Structural Subsumption Considered from an Automata-Theoretic Point of View

This paper compares two approaches for deriving subsumption algorithms for the description logic ALN: structural subsumption and an automata-theoretic characterization of subsumption. It turns out that structural subsumption algorithms can be seen as special implementations of the automata-theoretic characterization


Introduction
Description logics (DLs) and corresponding DL systems can be used to represent the terminological knowledge of a problem domain in a structured and well-de ned way.Relevant concepts of the domain are described by concept descriptions, which are formed from atomic concepts (unary predicates) and roles (binary predicates) using concept forming operators provided by the DL.One of the most important inference services of a DL system is to arrange the represented concepts of the domain in a superconcept/subconcept hierarchy.This reasoning task is based on the subsumption relation between concept descriptions.Intuitively, a concept D subsumes a concept C if the set of individuals represented by D is a superset of the one represented by C.
In the literature several approaches to subsumption have been investigated (see 8] for an overview).In this work, we are interested in the relation between two of these approaches for languages of small expressive power, namely, structural subsumption and an automata theoretic approach.
Structural subsumption algorithms are e cient methods for deciding subsumption in description logics without full negation, disjunction, and existential restrictions.The structural subsumption algorithm employed by the system classic 5,6] is based on a speci c data structure for representing concept descriptions, called description graphs.In this context, subsumption is reduced to a structural comparison of description graphs.
Another approach for deciding subsumption in sub-languages of classic can be obtained from the automata-theoretic characterizations of subsumption w.r.t.greatest xedpoint (gfp) semantics in cyclic terminologies 1,12], which reduce the subsumption prob-1 lem to an inclusion problem for certain regular languages.In the case of acyclic terminologies (and thus in particular for concept descriptions), these languages turn out to be nite.At rst sight, there is no connection between these two approaches since they are based on rather di erent normal forms for concept descriptions.Intuitively speaking, structural subsumption is based on a normal form that applies the equivalence 8R:(A u B) 8R:Au8R:B as a rewrite rule from right to left, i.e., the descriptions are grouped w.r.t.role names, whereas the nite languages considered in the automata-theoretic approach correspond to a normal form obtained by applying the above equivalence from left to right.
Another di erence between the two approaches is that they describe decision procedures for subsumption on two di erent levels of abstraction.The structural subsumption algorithm for Classic is presented in 5,6] on the level of the data structure (namely, description graphs) used in the implementation.This provides a description of the algorithm that is very close to its actual implementation.Consequently, both the formal description of the algorithm and the proof of its correctness are quite complex 5,6,13].In contrast, the automata-theoretic approach reduces the subsumption problem to a formal language problem (namely, inclusion of nite or regular languages), which means that the description of the subsumption algorithm (and thus also the proof of its correctness) can be split into two independent parts: (i) the characterization of subsumption on the abstract formal language level, and (ii) an algorithm that decides the formal language problem.
The goal of this report is to show that there is in fact a tight relation between the approaches.In order to illustrate this relation, we will rst construct an isomorphism between the data-structures both approaches are working on.Then we point out, that structural subsumption algorithms based on description graphs can be seen as \parallel" implementations of the language inclusion tests required by the automata-theoretic characterizations of subsumption.
The report is structured as follows.We rst introduce the description logics of interest, namely, the small language FL 0 , which allows for value restrictions and conjunction, and the more expressive language ALN, which additionally provides us with atomic negation and number restrictions.In Section 3 we rst describe the automata theoretic approach to subsumption in FL 0 as well as structural subsumption for FL 0 -concept descriptions.
Thereafter, we discuss the tight relation between both approaches in detail.An extension of the comparison to ALN is given in Section 6.Both approaches to subsumption of concept descriptions have to be extended in order to cope with inconsistencies that are expressible in ALN.We will show, that there is a 1-1-correspondence between the extensions done in the automata theoretic approach and the way inconsistency is handled in structural subsumption.In the last section we summarize our analysis and give an overview over future work concerned with di erent approaches to subsumption in description logics.

Preliminary
We rst introduce syntax and semantics of the description logics FL 0 and ALN as well as the inference problem subsumption.
De nition 1 (Syntax and semantics) Let C be a set of concept names and a set of role names.ALN-concept descriptions are inductively de ned as follows: P and :P are concept descriptions for each concept name P 2 C. Let C; D be concept descriptions, R 2 a role name and n 2 IN.Then { C u D (conjunction), { 8R:C (value restriction), { ( n R) and ( n R) (number restrictions) are concept descriptions as well.
An interpretation I = (dom(I); I ) exists of the domain dom(I), i.e., a set of individuals, and an interpretation function I that maps each concept name P 2 C to a subset P I of dom(I) and each role name R 2 to a binary relation R I dom(I) dom(I).The extension of I to arbitrary concept descriptions is inductively de ned as shown in Table 1.The concept descriptions > and ?denote the entire domain and the empty set, respectively.
