A new n-ary existential quantifier in description logics

Motivated by a chemical process engineering application, we introduce a new concept constructor in Description Logics (DLs), an n-ary variant of the existential restriction constructor, which generalizes both the usual existential restrictions and so-called qualified number restrictions. We show that the new constructor can be expressed in ALCQ, the extension of the basic DL ALC by qualified number restrictions. However, this representation results in an exponential blow-up. By giving direct algorithms for ALC extended with the new constructor, we can show that the complexity of reasoning in this new DL is actually not harder than the one of reasoning in ALCQ. Moreover, in our chemical process engineering application, a restricted DL that provides only the new constructor together with conjunction, and satisfies an additional restriction on the occurrence of roles names, is sufficient. For this DL, the subsumption problem is polynomial.


Introduction
Description Logics (DLs) [3] are a class of knowledge representation formalisms in the tradition of semantic networks and frames, which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. DL systems provide their users with inference services (like computing the subsumption hierarchy) that deduce implicit knowledge from the explicitly represented knowledge. For these inference services to be feasible, the underlying inference problems must at least be decidable, and preferably of low complexity. This is only possible if the expressiveness of the DL employed by the system is restricted in an appropriate way. Because of this restriction of the expressive power of DLs, various application-driven language extensions have been proposed in the literature (see, e.g., [4,10,23,17]), some of which have been integrated into state-of-the-art DL systems [16,14].
The present paper considers a new concept constructor that is motivated by a process engineering application [24]. This constructor is an n-ary variant of the usual existential restriction operator available in most DLs. To motivate the need for this new constructor, assume that we want to describe a chemical plant that has a reactor with a main reaction, and in addition a reactor with a main and a side reaction. Also assume that the concepts Reactor with main reaction and Reactor with main and side reaction are defined such that the first concept subsumes the second one. We could try to model this chemical plant with the help of the usual existential restriction operator as Plant ∃has part.Reactor with main reaction ∃has part.Reactor with main and side reaction.
However, because of the subsumption relationship between the two reactor concepts, this concept is equivalent to Plant ∃has part.Reactor with main and side reaction, and thus does not capture the intended meaning of a plant having two reactors, one with a main reaction and the other with a main and a side reaction.
To overcome this problem, we consider a new concept constructor of the form ∃r.(C 1 , . . . , C n ), with the intended meaning that it describes all individuals having n different r-successors d 1 , . . . , d n such that d i belongs to C i (i = 1, . . . , n). Given this constructor, our concept can correctly be described as The situation differs from other application-driven language extensions in that the new constructor can actually be expressed using constructors available in the DL ALCQ, which can be handled by state-of-the-art DL systems (Section 3).

Name
Syntax Semantics conjunction C D C I ∩ D I negation ¬C ∆ I \ C I at-least qualified number restriction n r.C {x | card({y | (x, y) ∈ r I ∧ y ∈ C I }) ≥ n} Table 1: Syntax and semantics of ALCQ.
Thus, the new constructor can be seen as syntactic sugar; nevertheless, it makes sense to introduce it explicitly since this speeds up reasoning. In fact, expressing the new constructor with the ones available in ALCQ results in an exponential blow-up. In addition, the translation introduces many "expensive" constructors (disjunction and qualified number restrictions). For this reason, even highly optimized DL systems like Racer [14] cannot handle the translated concepts in a satisfactory way. In contrast, the direct introduction of the new constructor into ALCQ does not increase the complexity of reasoning (Section 4). Moreover, in the process engineering application [24] mentioned above, the rather inexpressive DL EL (n) that provides only the new constructor together with conjunction is sufficient. In addition, only concept descriptions are used where in each conjunction there is at most one n-ary existential restriction for each role. For this restricted DL, the subsumption problem is polynomial (Section 5). If this last restriction is removed, then subsumption is in coNP, but the exact complexity of the subsumption problem in EL (n) is still open (Section 6). If one allows to impose disjointness statements between concept names (Section 7), then subsumption between restricted EL (n) -concept descriptions remains polynomial. In the case of unrestricted EL (n) -concept descriptions, subsumption can then be shown to be coNP-complete.

