\cal E\!\!\>LO}_{u}(\Sigma)$-concepts correspond to pointed $\Sigma$-ABoxes $\mathcal{A},x$ such that $\mathcal{A}$ is ditree-shaped modulo ${\sf N_{I}}\cap\Sigma$ and $\mathcal{ELO}(\Sigma)$-concepts correspond to rooted pointed $\Sigma$-ABoxes $\mathcal{A},x$ such that $\mathcal{A}$ is ditree-shaped modulo ${\sf N_{I}}\cap\Sigma$, where $\mathcal{A},x$ is called rooted if for every $y\in\text{ind}(\mathcal{A})$ there is a path from $x$ to $y$ in $G_{\mathcal{A}}$. We write $\mathcal{O},\mathcal{A}\models C(x)$ if $x^{\mathcal{I}}\in C^{\mathcal{I}}$ for every model $\mathcal{I}$ of $\mathcal{O}$ and $\mathcal{A}$.

Given an $\mathcal{ELRO}_{u}$-ontology $\mathcal{O}$ in normal form and a concept name $A$, one can construct in polynomial time the canonical model $\mathcal{I}_{\mathcal{O},A}$ of $\mathcal{O}$ and $A$ using the approach introduced in (?). More generally, the canonical model $\mathcal{I}_{\mathcal{O},\mathcal{A}}$ for an ABox $\mathcal{A}$ and ontology $\mathcal{O}$ can be constructed in polynomial time and is a model of both $\mathcal{O}$ and $\mathcal{A}$ such that for any $\mathcal{ELO}_{u}$-concept $C$ using symbols from $\mathcal{O}$ only and any $x\in\text{ind}(\mathcal{A})$,

• ($\dagger$)

  $\mathcal{O},\mathcal{A}\models C(x)$ iff $x\in C^{\mathcal{I}_{\mathcal{O},\mathcal{A}}}$,

details are given in the appendix of the full version. We let $\mathcal{I}_{\mathcal{O},A}=\mathcal{I}_{\mathcal{O},\mathcal{A}}$ with $\mathcal{A}=\{A(\rho_{A})\}$. Note that in (?) the condition ($\dagger$) is only stated for subconcepts $C$ of the ontology $\mathcal{O}$, thus ($\dagger$) requires a proof.

The directed unfolding of a pointed $\Sigma$-ABox $\mathcal{A},x$ into a pointed $\Sigma$-ABox $\mathcal{A}^{u},x$ that is ditree-shaped modulo $\Sigma\cap{\sf N_{I}}$ is defined in the standard way. In the rooted directed unfolding, nodes that cannot be reached from $x$ via role names are dropped.

Assume now that $\mathcal{O}$ is in normal form and $A$ a concept name. Let $\mathcal{A}_{\mathcal{O},A}^{\Sigma}$ be the $\Sigma$-reduct of the canonical model $\mathcal{I}_{\mathcal{O},A}$, regarded as an ABox. Denote by $\mathcal{A}_{\mathcal{O},A}^{\Sigma,u},\rho_{A}$ the directed unfolding of $\mathcal{A}_{\mathcal{O},A}^{\Sigma},\rho_{A}$, by $\mathcal{A}_{\mathcal{O},A}^{\downarrow\Sigma},\rho_{A}$ the sub-ABox of $\mathcal{A}_{\mathcal{O},A}^{\Sigma}$ rooted in $\rho_{A}$, and by $\mathcal{A}_{\mathcal{O},A}^{\downarrow\Sigma,u},\rho_{A}$ its rooted directed unfolding. Theorem 2 is a direct consequence of the following characterization of interpolants.

Note that the polynomial time decidability of interpolant existence follows from Point 2 of Theorem 5 (and the tractability of $\mathcal{ELRO}_{u}$ (?)).

The following example illustrates the difference between the existence of explicit definitions in $\mathcal{ELO}$ and $\mathcal{ELO}_{u}$ and thus the need for moving to the ABoxes $\mathcal{A}_{\mathcal{O},A}^{\downarrow\Sigma}$, and $\mathcal{A}_{\mathcal{O},A}^{\downarrow\Sigma,u}$ if one does not admit the universal role in explicit definitions.