Syntax Semantics > dom(I) ?; :P dom(I) n P I C u D C I \ D I 8R:C fx 2 dom(I) j 8y : (x; y) 2 R ! y 2 C I g ( n R) fx 2 dom(I) j jfy j (x; y) 2 R I gj ng ( n R) fx 2 dom(I) j jfy j (x; y) 2 R I gj ng The description logic FL 0 only allows for the constructors conjunction (C u D) and value restriction (8R:C), whereas ALN additionally allows for primitive negation and number restrictions.Notice that both constructors > and ?are expressible in ALN because of > ( 0 R) and ?(P u:P).W.l.o.g.we will use them only as abbreviations rather than allowing for these constructors in ALN-concepts explicitly.
For some induction we will need the role depth of a concept description C: De nition 3 (Terminology) Let A be a concept name and C an ALN-concept description.Then A = C is a concept de nition.A nite set T of concept de nitions is an ALN-terminology if each concept name in T occurs at most once as left-hand side of a concept de nition.We call concept names appearing on the left-hand side of some concept de nition in T de ned concepts and primitive otherwise.The set of all de ned concepts in T is denoted by D T .T is called FL 0 -terminology i each right hand side of a concept de nition in T is an FL 0 -concept description.
An interpretation I = (dom(I); I ) is called model of T i A I = D I for each concept de nition in T. The descriptive semantics of a terminology T is de ned by the set of all models of T.
Notice that De nition 3 allows for cyclic terminologies.Formally, cycles are de ned as follows: Let A be a de ned concept and B an atomic concept in T. The concept A directly uses B if B occurs on the right-hand side of the concept de nition A = C in T. Let uses be the transitive closure of `directly uses'.Then T is cyclic i there exists a de ned concept name A in T that uses itself.Otherwise T is called acyclic.
De nition 4 (Primitive interpretation and its extension) Let T be a terminology, P 1 ; : : : ; P m the primitive concepts, R 1 ; : : : ; R k the role names, and A 1 ; : : : ; A n the de ned concepts in T. A primitive interpretation J = (dom(J); J ) consists of the domain dom(J) as well as the interpretation of the primitive concepts (P 1 J ; : : : ; P J m ) and the interpretation of the role names (R J 1 ; : : : ; R J k ).
An interpretation I = ( ; I ) of T is an extension of J i P I 1 = P J 1 ; : : : ; P I m = P J m and R 1 I = R 1 J ; : : : ; R k I = R k J .
In 2] it has been pointed out that for cyclic terminologies one can not uniquely extend an primitive interpretation J to a model I of T in general.Therefore, beside the descriptive semantics also xed-point semantics are considered in case of cyclic terminologies.For acyclic terminologies, descriptive and xed-point semantics coincide since there is always a unique extension I of a primitive interpretation J to a model of T. Therefore, in case of acyclic terminologies we do not distinguish between the primitive interpretation J and its extension.In particular, A J denotes the extension of A de ned by the model of T which is uniquely determined by J and T. Furthermore, one can de ne subsumption of atomic concepts which are de ned in di erent terminologies over the same set of primitive concepts and roles.
De nition 5 (Subsumption w.r.t.terminologies) Let T A and T B be acyclic ALNterminologies where A is a de ned concept in T A and B is de ned in T B .Furthermore, we assume that T A and T B are de ned over the same set of primitive concepts and roles.
We say that A in T A is subsumed by B in T B (A v T A ;T B B for short) i for all primitive interpretations J it holds A J B J .A is equivalent to B (for short A T A ;T B B) i A J = B J for all primitive interpretations J .
In the sequel, C, D denote concept descriptions, A, B refer to de ned concepts, P, Q are used for primitive concepts, and R, S for roles.3 The automata theoretic approach to subsumption for FL 0 The automata theoretic approach has been proposed in order to gain a more profound understanding of cyclic terminologies.The idea of this approach is to assign a semiautomaton A T to a terminology T, and to characterize di erent semantics as well as important inference problems, e.g.subsumption, using this semi-automaton.More precisely, Baader 2] has given an automata theoretic characterization of the semantics and subsumption of cyclic FL 0 -terminologies.These results are extended in 12] to the language ALN.

Concept descriptions and automata
As mentioned in the introduction, we are interested in comparing the automata theoretic approach to subsumption of concept descriptions and structural subsumption for concept descriptions.Therefore we represent concept descriptions by de ned concepts in (acyclic) FL 0 -terminologies.The semi-automata corresponding to these terminologies are speci ed recursively in order to simplify the comparison presented in Section 5.