The DL ALCQ
Concept descriptions are inductively defined with the help of a set of constructors, starting with a set N C of concept names and a set N R of role names. The constructors determine the expressive power of the DL. In this section, we restrict the attention to the DL ALCQ, whose concept descriptions are formed using the constructors shown in Table 1. Using these constructors, several other constructors can be defined as abbreviations: • C D := ¬(¬C ¬D) (disjunction), • := A ¬A for a concept name A (top-concept), • ∃r.C := 1 r.C (existential restriction), • ∀r.C := ¬∃r.¬C (value restriction), • n r.C := ¬( (n + 1) r.C) (at-most restriction).
The semantics of ALCQ-concept descriptions is defined in terms of an interpretation I = (∆ I , · I ). The domain ∆ I of I is a non-empty set of individuals and the interpretation function · I maps each concept name A ∈ N C to a subset A I of ∆ I and each role r ∈ N R to a binary relation r I on ∆ I . The extension of · I to arbitrary concept descriptions is inductively defined, as shown in the third column of Table 1. Here, the function card yields the cardinality of the given set.
A general ALCQ-TBox is a finite set of general concept inclusions (GCIs) C D where C, D are ALCQ-concept descriptions. The interpretation I is a model of the general ALCQ-TBox T iff it satisfies all its GCIs, i.e., if C I ⊆ D I holds for all GCIs C D in T .
We use C ≡ D as an abbreviation of the two GCIs C D, D C. An acyclic ALCQ-TBox is a finite set of concept definitions of the form A ≡ C (where A is a concept name and C an ALCQ-concept description) that does not contain multiple definitions or cyclic dependencies between the definitions. Concept names occurring on the left-hand side of a concept definition are called defined whereas the others are called primitive.
Given two ALCQ-concept descriptions C, D we say that C is subsumed by D w.r.t. the general TBox T (C T D) iff C I ⊆ D I for all models I of T . Subsumption w.r.t. an acyclic TBox and subsumption between concept descriptions (where T is empty) are special cases of this definition. In the latter case we write C D in place of C ∅ D. The concept description C is satisfiable (w.r.t. the general TBox T ) iff there is an interpretation I (a model I of T ) such that C I = ∅.
The complexity of the subsumption problem in ALCQ depends on the presence of GCIs. Subsumption of ALCQ-concept descriptions (with or without acyclic TBoxes) is PSpace-complete and subsumption w.r.t. a general ALCQ-TBox is ExpTime-complete [25]. 1 These results hold both for unary and binary coding of the numbers in number restriction, but in this paper we restrict the attention to unary coding (where the size of the number n is counted as n rather than log n).
We call the DL whose concept descriptions are formed using the constructors conjunction, negation, and n-ary existential restriction EL (n) C. It is an immediate consequence of the semantics of n-ary existential restrictions that the at-least restriction n r.C can be expressed by the n-ary existential restriction ∃r.(C, . . . , C). 2 Consequently, all of ALCQ can be expressed within EL (n) C.
Conversely, can we express n-ary existential restrictions within ALCQ? We have seen in the introduction that, in general, ∃r.(C 1 , . . . , C n ) cannot be replaced by the conjunction ∃r.C 1 . . . ∃r.C n since this conjunction does not ensure the existence of n different r-successors. However, ALCQ provides us with the more expressive qualified number restriction constructor. Let us first consider the case n = 2. We claim that ∃r.(C 1 , C 2 ) can be expressed by the ALCQ-concept description It is clear that any individual belonging to ∃r.(C 1 , C 2 ) also belongs to D. Conversely, assume that x belongs to D. Then x has two distinct r-successors y 1 , y 2 , both belonging to C 1 C 2 . If one of them belongs to C 1 and the other to C 2 , then we are done. Otherwise, we have two cases: (i) both belong to C 1 ¬C 2 , or (ii) both belong to ¬C 1 C 2 . We restrict our attention to the first case (since the second is symmetric). Due to the conjunct 1 r.C 2 in D, x has an r-successor in C 2 , which is different from y 1 since y 1 does not belong to C 2 . Consequently, there are two distinct r-successors of x, one belonging to C 1 and the other belonging to C 2 , which shows that x belongs to ∃r.(C 1 , C 2 ).
This result can be extended to arbitrary n.
Theorem 3.1 The n-ary existential restriction constructor can be expressed within ALCQ, and thus ALCQ and EL (n) C have the same expressive power.
To prove this theorem we show that ∃r.(C 1 , . . . , C n ) can be expressed by the ALCQ-concept description It is again clear that any individual belonging to the concept ∃r.(C 1 , . . . , C n ) also belongs to D n . The other direction is an easy consequence of Hall's theorem [15]. Let F = (S 1 , . . . , S n ) be a finite family of sets. This family has a system of distinct representatives (SDR) iff there are n distinct elements s 1 , . . . , s n such that s i ∈ S i (i = 1, . . . , n).
Now, assume that the individual x belongs to D n . For i = 1, . . . , n, let S i be the set of r-successors of x that belong to C i . By the definition of D n , the family (S 1 , . . . , S n ) satisfies the condition of Hall's theorem, and thus it has an SDR. This SDR obviously shows that x belongs to ∃r.(C 1 , . . . , C n ).
The proof of Theorem 3.1 shows that the subsumption problem in EL (n) C can be reduced to the subsumption problem in ALCQ, and thus DL systems like Racer that can handle ALCQ can in principle be used to compute subsumption in EL (n) C. However, the translation from EL (n) C into ALCQ is obviously exponential. In addition, the constructs it introduces (disjunctions and qualified number restrictions) are hard to handle for tableau-based subsumption algorithms like the one used by Racer. In fact, faced with the ALCQ-translations of the EL (n) C-concept descriptions it takes Racer 3 57 minutes to find out that C D. For the 5-ary variant of this example, Racer did not finish its computation within 4 hours.
This problem can be due either to the inherently higher complexity of reasoning in EL (n) C, or to the translation. We will see in the next section that the latter is the culprit.

Complexity of reasoning in EL (n) C
The exponential translation of EL (n) C-concepts into ALCQ-concepts together with the known complexity of the subsumption problem in ALCQ (see Section 2) yields the following complexity upper-bounds for the subsumption problem in EL (n) C: ExpSpace for subsumption of concept descriptions and 2ExpTime for subsumption w.r.t. a general TBox. The next theorem shows that these upperbounds are not optimal.
Theorem 4.1 The subsumption problem in EL (n) C is PSpace-complete for subsumption between concept descriptions and ExpTime-complete for subsumption w.r.t. a general TBox.
The hardness results are an immediate consequence of the corresponding hardness results [12] for the subsumption problem in ALC (which allows for conjunction, negation, and existential restrictions). Since EL (n) C is closed under negation, it is enough to prove the upper bounds for the satisfiability problem. To show the PSpace-upper bound, we adapt the "witness algorithm" (also called K-worlds algorithm) commonly used in modal logics to show that satisfiability in the modal logic K is in PSpace (see, e.g., [8]). The ExpTime-upper bound is proved by an adaptation of Pratt's "elimination of Hintikka sets" approach to show that satisfiability in propositional dynamic logic (PDL) is in ExpTime (see also [8]). But first, we must introduce some notation.
In the following, we assume that all concept descriptions are built using only the constructors conjunction, negation, and n-ary existential restriction. We use sub(C) to denote the set of all subconcepts of C, sub(T ) to denote We use cl(C) as an abbreviation for cl(C, ∅). Let Γ be a set of concept descriptions. A set Ψ ⊆ Γ is a type for Γ iff it satisfies the following conditions: • for all C D ∈ Γ: C D ∈ Ψ iff {C, D} ⊆ Ψ; • for all ¬(C D) ∈ Γ: ¬(C D) ∈ Ψ iff {¬C, ¬D} ∩ Ψ = ∅; • for all ¬C ∈ Γ: Intiuitively, a type for cl(C, T ) can be used to describe to which subconcepts of C, T an individual of a given interpretation belongs or not. Individuals having identical types behave the same w.r.t. subconcepts of C, T , and thus, in the algorithms, types can be used to represent the relevant properties of individuals. Basically, the ExpTime-upper bound is due to the fact that there are only exponentially many types for cl(C, T ). In case T is empty, there are still exponentially many types, but the way one goes through them is such that only polynomially many of them need to be held in memory at the same time.
For each j, 1 ≤ j ≤ k we define the set Then (S 1 , . . . , S k ) has an SDR iff there are distinct indices i 1 , . . . , i k < n such that C j ∈ Φ i j for 1 ≤ j ≤ k. The existence of an SDR can be decided in polynomial time by a reduction to the maximum bipartite matching problem (see Section 5.2 for more details).  The following lemma shows that the procedure EL (n) C-World introduced in Fig. 1 decides the satisfiability of EL (n) C-concept descriptions. Since, with every recursive call of the procedure, the maximal role depth of concept descriptions occurring in its arguments decreases, the resursion depth of the algorithm is bounded polynomially. 4 Thus, EL (n) C-World is a non-deterministic polynomial space algorithm for EL (n) C-satisfiability. Because of Savitch's theorem, which says that PSpace = NPSpace, this yields the desired PSpace upper-bound.
Let T be the recursion tree of a successful run of EL (n) C-World(Ψ, cl(C 0 )), i.e., T = (V, E, ∆ , Γ ) is a finite tree where the node labelling function ∆ ( Γ ) assigns, to each node, the first (second) argument of the corresponding recursive call. Additionally, we assume that, for each node v ∈ V except the root, R (v) returns the role name that the for all loop was processing when making the recursion call that generated v. We define an interpretation I as follows: We prove by structural induction on C that, for all v ∈ ∆ I and all C ∈ Γ (v), The base case is straightforward by the definition of I. The Boolean cases are easy since, for each v ∈ V , ∆ (v) is a type and Γ (v) is closed under building subconcepts. The remaining case concerns the n-ary existential restriction constructor.
For the "if" direction, let v ∈ V and ∃r.(C 1 , . . . , C k ) ∈ ∆ (v). Then r ∈ rol ∃ ( ∆ (v)), and there exists a successor candidate Ψ 0 , . . . , By the construction of I, the nodes v 1 , . . . , v k are (distinct) successors of v in T and R (v j ) = r for 1 ≤ j ≤ k. It follows that r ∈ rol ∃ (v), and the definition of the procedure EL (n) C-World implies that there exists a successor candidate Ψ 0 , . . . , Ψ n−1 for ∆ (v) w.r.t. r and Γ (v) and distinct indices i 1 , . . . , i k such that Now assume that C 0 is satisfiable, let I be a model of C 0 , and x 0 ∈ C I 0 . For x ∈ ∆ I and Γ a set of concepts, we define We use I to guide the non-deterministic choices of EL (n) C-World. To describe this in more detail, it is convenient to pass an element of ∆ I as a "virtual" third argument to EL (n) C-World. Initially, we call EL (n) C-World with arguments Now, assume EL (n) C-World is called with arguments (∆, Γ, x). By induction, we assume that ∆ = tp Γ (x). For every role r ∈ rol ∃ (∆) we must execute the body of the for all loop. First, we must determine the number n of components of the successor candidate to be chosen. For every ∃r.
For a given such concept description ∃r.(C 1 , . . . , C k ) ∈ ∆ there may be more than one such tuple of r-successor; then we just select an arbitrary one of them. Let y 0 , . . . , y n−1 be the collection of all r-successors of x that are selected if we go through all ∃r.(C 1 , . . . , C k ) ∈ ∆ in this way. By the definition of N r (Γ), we have n ≤ N r (Γ), and thus n is an eligible choice for the size of the successor candidate. The components Ψ 0 , . . . , Ψ n−1 of the successor candidate are obtained by setting Ψ i := tp r-cl(∆) (y i ) for i < n. As the additional third argument, we pass y i to the recursive call of EL (n) C-World with first two arguments Ψ i and r-cl(∆). It is routine to show that, when guided in this way, the algorithm returns true.
Let us now turn to the case of satisfiability w.r.t. a general TBox. Let C be a concept and T a TBox. A set Ψ ⊆ cl(C, T ) is a type for C and T if it is a type for cl(C, T ) and additionally satisfies the following property: for all D E ∈ T , A type Γ is called moribund w.r.t. a set of types T if there exists a role name r ∈ rol ∃ (Γ) such that there is no sequence Φ 0 , . . . , Φ n−1 ∈ T with n ≤ N r (Γ) that is a successor candidate for Γ w.r.t. r and cl(C, T ).