We next sketch the proof idea for Theorem 5 for the case with universal role in interpolants. We show “1. $\Rightarrow$ 2.”, observe that “3. $\Rightarrow$ 1.” is trivial, and then sketch the proof of “2. $\Rightarrow$ 3.” and the exponential time algorithm computing interpolants, details are provided in the appendix of the full version. For “1. $\Rightarrow$ 2.” assume that $C$ is an $\mathcal{ELO}_{u}(\Sigma)$-concept with (i) $\mathcal{O}_{1}\cup\mathcal{O}_{2}\models A\sqsubseteq C$ and (ii) $\mathcal{O}_{1}\cup\mathcal{O}_{2}\models C\sqsubseteq B$. By ($\dagger$) and (i), $\mathcal{A}^{\Sigma}_{\mathcal{O}_{1}\cup\mathcal{O}_{2},A}\models C(\rho_{A})$. But then by (ii) $\mathcal{O}_{1}\cup\mathcal{O}_{2},\mathcal{A}^{\Sigma}_{\mathcal{O}_{1}\cup\mathcal{O}_{2},A}\models B(\rho_{A})$, as required.

If one does not impose a bound on the size of $\mathcal{A}$ in Point 3, then one can prove “2. $\Rightarrow$ 3.” using compactness and a generalization of unraveling tolerance according to which $\mathcal{O}_{1}\cup\mathcal{O}_{2},\mathcal{A}_{\mathcal{O}_{1}\cup\mathcal{O}_{2},A}^{\Sigma}$ and $\mathcal{O}_{1}\cup\mathcal{O}_{2},\mathcal{A}_{\mathcal{O}_{1}\cup\mathcal{O}_{2},A}^{\Sigma,u}$ entail the same $C(\rho_{A})$ (?; ?). As we are interested in an exponential bound on the size of $\mathcal{A}$ (and a deterministic exponential time algorithm computing it) we require a more syntactic approach. Our proof of “2. $\Rightarrow$ 3.” is based on derivation trees which represent a derivation of a fact $C(a)$ from an ontology $\mathcal{O}$ and ABox $\mathcal{A}$ using a labeled tree. Our derivation trees generalize those introduced in (?; ?) to languages with nominals and role inclusions. Reflecting the use of individual names and concept names in the construction of the domain of the canonical model (?), we assume $a\in\Delta:=\text{ind}(\mathcal{A})\cup(({\sf N_{C}}\cup{\sf N_{I}})\cap\textup{sig}(\mathcal{O}))$ and $C\in\Theta:=\{\top\}\cup({\sf N_{C}}\cap\text{sig}(\mathcal{O}))\cup\{\{a\}\mid a\in{\sf N_{I}}\cap\textup{sig}(\mathcal{O})\}$. Then a derivation tree $(T,V)$ for $(a,C)\in\Delta\times\Theta$ is a tree $T$ with a labeling function $V:T\rightarrow\Delta\times\Theta$ such that $V(\varepsilon)=(a,C)$ and $(V,T)$ satisfies rules stating under which conditions the label of $n$ is derived in one step from the labels of the successors of $n$. To illustrate, the existence of successors $n_{1},n_{2}$ of $n$ with $V(n_{1})=(a,C_{1})$ and $V(n_{2})=(a,C_{2})$ justifies $V(n)=(a,C)$ if $\mathcal{O}\models C_{1}\sqcap C_{2}\sqsubseteq C$. The rules are given in the appendix of the full version, we only discuss the rule used to capture derivations using RIs: $V(n)=(a_{1},C)$ is justified if there are role names $r_{2},\ldots,r_{2k-2},r$ such that $(a_{2k},C^{\prime})$ is a label of a successor of $n$, $\mathcal{O}\models\exists r.C^{\prime}\sqsubseteq C$, $\mathcal{O}\models r_{2}\circ\cdots\circ r_{2k-2}\sqsubseteq r$, and the situation depicted in Figure 4 holds, where the “dotted lines” stand for ‘either $a_{i}=a_{i+1}$ or some $(a_{i},\{c\}),(a_{i+1},\{c\})$ with $c\in{\sf N_{I}}$ are labels of successors of $n$’, and $\hat{r}_{i}$ stands for ‘either $r(a_{i},a_{i+1})\in\mathcal{A}$ or some $(a_{i},C_{i})$ is a label of a successor of $n$ and $\mathcal{O}\models C_{i}\sqsubseteq\exists r_{i}.\{a_{i+1}\}$ if $a_{i+1}\in{\sf N_{I}}$ and $\mathcal{O}\models C_{i}\sqsubseteq\exists r_{i}.a_{i+1}$ if $a_{i+1}\in{\sf N_{C}}$’. Moreover, for all $a_{i}\not=a_{1}$, $1\leq i\leq 2k$, there exists a successor of $n$ with label $(a_{i},D)$ for some $D$. The soundness of this rule should be clear, completeness can be shown similarly to the analysis of canonical models.