De nition 6 (Terminology of C) Let C = P 1 u: : :uP n u8R 1 :C 1 u: : :u8R m :C m be an FL 0 -concept description.We represent C by the de ned concept A in the terminology T C of C. We refer to A as the de ned concept of C in T C .The terminology T C is recursively de ned by If C = P 1 u : : : u P n , then T C := fA = P 1 u : : : u P n g.If C = P 1 u: : :uP n u8R 1 :C 1 u: : :u8R m :C m , then let T C i be the recursively de ned terminologies of C i , 1 i m, and A i the de ned concept of C i in T C i , respectively.
W.l.o.g. the sets D T C i are pairwise disjoint.T C := fA = P 1 u : : : u P n u 8R 1 :A 1 u : : : u 8R m :A m g S 1 i m T C i Note that the set of primitive concepts and roles in T C coincides with the concept names occuring in C. Thus a primitive interpretation J of T C is also an interpretation for C. Furthermore, it is easy to see that T C is acyclic.By induction on the role depth of C it is not hard to show that A J = C J holds for all primitive interpretations J .
After introducing the terminologies representing concept descriptions, we now assign a nite semi-automaton A T = ( ; Q; ) to an FL 0 -terminology T 2].The alphabet denotes the set of role names in T and the concept names in T yield the set of states Q of A T .The transitions Q ( f"g) Q of A T are induced by the value restrictions in T, i.e., every concept de nition A = P 1 u : : : u P n u 8R 1 :A 1 u : : : u 8R m :A m in T gives rise to the transitions (A; "; P i ), 1 i n and (A; R i ; A i ), 1 i m.Notice that for acyclic terminologies the corresponding semi-automata are acyclic as well.In the general case of a cyclic terminology T the corresponding automaton A T is cyclic (see 2] for an example).
The semi-automaton A T C corresponding to the concept description C can be constructed recursively.
De nition 7 (Semi-Automaton of C) Let C be an FL 0 -concept description.The semi-automaton A T C of C is recursively de ned by depth(C) = 0: C = P 1 u : : : u P n : Let T C = fA = P 1 u : : : u P n g be the terminology of C. Then A T C := (;; fA; P 1 ; : : : ; P n g; ) where = f(A; "; P i ) j 1 i ng.depth(C) > 0: C = P 1 u : : : u P n u 8R 1 :C 1 u : : : u 8R m :C m : Let T C be the terminology of C with A = P 1 u: : :uP n u8R 1 :A 1 u: : :u8R m :A m 2 T C and A i the de ned concept name of C i , 1 i m.W.l.o.g. the sets D T C i , 1 i m, and D T C are pairwise disjoint.Further, let A T C i = ( i ; Q i ; i ) be the recursively de ned semi-automaton of C i , 1 i m.
Then A T C := ( ; Q; ) where := fR 1 ; : : : ; R m g S 1 i m i , Q := fA; P 1 : : : ; P n g S 1 i m Q i and := f(A; "; P i ) j 1 i ng f(A; R i ; A i ) j 1 i mg S 1 i m i .
In order to de ne the power set automaton in Section 3 and to compare automata and description graphs in Section 5, we need the following de nitions.Let A = ( ; Q; ) be a semi-automaton, p; q 2 Q, and I Q.There exists a path from p to q with label W 2 in A i there are states p 0 ; : : : ; p n 2 Q and R 1 ; : : : ; R n 2 f"g such that p 0 = p, p n = q, (p i 1 R i p i ) 2 for 1 i n, and R 1 : : : R n = W.The "-closure of a set I Q of states is de ned by "-closure(I) := fq 0 2 Q j there exists q 2 I and a (possibly empty) path from q to q 0 with label " in Ag: The W-successor set of I Q in A w.r.t.W 2 is de ned by  next A (I; W) := fq 0 2 Q j there exists q 2 "-closure(I) and a path from q to q 0 with label W in Ag.Example 8 Consider the concept descriptions C := 8R:P u8R:Qu8R:8S:P u8S:Q and D := 8R:8S:8R:P u8S:Q.These descriptions can be represented by the de ned concepts A and B in the following acyclic FL 0 -terminologies: T C : A = 8R:A 1 u 8R:A 2 u 8R:A 3 u 8S:A 4 ; T D : B = 8R:B 1 u 8S:B 2 ; A 1 = P; The terminologies T C and T D yield the semi-automata A T C and A T D of Figure 1.
For a de ned concept A and a primitive concept P in T, the language L A T (A; P) is the set of all words labeling paths in A T from A to P. Since for acyclic terminologies T the corresponding semi-automata A T are also acyclic, the languages L A T (A; P) are nite.
In the case of cyclic terminologies, these languages can be in nite.

The automata theoretic characterization of subsumption
In 2], an automata theoretic characterization of subsumption has been proved for cyclic FL 0 -terminologies w.r.t. to both descriptive semantics and xed-point semantics.We restrict our attention to acyclic terminologies.Thus, the subsumption relation coincides for these semantics.