Lemma 4.4
The procedure EL (n) C-Elim introduced in Fig. 2 decides satisfiability of C 0 w.r.t. T in exponential time.
Proof. The repeat loop of EL (n) C-Elim terminates after at most exponentially many steps since there are exponentially many types and, in each elimination define procedure EL (n) C-Elim(C, T ) Set i := 0 and T 0 to the set of all types for C and T round, at least one type is eliminated. Checking whether a type is moribund can be done in exponential time since there are at most exponentially many sequences of types of length at most N r (Γ). By Lemma 4.2, for each such sequence, it can be checked in polynomial time whether it is a successor candidate. Thus, EL (n) C-Elim is a (deterministic) exponential time procedure.
Assume that EL (n) C-Elim terminates returning true, let T be the set of types that have not been eliminated, and let Γ C 0 ∈ T be such that C 0 ∈ Γ C 0 . Let Γ ∈ T and r ∈ rol ∃ (Γ). Since Γ was not eliminated, it has a successor candidate Ψ 0 , . . . , Ψ n−1 where all the components Ψ i also belong to T. It should be noted, however, that these types need not be pairwise distinct. For this reason, it is not enough to take just the types in T as the elements of our model. To have enough copies of each type available, we define and generate N copies of each type in T. Now, the interpretation I is defined as follows: • Let (Γ, i) ∈ ∆ I and r ∈ rol ∃ (Γ), and assume that Γ contains m existential restrictions for r. Since Γ was not eliminated, these restrictions have suc- to be the set of r-successors of Γ in I.
It is straightforward to prove by structural induction C that, for all (Γ, i) ∈ ∆ I and all C ∈ cl(C, T ), we have (Γ, i) ∈ C I iff C ∈ Γ.
It follows that (Γ C 0 , 1) ∈ C I 0 . In addition, if D E ∈ T and (Γ, i) ∈ D I , then D ∈ Γ, and thus E ∈ Γ by the definition of the notion "type for C 0 and T ." Thus, we also have (Γ, i) ∈ E I . To sum up, we have shown that I is a model of T that interprets C 0 as a non-empty set.
Conversely, assume that C 0 is satisfiable w.r.t. T , and let I be a model T such It is readily checked that no type in T :

A tractable sublanguage
In the chemical process engineering application mentioned above [24], the full expressive power of EL (n) C is actually not needed. This application is concerned with supporting the construction of mathematical models of process systems by storing building blocks for such models in a class hierarchy. In order to retrieve building blocks, one can then either browse the hierarchy or formulate query classes. In both cases, the existence of efficient algorithms for computing subsumption between class descriptions is an important prerequisite.