Let T denote an ayclic FL 0 -terminology, A a de ned concept in T, P a primitive concept, and W a nite word over .It can be shown 2] that A v T 8W.P i W 2 L A T (A; P).Thus, roughly speaking, the language L A T (A; P) represents exactly those value restrictions that are satis ed by A. Consequently, an atomic concept A is subsumed by the atomic concept B i the set of value restrictions which have to be satis ed by A is a superset of the value restrictions which have to be satis ed by B. Formally, this fact is stated in Theorem 10 (Characterizing subsumption for FL 0 ) Let T A and T B be acyclic FL 0terminologies de ned over the same set of primitive concepts and roles.Furthermore, let A be de ned in T A and B in T B .Then A v T A ;T B B i for all primitive concepts P it holds that L A T B (B; P) L A T A (A; P): Proof.See 2].
In the example we have L A T D (B; P) = fRSRg 6 L A T C (A; P).By Theorem 10 this implies A 6 v T C ;T D B.
In order to decide subsumption based on Theorem 10, one must decide the inclusion problem for regular languages.Note, however, that in case of acyclic terminologies the considered languages are merely nite.
In the next section we recall the automata theoretic approach of deciding inclusion of regular languages given by nite automata (see 10] for details).

Deciding inclusion of regular languages
A nite automaton A is a semi-automaton with initial and nite states, i.e., A = ( ; Q; I; ; F) where denotes the nite alphabet, Q the nite set of states, I Q the set of initial states, Q ( f"g) Q the transition set, and F Q the set of nal states.Note that w.l.o.g.we can assume I to be a singleton.
In automata theory, the inclusion problem for regular languages L 1 and L 2 , de ned by A 1 and A 2 , respectively, is reduced to the emptiness problem: L 2 L 1 i L 2 \L 1 = ;.In order to decide this problem one rst computes a deterministic automaton B 1 for L 1 , i.e., one constructs the powerset automaton of A 1 .This automaton is de ned as follows: De nition 11 (Powerset automaton) Let A = ( ; Q; q 0 ; ; F) be a nite automaton.
Then the powerset automaton P(A) of A is de ned by P(A Q j G \ F 6 = ;g.From the powerset automaton B 1 of A 1 we obtain a nite automaton B 1 for L 1 by permuting the set of nal and non-nal states in B 1 .Note that this permutation only leads to an automaton accepting the complement of L 1 since B 1 is deterministic.Now, one can construct the product automaton of B 1 and A 2 (without "-transitions), which accepts the language L 2 \ L 1 .Emptiness of this language can be tested by deciding if the language accepted by the product automaton is empty.An algorithm deciding this problem can be described as follows: Search for a word W that is accepted by A 2 and B 1 .
If such a word exists, then the language accepted by the product automaton is not empty.Using the `Pumping-Lemma ' 10] it can be shown that it is su cient to consider words W up to a certain length, namely, the product of the size of A 2 and B 1 , which corresponds to the size of the product automaton, i.e., the number of states of the product automaton.
In the general case of arbitrary nite automata, the powerset automaton B 1 is of exponential size in the worst case, and inclusion of regular languages is PSPACE-complete 9].In case of acyclic nite automata, the powerset automaton still may be exponential in the size of A 1 , whereby inclusion is coNP-complete (see 14]).However, for terminologies constructed from concept descriptions it can be shown that this exponential blow-up cannot occur (see Section 3.3.1).
Example 12 (Example 8 continued) Consider the language L A T C (A; P) (see Example 8).L A T C (A; P) is accepted by the semi-automaton A T C if A is the initial state and P the nal state.The corresponding powerset automaton is shown in Figure 2 where fAg is the initial state and the states containing P are nal states.This automaton accepts the complement L A T C (A; P) of the language L A T C (A; P) if we specify fAg as initial state and all states not containing P as nal states, i.e., if we permute nal and non-nal states.
Hence, we have L A T D (B; P) \ L A T C (A; P) 6 = ; i there exists a word W such that (1) there is a path with label W in P(A T C ) leading from fAg to a state not containing P, and (2) there is a path with label W in A T D leading from B to P. In the example, RSR is such a word.Consequently, the inclusion L A T D (B; P) L A T C (A; P) does not hold.

The complexity of the powerset construction
In this section we are concerned with the complexity of the powerset construction for nite automata.As already mentioned, even for an acyclic nite automaton A the corresponding powerset automaton can be exponential in the size of A. The automata obtained from concept descriptions are acyclic but have a certain structure, namely a weak tree structure.We will show, that for this class of automata the powerset construction yields automata of linear size.
Intuitively speaking, a nite automaton A = ( ; Q; q 0 ; ; F) has a tree structure if there are no "-transitions in , q 0 has no predecessor, i.e., there is no transition of the form (q; R; q 0 ) 2 , and each state q 2 Q n fq 0 g is reachable from q 0 and has exactly one predecessor, i.e., for each q 2 Q n fq 0 g there exists a unique state q 0 2 Q and a unique symbol R 2 such that (q 0 ; R; q) 2 .Thus, the graphical notation of A yields a tree with root q 0 .Furthermore, for each q 2 Q there is exactly one path from q 0 to q in A and each q has a unique level, namely the length of the label of this path from q 0 to q in A.