Restricted EL (n) -concept descriptions
The frame-like formalism for describing classes of such building blocks introduced in [24] can be expressed in the sublanguage EL (n) of EL (n) C, which allows for conjunction, n-ary existential restrictions, and the top concept. Moreover, since in each frame a given slot-name can be used only once, it is sufficient to consider restricted EL (n) -concept descriptions where in each conjunction there is at most one n-ary existential restriction for each role: an EL (n) -concept description is restricted iff it is of the form As in the case of EL [5], the corresponding DL with unary existential restrictions, each restricted EL (n) -concept description C can be translated into an EL (n)description tree T C , where the nodes are labeled with sets of concept names and the edges are labeled with role names. Formally, this tree is described by is the root, and : V −→ 2 N C is the node labeling function. The set of all concept names occurring in the top-level conjunction of C yields the label (v 0 ) of the root v 0 , and each existential restriction ∃r.(C 1 , . . . , C n ) in this conjunction yields n r-successor of v 0 that are the roots of the trees corresponding to C i . For example, the restricted EL (n) -concept descriptions  x 8 x 3 x 7 x 2 Figure 3: Two EL (n) -description trees.
In [5], it was shown that subsumption between EL-concept descriptions corresponds to the existence of a homomorphism between the corresponding description trees. In EL (n) we must additionally require that the homomorphism is injective.
This homomorphism is an embedding iff the mapping ϕ : For example, mapping y i to x i for i = 1, . . . , 6 yields an embedding from the description tree on the right-hand side of Fig. 3 to the description tree on the left-hand side. If we changed the label of x 6 to {B}, then there would still exist a homomorphism between the two trees (mapping both y 5 and y 6 onto x 5 ), but not an embedding.
Theorem 5.2 Let C, D be restricted EL (n) -concept descriptions and T C , T D the corresponding description trees. Then C D iff there exists an embedding from The proof of this theorem is similar to the proof of the corresponding result for EL [5].
First, note that any interpretation can be viewed as a graph. An • given a node u ∈ ∆ I , its label is (u) = {A | A is a concept names such that u ∈ A I }, • and E = {(u, r, v) | (u, v) ∈ r I }.
Conversely, any EL (n) -graph G obviously represents an interpretation I G . For example, the EL (n) -graph depicted on the left-hand side of Fig. 3 represents the interpretation I = (∆ I , · I ), where it satisfies 2. and 3. of Definition 5.1, and the following local injectivity condition: Obviously, any embedding between EL (n) -description trees is also an EL (n) -homomorphism. Conversely, if ϕ : T 1 −→ T 2 is an EL (n) -homomorphism between the EL (n) -description trees T 1 , T 2 that maps the root of T 1 onto the root of T 2 , then ϕ is an embedding between these trees. In addition, if ϕ 1 : T 1 −→ T 2 is an embedding between EL (n) -description trees and ϕ 2 : T 2 −→ G is an EL (n)homomorphism, then their composition ϕ 1 • ϕ 2 : T 1 −→ G is also an EL (n)homomorphism.
Lemma 5.4 Let C be a restricted EL (n) -concept description, I an interpretation, and d 0 ∈ ∆ I . Then d 0 ∈ C I iff there is an EL (n) -homomorphism ϕ : T C −→ G I that maps the root of T C onto d 0 .
Proof. The restricted EL (n) -concept description C is of the form • it has a root v 0 with label (v 0 ) = {A 1 , . . . , A n }; • for 1 ≤ i ≤ m and 1 ≤ j ≤ k i , this root has an r i -successor v i,j that is the root of the EL (n) -description tree T B i,j corresponding to B i,j .
Let G I = (∆ I , E I , I ).
First, assume that d 0 ∈ C I . Then d 0 ∈ A I i for i = 1, . . . , n, which shows that (v 0 ) ⊆ I (d 0 ). Thus, if we define ϕ(v 0 ) = d 0 , then 2. of Definition 5.1 is satisfied. In addition, for 1 ≤ i ≤ m and 1 ≤ j ≤ k i there are elements d i,j ∈ ∆ I such that By induction, there are EL (n) -homomorphisms ϕ i,j : We define ϕ : T C −→ G I as follows: It is easy to see that ϕ is indeed a well-defined EL (n) -homomorphism.
Second, assume that there is an EL (n) -homomorphism ϕ : T C −→ G I such that ϕ(v 0 ) = d 0 . By 2. of Definition 5.1, (v 0 ) = {A 1 , . . . , A n } ⊆ I (d 0 ), which shows that d 0 ∈ A I i for i = 1, . . . , n. To show d 0 ∈ C I , it remains to be shown that there are d i,j ∈ ∆ I (for 1 ≤ i ≤ m and 1 ≤ j ≤ k i ) such that If we define d i,j := ϕ(v i,j ), then the fact that ϕ is an EL (n) -homomorphism implies that the first and the second point are satisfied. In addition, the restriction ϕ i,j of ϕ to T B i,j is an EL (n) -homomorphism such that ϕ i,j (v i,j ) = d i,j . By induction, this shows that the third point is satisfied as well.
We are now ready to prove Theorem 5.2.
First assume that there is an embedding ϕ : where u 0 is the root of T C and v 0 is the root of T D . To show C D, let I be an interpretation, and assume that d 0 ∈ C I . By Lemma 5.4, there is an EL (n)homomorphism ϕ : Second, assume that C D. The EL (n) -description tree T C is an EL (n) -graph, and thus represents an interpretation I. Let u 0 be the root of T C . Since the identity map is an EL (n) -homomorphism from T C into T C that maps u 0 onto u 0 , Lemma 5.4 yields u 0 ∈ C I . But then C D implies u 0 ∈ D I . By Lemma 5.4, this means that there is an EL (n) -homomorphism ϕ : T D −→ T C such that ϕ(v 0 ) = u 0 where v 0 is the root of T D . As noted above, such an EL (n) -homomorphism is actually an embedding. This completes the proof of Theorem 5.2.

Deciding the existence of an embedding
To show that subsumption between restricted EL (n) -concept descriptions is a polynomial-time problem, it remains to be shown that the existence of an embedding can be decided in polynomial time. First, let us recall the well-known bottom-up approach for testing for the existence of a homomorphism [22,5].
Let T 1 = (V 1 , E 1 , v 0,1 , 1 ) and T 2 = (V 2 , E 2 , v 0,2 , 2 ) be two EL (n) -description trees, and assume that we want to check whether there is a homomorphism from T 1 to T 2 . The idea underlying the polynomial time test is to compute, for each v ∈ V 1 , the set δ(v) of all nodes w ∈ V 2 such that there is a homomorphism from the subtree of T 1 with root v to the subtree of T 2 with root w. Once these sets δ are computed for all nodes of T 1 , we can simply check whether v 0,2 belongs to δ(v 0,1 ). The sets δ(v) are computed in a bottom-up fashion, where a node is treated only after all its successor nodes have been considered: 6 1. If v is a leaf of T 1 , then δ(v) simply consists of all the nodes w ∈ V 2 such that 1 (v) ⊆ 2 (w).