The automata corresponding to concept descriptions do not have a tree structure, because they contain "-transitions (see Figure 1).But in these automata, we have only "-transitions of a special kind, i.e., each state that is reached via "-transitions has no outgoing transitions.In other words, these states, namely the primitive concepts occuring in the concept descriptions, can be seen as special leaves in these automata.
De nition 13 (weak tree structure) A nite automaton A = ( ; Q; q 0 ; ; F) has a weak tree structure i q 0 has no predecessor, i.e., there is no transition of the form (q; R; q 0 ) 2 , R 2 f"g, for each q 2 Q n fq 0 g, there exists at least one path from q 0 to q in A, { each q 2 Q " is only reached via "-transitions and has no outgoing transitions, i.e., for all q 2 Q " : (q 0 ; R; q) 2 =) q 0 2 Q and R = ", and there exists no transition of the form (q; R; q 0 ) 2 , R 2 f"g, { for each q 2 Q n fq 0 g there exists a unique state q 0 2 Q and a unique symbol R 2 such that (q 0 ; R; q) 2 .We de ne the level of each state q 2 Q , level(q) 2 IN, as the length of the unique path from q 0 to q in A.
Notice that if A has a weak tree structure then A is acyclic.Consequently, A is not complete, i.e., it exists q 2 Q; R 2 such that there exists no transition of the form (q; R; q 0 ) 2 .So one must introduce a sink state ; within the powerset construction.But due to the weak tree structure of A, each state in the powerset automaton has a special form.
To make this more precise, let P(A) = ( ; b Q; b q 0 ; b ; b F) be the powerset automaton obtained from A by De nition 11.We refer to the set of all states q 2 Q with level(q) = l by Q l := fq 2 Q j level(q) = lg, and the set of all states in P(A) that are reached by words of length l by b Q l := fnext A (fq 0 g; W) j W 2 l g n f;g.
Intuitively speaking, each state in the powerset automaton beside the sink has a certain level.Further, the states on one level yield a partition of the set Q l w.r.t.Q .Therefore, the number of states on level l in P(A) can be bounded by jQ l j and we get j b Consequently, we obtain the desired result, i.e., the size of the powerset automaton of an automaton A with weak tree structure is linear in the size of A. Formally, we show by induction on the level l 2 IN that for b There exists a word W of length l +1 such that W 2 L A (q 0 ; q).Thus, q 2 next A (fq 0 g; W).By de nition of b Q l+1 we get q 2 S I2 b There exists I 0 2 b Q l , R 2 such that I = next A (I 0 ; R).Let q 2 I.For q 2 Q " nothing has to be shown.Assume q 2 Q .We want to show q 2 Q l+1 .Since A has a weak tree structure q has a unique R-predecessor q 0 2 I 0 .By induction we get q 0 2 Q l and hence q 2 Q l+1 .Thus we have S I2 b So the second condition is satis ed.In order to prove claim (1) for l + 1 we show As already shown it is q 2 Q l+1 .This implies that there exists a unique q 0 2 Q l and a unique R 2 with (q 0 ; R; q) 2 .It follows R = R 1 = R 2 and q 0 2 I 0 1 , q 0 2 I 0 2 .Hence q 0 2 I 0 1 \ I 0 2 n Q " .By induction we get I 0 1 = I 0 2 .Since R 1 = R 2 this implies I 1 = I 2 .We sum up the complexity result for the powerset construction for nite automata with weak tree structure in 1 Notice that the upper bound 1 + jQj is reached, if A is deterministic and has a weak tree structure without "-transitions.In this case, the powerset construction would add the sink state and transitions leading to the sink in order to obtain a deterministic and complete automaton.
Lemma 14 Let A = ( ; Q; q 0 ; ; F) be a nite automaton with weak tree structure and P(A) = ( ; b Q; b q 0 ; b ; b F) the powerset automaton of A. Then the size of P(A) is linear in the size of A.
As an easy consequence of De nition 6 and De nition 7 we get that the automata corresponding to FL 0 -concept descriptions have a weak tree structure 2 .By Lemma 14 this implies that the size of the corresponding powerset automata is linear in the size of the concept descriptions.So the language inclusion tests required by the automata theoretic characterization of subsumption can be decided in time polynomial in the size of the concept descriptions.De nition 15 (FL 0 -description graphs) Let C be a set of primitive atomic concepts and a set of role names.An FL 0 -description graph over C and is a tuple G = (V; E; v 0 ; l) where V = fv 0 ; : : : ; v n g is a set of nodes, E V V a set of edges and v 0 2 V the root of G such that there exists no edge vRv 0 in E, for each v 2 V n fv 0 g there exists exactly one v 0 2 V and exactly one R 2 with v 0 Rv 2 E, each v 2 V is reachable from v 0 , i.e., there is a path v 0 R 1 v 1 : : : v n 1 R n v in E, and the label l(v) of a node v 2 V is a nite subset of C.