2.
Let v be a node of T 1 and let (v, r 1 , v 1 ), . . . , (v, r k , v k ) be all the edges in E 1 with first component v. Since we work bottom up, we know that the sets δ(v 1 ), . . . , δ(v k ) have already been computed. The set δ(v) consists of all the nodes w ∈ V 2 such that (a) 1 (v) ⊆ 2 (w) and It is easy to show that this indeed yields a polynomial-time algorithm for checking the existence of a homomorphism between two EL (n) -description trees.
If we want to test for the existence of an embedding, we must modify Step 2 of this algorithm. In fact, we must ensure that distinct r-successors of v can be mapped to distinct r-successors of w. This can be achieved as follows: 2 . Let v be a node of T 1 , and for each role r let (v, r, v 1,r ), . . . , (v, r, v kr,r ) be the edges in E 1 with first component v and label r. Since we work bottom up, we know that the sets δ(v 1,r ), . . . , δ(v kr,r ) have already been computed. The set δ(v) consists of all the nodes w ∈ V 2 satisfying the following two properties: (a) 1 (v) ⊆ 2 (w), (b) for all roles r, the family F r (w) := (S 1,r (w), . . . , S kr,r (w)) has an SDR, where the members of this family are defined as Obviously, the existence of an SDR for F r (w) allows us to map the r-successors of v to distinct r-successors of w, and thus construct an embedding. For this algorithm to be polynomial, it remains to be shown that the existence of an SDR can be decided in polynomial time. Note that Hall's characterization of the existence of an SDR obviously does not yield a polynomial-time procedure. However, checking for the existence of an SDR is basically the same as solving the maximum bipartite matching problem, which can be done in polynomial time since it can be reduced to a network flow problem [11].
To be more precise, let (L ∪ R, E) be a bipartite graph, i.e., L ∩ R = ∅ and E ⊆ L × R. A matching is a subset M of E such that each node in L ∪ R occurs at most once in M . This matching is called maximum iff there is no other matching having a larger cardinality. As shown in [11], such a maximum matching can be computed in time polynomial in the cardinality of V and E.
Let F = (S 1 , . . . , S n ) be a finite family of finite sets, and let L := {1, . . . , n} and R = S 1 ∪ . . . ∪ S n . 7 We define the set of edges of the bipartite graph G F = (L ∪ R, E) as follows: It is easy to see that the family F has an SDR iff the corresponding bipartite graph G F has a maximum matching of cardinality n. In fact, (1, s 1 ), . . . , (n, s n ) is a maximum matching iff s 1 , . . . , s n is an SDR.
Thus, we have shown that the existence of an embedding can be decided in polynomial time. Together with Theorem 5.2, this yields the following tractability result: Corollary 5.5 Subsumption between restricted EL (n) -concept descriptions can be decided in polynomial time.
A first implementation of this polynomial-time algorithm behaves much better than the translation approach on the example concept descriptions C, D from Section 3 and their obvious extensions to larger n. For small n, the subsumption relationship is found immediately (i.e., with no measurable run-time), and even for n = 100, the runtime (of our unoptimized implementation) is just 1 second. One could argue that the comparison of these results with the performance of Racer on the ALCQ-translations of C, D and their extensions to larger n is unfair since the culprit is the exponential translation rather than Racer. However, this is the only known translation of EL (n) -concept descriptions into a DL that can be handled by Racer, and it is the one originally used in the process engineering application.

Acyclic TBoxes
The frame-like formalism employed in the process engineering application allows to inherit properties from other frames. To represent this feature within our DL approach, a TBox is needed. However, it is sufficent to consider only acyclic EL (n) -TBoxes that are restricted in a similar way as restricted EL (n) -concept descriptions. Formally, an acyclic EL (n) -TBox is called restricted iff its concept definitions are of the form In the presence of TBoxes, it is obviously sufficient to have an algorithm that decides subsumption between defined concepts. In principle, subsumption between the defined concepts A and B w.r.t. an acyclic and restricted TBox can be decided by first expanding A and B into EL (n) -concept descriptions by replacing defined concept names by their definitions until no more defined concepts occur. Then, subsumption between the expanded concept descriptions can be decided without reference to a TBox. The definition of a restricted EL (n) -TBox makes sure that the expanded EL (n) -concept descriptions are actually restricted, and thus one can use the algorithm described in Section 5.2 to decide subsumption between them. However, it is well-know that the expansion process may lead to an exponential blow-up of the concept descriptions it is applied to [20]. Thus, the approach described above yields a subsumption algorithm that may need exponential time.
In this section we show how to obtain a polynomial-time subsumption algorithm in the presence of restricted acyclic EL (n) -TBoxes. To formulate this algorithm, it is convenient to assume that TBoxes are in a certain form: an EL (n) -TBox T is in normal form if it is acyclic, restricted, and, for all concept definitions A .
= C ∈ T , each defined concept name occurs at most once in C. It is not hard to see that every restricted acyclic EL (n) -TBox can be converted into normal form by introducing additional defined concept names. For example, the TBox can be rewritten into This conversion can be carried out in polynomial time, and it causes an at most quadratic blowup in size. In the following we assume that all TBoxes are in normal form.
Similar to our representation of restricted EL (n) -concept descriptions as trees, we represent EL (n) -TBoxes in normal form as EL (n) -directed acyclic graphs (DAGs), where the nodes (which are the defined concept names) are labelled with sets of primitive concept names, and the edges are labelled with role names. Formally, an EL (n) -DAG is given by a tuple G T = (V, E, ), where V is a set of nodes, E ⊆ V × N R × V is a set of N R -labeled edges that form a directed acyclic graph, and : V −→ 2 N C is the node labelling function. A given TBox T in normal form can be translated into the following EL (n) -DAG G T = (V T , E T , T ): • V T is the set of defined concept names in T ; • if A ≡ P 1 . . . P n ∃r 1 . (A 1,1 , . . . , A 1, 1  Note that EL (n) -DAGs are a special kind of EL-graphs as introduced for cyclic EL-TBoxes in [1]. The fact that the TBox T is assumed to be in normal form makes sure that, for every node A of G T , its successor nodes are distinct defined concepts of T . For a node v in an EL (n) -DAG G = (V, E, ) we write S G (v) to denote the set {u | (v, r, u) ∈ E for some r ∈ N R } of its successor nodes.
2. there exists an injection ϕ : It is easily seen that being embeddable is well-defined as the recursive use of "embeddable" in the definition refers only to nodes for which the maximum length of a path to a sink (i.e., a node without successor nodes) is strictly smaller.
Theorem 5.7 Let T be an EL (n) -TBox in normal form and A, B defined concepts in T .
Proof. Let A and B be the results of expanding A and B w.r.t. T . It is wellknown that A T B iff A B. By Theorem 5.2, the latter holds iff there exists an embedding from T B into T A . For proving Theorem 5.7, it thus suffices to show that there exists an embedding from The proof of the if-direction is by induction on the depth of T B . Assume that B is embeddable into A in G T , and let ϕ be the injection witnessing Property 2 in the definition of embeddable. For the induction step, let A ≡ C and B ≡ D be the definitions of A and B in T . Since B is embeddable into A in G T , by Property 2a of embeddable, and by construction of G T , every role occurring in an existential restriction in D also occurs in an existential restriction in C. Thus, C and D can be written as C = P 1 . . . P n ∃r 1 . (A 1,1 , . . . , A 1, 1 ) . . . ∃r m . (A m,1 , . . . , A m, m ), D = Q 1 . . . Q n ∃r 1 . (B 1,1 , . . . , B 1, 1 ) . . . ∃r m .(B m ,1 , . . . , B m , m ), with m ≤ m. Let Because T is assumed to be in normal form, the following holds for G T : (a) For (i, j) ∈ I C , the node A is connected (only) via r i to A i,j . Moreover, mapping (i, j) to A i,j yields a bijection between I C and {A i,j | (i, j) ∈ I C }.
(b) For (i, j) ∈ I D , the node B is connected (only) via r i to B i,j . Moreover, mapping (i, j) to B i,j yields a bijection between I D and {B i,j | (i, j) ∈ I D }.