In the sequel, we will use the following notions referring to paths and subgraphs.p = w 0 R 1 w 1 : : : w n 1 R n w n is called path from w 0 to w n with label W = R 1 : : : R n in G i w i 1 R i w i 2 E for all 1 i n.The path p is called rooted path if w 0 = v 0 , i.e., p starts at the root of G. G j v denotes the subgraph of G with root v 2 V , i.e., G j v = (V 0 ; E 0 ; v; l 0 ) with V 0 := fw 2 V j exists path from v to w in Gg, E 0 := E \ V 0 V 0 , and l 0 (w) := l(w) for w 2 V 0 .The size of a description graph G = (V; E; v 0 ; l) is de ned as the sum of the number of nodes and edges and the sum of the size of all labels, i.e., jGj := jV j + jEj + X v2V jl(v)j: After introducing the syntax of description graphs and thus the data structure our structural subsumption test is working on, we now have to de ne the semantics: which set of individuals G I is determined by a description graph G under an interpretation I. De nition 16 (Extension of Description Graphs) Let I be an interpretation of C and , G = (V; E; v 0 ; l) a description graph over C and .The extension of a node v 2 V is recursively de ned by x 2 v I i x 2 P I for all P 2 l(v) and for all vRv 0 2 E and y 2 dom(I) with (x; y) 2 R I it holds that y 2 v 0 I .The extension of G is de ned as G I := v I 0 .We have introduced syntax and semantics of description graphs.Our aim is to use this representation formalism to characterize subsumption of FL 0 -concept descriptions.Therefore, we rst have to translate FL 0 -concept descriptions into (equivalent) description graphs.

Translating concept descriptions into description graphs
The translation of FL 0 -concept descriptions into description graphs is formalized by the algorithm in Figure 3. Obviously, the size of G C is linear in the size of C. In the sequel, G C denotes the description graph of C where C is an FL 0 -concept description and G C is obtained from C by the algorithm in Figure 3.
The translation is sound in the following sense: Lemma 17 (Equivalence of concepts and description graphs) Let C be an arbitrary FL 0 -concept description and G C the description graph of C. Then for all interpretations I it holds that C I = G I C .
Input: An FL 0 -concept C = P 1 u : : : u P n u 8R 1 :C 1 u : : : u 8R m :C m Output: The corresponding description graph G C = (V; E; v 0 ; l) m = 0 : G C := (fv 0 g; ;; v 0 ; l) where l(v 0 ) := fP 1 ; : : : ; P n g. m > 0 : Let G Ci = (V i ; E i ; v 0i ; l i ) be the recursively de ned description graph of C i ; 1 i m, where w.l.o.g. the V i are pairwise disjoint and v 0 6 2 S 1 i m V i .G C := (V; E; v 0 ; l) is de ned by := fP 1 ; : : : The edge labeled S from the root to a node labeled Q says that there is a value restriction 8S:C 0 in the top-level conjunction of C such that Q is the only primitive concept occurring in the top-level conjunction of C 0 , etc.

Structural subsumption for FL 0
Before we can decide whether C is subsumed by D based on a structural comparison of the description graphs, the graph for the subsumee C must be normalized by merging successor nodes reached by edges labeled by the same role name.This corresponds to applying the rewrite rule 8R:A u 8R:B !8R:(A u B) to the descriptions.Formally, we apply the normalization rule shown in Figure 5 as long as possible to an FL 0 -description graph G C .Notice that each iterated application of the rule in Figure 5 terminates since jGj > jG 0 j if G 0 is obtained from G by one application of the rule.Furthermore, it is not Let G = (V; E; v 0 ; l) be an FL 0 -description graph.G 0 = (V 0 ; E 0 ; v 0 ; l 0 ) is obtained from G by: Let v 2 V with n > 1 R-successors v 1 ; : : : ; v n and v 0 a new node not occuring in V .Then G 0 is de ned by merging v 1 ; : : : ; v n to one R-successor v 0 of v: V 0 := V n fv 1 ; : : : ; v n g fv 0 g, E 0 := E v i =v 0 j i = 1 : : : n] (each v i is replaced by v 0 ), l 0 (v 0 ) := S i=1:::n l(v i ) and l 0 (w) := l(w), w 2 V 0 n fv 0 g.De nition 20 (More speci c paths) Let G = (E; V; v 0 ; l) and G 0 = (V 0 ; E 0 ; v 0 0 ; l 0 ) be FL 0 -description graphs.A node v 2 V is more speci c than a node v 0 2 V 0 i l 0 (v 0 ) l(v).A rooted path p = v 0 R 1 v 1 : : : v n 1 R n v n in G is more speci c than the rooted path p 0 = v 0 . for all 0 i m it is v i more speci c than v 0 i .5 Comparing the approaches for FL 0

Now
We rst illustrate the tight relation between both approaches by Example 8.If we compare Figure 1 with Figure 4, then we see that the description graphs G C and G D essentially agree with the semi-automata A T C and A T D .The only di erence is that in the semiautomata there is only one state for every atomic concept and the primitive concepts P and Q are only reached from de ned concepts, e.g.A 1 , by "-transitions.In general, there exists an \isomorphism" between the semi-automaton A T C and the description graph G C corresponding to C. Roughly speaking, we can de ne a bijective mapping ' from the set of nodes in G C to the set of de ned concepts in A T C such that the label of a node v is the same as the set of all primitive concepts reached from '(v) by "-transitions.