Similar properties are satisfied in T A and T B :
(c) The root v A of T A has exactly one successor v i,j for each (i, j) ∈ I C . Moreover, v A is connected to v i,j (only) via r i , and v i,j is the root of the EL (n) -tree T A i,j obtained by expanding A i,j .
(d) The root u B of T B has exactly one successor u i,j for each (i, j) ∈ I D . Moreover, u B is connected to u i,j (only) via r i , and u i,j is the root of the EL (n) -tree T B i,j obtained by expanding B i,j .
Let (i, j) ∈ I D . By Property 2a of embeddable and due to the first part of (a) and (b) above, ϕ(B i,j ) = A i,k for some (i, k) ∈ I C . By Property 2b of embeddable, B i,j is embeddable into A i,k in G T . The induction hypothesis thus yields embeddings Now define the mapping ψ : V B → V A by setting ψ(u B ) := v A and taking the union of all the mappings ψ i,j . We claim that ψ is an embedding from T B into T A . As the ψ i,j are embeddings and their domains are disjoint, it suffices to consider u B and its successors. Property 1 of homomorphisms (mapping of root to root) is clearly satisfied. For Property 2 (inclusion of node labels), we can show as in the induction start that B (u B ) ⊆ A (v A ). Concerning Property 3 (edge labels), let (u B , r i , u i,j ) ∈ E B . Then ψ i,j (u i,j ) = v i,k for some k with 1 ≤ k ≤ i . By (c) above, we have (v A , r i , v i,k ) ∈ E A as required.
It remains to be shown that ψ is an embedding. Thus, let u i,j , u i ,j be distinct successors of u A . We must show that v i,k := ψ i,j (u i,j ) = ψ i ,j (u i ,j ) =: v i ,k . Since u i,j = u i ,j , the index pairs (i, j), (i , j ) are distinct elements of I D , and thus B i,j , B i ,j are distinct defined concepts. By the definition of the mappings ψ i,j and ψ i ,j , we have ϕ(B i,j ) = A i,k and ϕ(B i ,j ) = A i ,k , and thus A i,k , A i ,k are distinct defined concepts by the definition of embeddable. This shows that the index pairs (i, k), (i , k ) are distinct elements of I C , and thus v i,k = v i ,k by Property (c) above. Note that this also implies that the codomains of ψ i,j and ψ i ,j are disjoint. The First, we show that ϕ is an injection. If B i 1 ,j 1 = B i 2 ,j 2 , then (i 1 , j 1 ) = (i 2 , j 2 ), and thus u i 1 ,j 1 = u i 2 ,j 2 . Since ψ is an embedding, this implies that v i 1 ,j 1 := ψ(u i 1 ,j 1 ) = ψ(u i 2 ,j 2 ) =: v i 2 ,j 2 . Finally, this implies A i 1 ,j 1 = A i 2 ,j 2 by Property (a) above.
It thus remains to be shown that ϕ satisfies Properties 2a and 2b of embeddable. For Property 2a, let (i, j) ∈ I D . By Property (b), (B, r, B i,j ) ∈ E T implies r = r i . Due to Property 3 of homomorphisms, ψ(u i,j ) = v i,k for some (i, k) ∈ I C . Thus, ϕ(B i,j ) = A i,k . By Property (a) above, we have (A, r i , A i,k ) ∈ E T as required. To show Property 3a, consider again a tuple (i, j) ∈ I D . We must show that . It follows by the induction hypothesis that B i,j is embeddable into A i,k in G T , as required by Property 3a.
It remains to note that, to deciding whether B is embeddable into A in G T , we can use the marking algorithm for testing for the existence of an embedding between description trees presented in Section 5.2: the bottom-up labelling strategy of "treating a node only after all its successor nodes have been considered" works also on DAGs, and it is not hard to verify that B is embeddable into A in G T iff A occurs in the marking δ(B) of B.
Corollary 5.8 Subsumption in EL (n) w.r.t. a restricted acyclic TBox can be decided in polynomial time.

Unrestricted EL (n) -concept descriptions
In such concept descriptions, several n-ary existential restrictions for the same role r can occur in a conjunction, such as in the description If we translate this unrestricted EL (n) -concept description into a description tree, then we obtain the tree on the right-hand side of Fig. 3, which is also obtained as a translation of the restricted EL (n) -concept description C r := A ∃r. (A, B, ∃r.(A, A)).
To distinguish between these two descriptions, we introduce distinctness classes: for each node x in the tree and each role r, the r-successors of x are partitioned into such classes. For example, in the tree corresponding to C u , the r-successors of y 1 are partitioned into the sets {y 2 , y 3 }, {y 4 }, whereas there is only one distinctness class {y 2 , y 3 , y 4 } for these nodes in the tree corresponding to C r .
The notion of an embedding must take these distinctness classes into account. Instead of requiring that the homomorphism ϕ is injective, we require that it is injective on distinctness classes. Definition 6.1 Given two EL (n) -description trees T 1 , T 2 that are equipped with distinctness classes, a homomorphism ϕ : T 1 −→ T 2 is called an embedding iff for each node x in T 1 and each distinctness class {x 1 , . . . , x k } of the r-successors of x, the nodes ϕ(x 1 ), . . . , ϕ(x k ) are distinct r-successors of ϕ(x).
However, if we just change the notion of an embedding in this way, then Theorem 5.2 obviously does not hold for unrestricted EL (n) -concept descriptions. In fact, if ϕ(x 1 ), . . . , ϕ(x k ) do not belong to the same distinctness class in T 2 , then we cannot be sure that they really represent distinct individuals. For example, if C = ∃r.A ∃r.B and D = ∃r. (A, B), then there is an embedding from T D into T C , but D does not subsume C. Thus, an obvious conjecture could be that the embedding must respect distinctness classes, i.e., we must require ϕ(x 1 ), . . . , ϕ(x k ) to belong to the same distinctness class. However, the following example shows that this requirement is too strong. Let C = ∃r.A ∃r. (B, B) and D = ∃r. (A, B). There is no embedding from T D to T C that respects distinctness classes, but it is easy to see that D subsumes C.
Before we can formulate a correct characterization of subsumption between unrestricted EL (n) -concept descriptions, we must introduce some notation.
Definition 6.2 Given an EL (n) -description tree T = (V, E, v 0 , ) where role successors are partitioned into distinctness classes, an identification on T is an equivalence relation ∼ on V such that v 1 ∼ v 2 implies that • there are u 1 , u 2 ∈ V and a role r such that v 1 is an r-successor of u 1 , v 2 is an r-successor of u 2 , and u 1 ∼ u 2 ; • if v 1 = v 2 , then v 1 , v 2 do not belong to the same distinctness class.
Any identification ∼ on T induces a description tree T /∼ whose nodes are the ∼-equivalence classes [v] ∼ := {u ∈ V | u ∼ v}, whose root is [v 0 ] ∼ , and whose edges and node labels are defined as follows: Note that the first condition on identifications in the above definition ensures that the graph defined this way is indeed a tree with root Before proving this theorem, let us point out that it yields an NP-algorithm for testing non-subsumption of unrestricted EL (n) -concept descriptions: guess in non-deterministic polynomial time an identification ∼ of T C , and then check in polynomial time (by a simple adaptation of the algorithm described in Section 5.2) whether there is an embedding from T D into T C /∼. Corollary 6.4 The subsumption problem for (unrestricted) EL (n) -concept descriptions is in coNP.
Before we can prove Theorem 6.3, we must first show that the auxiliary definitions and results from Section 5.1 can be adapted to the case of unrestricted EL (n)concept descriptions.
If G 2 is also an EL (n) -description tree and ϕ maps the root of T 1 onto the root of G 2 , then it is easy to see that ϕ is an embedding in the sense of Definition 6.1.
With this adapted notion of an EL (n) -homomorphism, the following analogon of Lemma 5.4 can easily be proved. Lemma 6.6 Let C be an (unrestricted) EL (n) -concept description, I an interpretation, and d 0 ∈ ∆ I . Then d 0 ∈ C I iff there is an EL (n) -homomorphism ϕ : T C −→ G I that maps the root of T C onto d 0 .
In order to prove "⇒" of Theorem 6.3, we assume that C D. Let u 0 be the root of T C , and ∼ an identification on T C . The EL (n) -description tree T C /∼ represents an interpretation I. It is easy to see that the mapping is an EL (n) -homomorphism with θ(u 0 ) = [u 0 ] ∼ . By Lemma 6.6, this implies [u 0 ] ∼ ∈ C I , and thus [u 0 ] ∼ ∈ D I . But then Lemma 6.6 also implies that there is where v 0 is the root of T D . As noted above, this homomorphism is in fact an embedding.
In order to prove "⇐" of Theorem 6.3, we assume that for every identification ∼ on T C there exists an embedding from T D into T C /∼. To show that this implies C D, let I be an interpretation, and assume that d 0 ∈ C I . By Lemma 6.6, this implies that there is an EL (n) -homomorphism ϕ : T C −→ G I such that ϕ(u 0 ) = d 0 , where u 0 is the root of T C . This homomorphism induces a binary relation ∼ ϕ on the nodes of T C , which we define by induction on the depth of nodes: • The root u 0 of T is the only node on depth 0, and we have u 0 ∼ ϕ u 0 .
• Assume that ∼ ϕ is already defined on nodes of depth n for n ≥ 0. If v 1 , v 2 are nodes on depth n + 1, then v 1 ∼ ϕ v 2 iff there are nodes u 1 ∼ ϕ u 2 on depth n and a role r such that v 1 is an r-successor of u 1 , v 2 is an r-successor of u 2 , and ϕ(v 1 ) = ϕ(v 2 ).
It is easy to see that ∼ ϕ is an identification on T C .
The EL (n) -homomorphism ϕ : T C −→ G I induces the following mapping from the nodes of T C /∼ ϕ to the nodes of G I : Note that the definition of ∼ ϕ implies that ϕ is well-defined.
By our assumption, there is an embedding ψ : T D −→ T C /∼ ϕ . We claim that the composition ψ • ϕ is an EL (n) -homomorphism from T D into G I such that where v 0 is the root of T D . By Lemma 6.6, this implies d 0 ∈ D I , which completes the proof of Theorem 6.3.
Second, let v be a node of T D and assume that A belongs to its label in T D . Since ψ is an embedding, this implies that A belongs to the label of [u] ∼ϕ := ψ(v). Thus, there is u ∼ ϕ u such that A belongs to the label of u in T C . Since ϕ is an EL (n) -homomorphism, this implies that A belongs to the label of ϕ(u ) in G I . However, since u ∼ ϕ u we know that ϕ(u ) = ϕ(u) = ϕ([u] ∼ϕ ) = ϕ(ψ(v)). This shows that A belongs to the label of ψ • ϕ(v) in G I .
Finally, assume that v 1 , v 2 are distinct r-successors of a common parent node v in T D that belong to the same distinctness class. Let [u] ∼ϕ := ψ(v), [u 1 ] ∼ϕ := ψ(v 1 ), and [u 2 ] ∼ϕ := ψ(v 2 ). Since ψ is an embedding, we know that ψ(v 1 ) = ψ(v 2 ), and thus u 1 ∼ ϕ u 2 . If we can show that this implies ϕ(u 1 ) = ϕ(u 2 ), then we are done: since, for i = 1, 2, we have ϕ( To show that u 1 ∼ ϕ u 2 implies ϕ(u 1 ) = ϕ(u 2 ), it is sufficient to show that, in T C , there are nodes w 1 ∼ ϕ w 2 such that u 1 is an r-successor of w 1 and u 2 is an r-successor of w 2 . Since ψ is an embedding, we know that, in T C /∼ ϕ , the node By the definition of T C /∼ ϕ , this implies that there are nodes w i , u i such that w i ∼ ϕ u and u i ∼ ϕ u i , and u i is an r-successor of w i . By the definition of ∼ ϕ , u i ∼ ϕ u i implies that there is a node w i ∼ ϕ w i such that u i is an r-successor of w i . Transitivity of ∼ ϕ yields w 1 ∼ ϕ w 2 .
This finishes the proof of Theorem 6.3.