Another obvious similarity between the automata-theoretic and the structural approach is that in both cases the automaton/graph for the subsumee C must be modi ed.
A closer look at Figure 2  G C can be obtained from P(A T C ) by ( 1) removing the names of de ned concepts from the states, and (2) by removing the sink state ; and the edges leading to this sink.
First, we consider the relationship between G C and A T C .Lemma 23 Let C be an FL 0 -concept description, G C = (V; E; v 0 ; l) the description graph of C, A T C = ( ; Q; ) the semi-automaton of C and D T C the set of de ned atomic concepts in A T C .Then there exists a bijective mapping ' : V !D T C such that 1. l(v) = "-closure('(v)) n f'(v)g for all v 2 V and 2. ('(v); R; '(w)) 2 i vRw 2 E for all R 2 .Proof.By induction on the role depth of C. depth(C) = 0: Then we have C = P 1 u: : : u P n .Furthermore, it is G C = (fv 0 g; ;; v 0 ; l) with l(v 0 ) = fP 1 ; : : : ; P n g and A T C = (;; fA; P 1 ; : : : ; P n g; ) with = f(A; "; P i ) j 1 i ng and D T C = fAg.We de ne ' : fv 0 g !fAg with '(v 0 ) := A. By construction it is "-closure(A) = fA; P 1 ; : : : ; P n g and hence l(v 0 ) = "-closure('(v 0 )) n f'(v 0 g.So, ' satis es the rst condition of Lemma 23.The second condition is satis ed trivially because there is no R-successors of v 0 in G C or of A in A T C .depth(C) > 0: Then we have C = P 1 u: : :uP n u8R 1 :C 1 u: : :u8R m :C m .By the algorithm in Figure 3 we have the description graph of C, G C = (V; E; v 0 ; l), the recursively de ned description graphs of C i , G C i = (V i ; E i ; v 0i ; l i ), 1 i m.For G C and G C i it holds that V i ; V j are pairwise disjoint, V = fv 0 g V 1 : : : V m , and E = fv 0 R i v 0i j 1 i mg E 1 : : : E m .
By De nition 6 and De nition 7 we have the recursively de ned terminologies of C and C i , 1 i m, T C and T C i with D T C = fAg D T C 1 : : : D T Cm , the de ned concept names A and A i of C and C i , 1 i m, and the recursively de ned semi-automata of C and C i , A T C = ( ; Q; ) and A T C i = ( i ; Q i ; i ).It is depth(C i ) < depth(C) for 1 i m.By induction there exist bijective mappings ' i : V i !D T C i such that ' i satis es 1 and 2 of Lemma 23 for G C i and A T C i .We de ne ' : V !D T C by '(v 0 ) := A and '(v) := ' i (v) for v 2 V i : Since by construction the V i as well as the D T C i are pairwise disjoint and each ' i is bijective, ' is a bijective mapping from V = fv 0 g V 1 : : : V m to D T C = fAg D T C 1 : : : D T Cm .Because of E = fv 0 R i v 0i j 1 i mg E 1 : : : E m and := f(A; "; P i ) j 1 i ng f(A; R i ; A i ) j 1 i mg 1 : : : m and the induction hypothesis it is ('(v); R; '(w)) 2 i vRw 2 E for all R 2 .As an easy consequence of the construction we have l(v 0 ) = "-closure(A) n fAg and by the induction hypothesis it is l(v) = "-closure('(v)) n f'(v)g for all v 2 V 1 : : : V m .So, ' satis es 1 and 2 of Lemma 23.
Example 24 (Example 8 and Example 18 continued) Consider the description graph G C = (V; E; v 0 ; l) in Figure 4 and the semi-automaton A T C in Figure 1  unde ned ; otherwise: The relation between the canonical description graph of C and the deterministic automaton becomes obvious, if we compare the normalization rule in Figure 5 and the de nition of the transition function 0 of B T C .