Adding disjointness statements
In the chemical process engineering application motivating this paper, the realworld concepts expressed by concept names are often disjoint. For example, an object cannot be both an apparatus and a plant. Disjointness statements of the form dis(P, Q), where P, Q are concept names, allow us to express such additional knowledge. An interpretation I is a model of a set of disjointness statements D iff P I ∩ Q I = ∅ for all statements dis(P, Q) in D. Satisfiability and subsumption w.r.t. D are defined in the usual way: C is satisfiable w.r.t. D iff there is a model I of D such that C I = ∅; and C is subsumed by D w.r.t. D (C D D) iff C I ⊆ D I for all models I of D.
The following lemma shows that (un)satisfiability w.r.t. a set of disjointness statements is easy to decide.
The EL (n) -concept description C is unsatisfiable w.r.t. the set of disjointness statements D iff there is a disjointness statement dis(P, Q) in D and a node v in T C whose label contains P and Q.
Proof. If there is a disjointness statement dis(P, Q) in D and a node v in T C containing P and Q, then C is obviously unsatisfiable w.r.t. D.
Conversely, assume that, for all disjointness statement dis(P, Q) in D and all nodes v in T C , the label of v does not contain both P and Q. The EL (n) -description tree T C represents an interpretation I. Because of our assumption, this interpretation is in fact a model of D. The identity map is an EL (n)homomorphism from T C into T C that maps the root u 0 of T C onto u 0 . By Lemma 6.6, this implies that u 0 ∈ C I , and thus C is satisfiable.
How does adding disjointness statements influence the complexity of the subsumption problem? Both for restricted and for unrestricted EL (n) -concept descriptions, the characterization of the subsumption problem (Theorem 5.2 and Theorem 6.3) can easily be extended to deal with disjointness statements.
For restricted EL (n) -concept descriptions, the only effect that disjointness statements have is that they can make concepts unsatisfiable. Proof. The "if" direction of the theorem is trivial.
To show the "only-if" direction, assume that C D D and that C is satisfiable w.r.t. D. The EL (n) -description tree T C represents an interpretation I, and the assumption that C is satisfiable w.r.t. D implies that I is a model of D (see the proof of Lemma 7.1). In addition, as shown in the proof of Lemma 7.1, the root u 0 of T C satisfies u 0 ∈ C I . Thus, C D D yields that u 0 ∈ D I . By Lemma 5.4, this implies that there is an EL (n) -homomorphism ϕ : T D −→ G I = T C such that ϕ(v 0 ) = u 0 . As noted in Section 5.1, such a homomorphism is actually an embedding. By Theorem 5.2, this implies that C D.

Corollary 7.3
For restricted EL (n) -concept descriptions, subsumption w.r.t. disjointness statements can be decided in polynomial time.
The effect of disjointness statements is less trivial if we consider unrestricted EL (n) -concept descriptions. The reason is that disjointness statements can enforce r-successors to be interpreted by distinct objects even though they do not belong to the same distinctness class. This problem does not occur for restricted EL (n)concept descriptions since there all r-successors of a given node belong to the same distinctness class.
Before we can formulate a characterization of subsumption w.r.t. disjointness statements in the unrestricted case, we must modify the definition of an identification such that it takes disjointness statements into account. Let C be an (unrestricted) EL (n) -concept description. If the identification ∼ on T C is compatible with D, then the interpretation I represented by the tree T C /∼ is a model of D. In particular, the identity relation is compatible with D iff C is satisfiable w.r.t. D. If C is unsatisfiable w.r.t. D, then no identification on T C is compatible with D.
Theorem 7.5 Let D be a set of disjointness statements, C, D (unrestricted) EL (n) -concept descriptions, and T C , T D the corresponding description trees. Then C D D iff for every identification ∼ on T C that is compatible with D there exists an embedding from T D into T C /∼.
Proof. The proof of "⇒" is basically identical to the proof of "⇒" of Theorem 6.3. The only additional fact to note is that the compatibility of ∼ with D implies that the interpretation I represented by the tree T C /∼ is a model of D.
The proof of "⇐" is also very similar to the proof of "⇐" of Theorem 6.3. Here the only additional thing to note is that the fact that I is a model of D implies that ∼ ϕ is compatible with D. In fact, assume to the contrary that there are nodes v 1 ∼ ϕ v 2 in T C and a disjointness statement dis(P, Q) in D such that {P, Q} ⊆ (u) ∪ (v). But then the label of ϕ(v 1 ) = ϕ(v 2 ) in G I contains both P and Q, which shows that I does not satisfy the disjointness statement dis(P, Q).
Again, this theorem yields an NP-algorithm for non-subsumption, and thus the subsumption problem is in coNP. In the presence of disjointness constraints, we can also prove the matching lower bound. 8 Corollary 7.6 The subsumption problem for (unrestricted) EL (n) -concept descriptions w.r.t. disjointness statements is coNP-complete.
Proof. The coNP-upper bound can be show as in the case of unrestricted EL (n)concept descriptions without disjointness statements.
We show coNP-hardness by a reduction of graph 3-colorability to non-subsumption. A given undirected graph G = (V, E) is 3-colorable iff there is a mapping f : V −→ {1, 2, 3} such that {u, v} ∈ E implies f (u) = f (v). It is well-known (see

Related and future work
Polynomiality of the subsumption problem in EL was shown in [5] as a by-product of the characterization of subsumption via the existence of homomomorphisms between the corresponding description trees. This result can also be obtained as a consequence of the fact that the containment problem Q 1 ⊆ Q 2 for conjunctive queries is polynomial if Q 2 is acyclic [26,21]. Since it is easy to see that EL (n) -concept descriptions can be expressed by acyclic conjunctive queries with disequations [18], one might conjecture that polynomiality of subsumption in EL (n) follows from the corresponding result for acyclic conjunctive queries with disequations. This is not true, however. In fact, the containment problem for conjunctive queries becomes considerably harder if disequations (i.e., atoms of the form x = y for variables x, y) are allowed to occur in the conjunctive queries. For general conjunctive queries with disequations, the containment problem is Π p 2complete rather than NP-complete as in the case of conjunctive queries without disequations. Surprisingly, the problem remains Π p 2 -complete if Q 2 is restricted to being acyclic [18]. And even if both queries contain only disequations (and no database predicates), it is not hard to show by a reduction of the complement of the graph homomorphism problem that the containment problem is coNP-hard. Thus, the polynomiality result shown in the present paper does not follow from known results for containment of conjunctive queries with disequations.
In [9], it was shown that subsumption in EL remains polynomial even in the presence of GCIs, and this result was recently extended to a DL extending EL by several other interesting constructors [2]. Unfortunately, the results in [2] imply that subsumption in EL (n) becomes ExpTime-hard in the presence of GCIs.
The most interesting topics for future research are, on the one hand, to show that the exponential translation from EL (n) C into ALCQ given in Section 3 is optimal, i.e., to prove that there is no polynomial translation. On the other hand, the exact complexity of subsumption between unrestricted EL (n) -concept descriptions is not yet known. The best complexity upper-bound that we currently have is coNP (see Corollary 6.4). We conjecture that the problem is coNP-hard, but have not yet found an appropriate reduction from a coNP-complete problem.