Let G C = (V; E; v 0 ; l) be the description graph of C, A T C = ( ; Q; ) the semiautomaton of C, and A the de ned concept of C. Let further fv 1 ; : : : ; v n g V be the non-empty set of all R-successors of v 0 in G C .The application of the normalization rule to v 0 and R in G C yields a description graph G 0 = (V 0 ; E 0 ; v 0 ; l 0 ) where v 1 ; : : : ; v n are merged to one new R-successor v new of v 0 .
On the other hand, de ning the R-successor 0 ("-closure(A); R) of "-closure(A) in B T C means merging all states that can be reached from a state in "-closure(A) by a path labeled with R to one new state in B T C .By Lemma 23 we know that the set f'(v 1 ); : : : ; '(v n )g is the set of all R-successors of A in A T C .By de nition of A T C , primitive concepts are only reached from de ned concepts by "-transitions and there are no edges leaving from . This shows that merging all R-successors of v 0 in G C to one new R-successor is the same as de ning 0 ("-closure(A); R).As a consequence of Lemma 23, property 1 we get More general, let G 0 C be the description graph obtained from G C by applying the normalization rule to each non-empty set of R-successors of v 0 , R 2 , and let v be an R-successor of v 0 in G 0 C .Generating G 0 C is the same as de ning the transition function 0 for "-closure(A) and each R 2 such that there is at least one R-successor of A in A T C .Furthermore, applying the normalization rule recursively to the subgraph G 0 C j v is the same as de ning 0 for each RW-successor set of "-closure(A), W 2 .Thus, So far, we have pointed out the tight relation between both approaches on the level of the data structures they are working on.In the sequel, we will argue that due to the similarity between the automata and description graphs structural subsumption algorithms can be seen as a special implementation of the language inclusion tests required by the automata theoretic characterization of subsumption of concept descriptions.
To make this more precise, let C; D (1) There is no path with label W in b G C .Without loss of generality we may assume that the path p in G D ends in a node with non-empty label set.Otherwise, the path could be extended appropriately because each leaf node in a description graph corresponding to an FL 0 -concept description has a non-empty label set.Obviously, such an extended path has still no more speci c path in b G C .Now assume that the primitive concept P is contained in the label set of the last node in p. Then we have W 2 L A T D (B; P) \ L A T C (A; P).In fact, W 2 L A T D (B; P) because the path p in G D to a node containing P yields a path in A T D from B to P. The fact that there is no (rooted) path with label W in b G C implies that the path with label W in P(A T C ) leads from the initial state to the sink state ;.Since ; does not contain P, it is a nal state for the automaton accepting L A T C (A; P).
(2) For p and the (unique) path b p with label W in b G C , the inclusion condition between the labels is violated by some primitive concept P, i.e., P belongs to the label of a node in p, but not to the label of the corresponding node in b p. Again, we may assume that p ends in the node v for which the inclusion condition is violated.(Obviously, the pre x of p that ends in v has also no more speci c path in b G C .)An argument similar to the one employed in the rst case can be used to show that W 2 L A T D (B; P) \ L A T C (A; P).
To sum up, we have shown that the existence of a rooted path in G D without a more speci c path in b G C implies that there is a primitive concept P such that L A T D (B; P) 6 L A T C (A; P).The converse of this implication can be shown analogously.

Subsumption for ALN
ALN is an extension of FL 0 which additionally provides us with primitive negation and number restrictions, i.e., atomic concepts of the form :P, ( n R), and ( n R) where P is a primitive concept, n a nonnegative integer, and R a role name.
In both approaches, number restrictions and negated primitive concepts are treated like new primitive concepts.In the description graphs, they may also occur in node labels.In the automata-theoretic approach, they give rise to new states in the semi-automaton of C, and to additional inclusion conditions.Notice that the automaton A T C corresponding to an ALN-concept description C also has a weak tree structure (see De nition 13).Thus, In the next two sections, we describe in detail the extensions that are made in both approaches to obtain a sound and complete subsumption algorithm.Thereafter, we extend our comparison from Section 5 to ALN and illustrate the 1-1-correspondence between these extensions.
6.1 The automata theoretic approach for ALN In 12], subsumption has been characterized for cyclic ALN-terminologies both for descriptive and xed-point semantics.As for FL 0 , we restrict our attention to acyclic ALN-terminologies.Thus, we do not have to distinguish di erent semantics.
In Section 3.2, we have seen that for an FL 0 -terminology T the languages L A T (A; P) represent exactly those value restrictions for P subsuming A, i.e., A v T 8W.P i W 2 L A T (A; P).Since inconsistencies are expressible in ALN, this equivalence does not hold for ALN-terminologies.There may be words such that A v T 8W.? (RS in Example 25).
These words are called A-excluding words; let E(A) denote the set of these words.Aexcluding words imply value restrictions that are not explicitly represented in L A T (A; P), i.e., even if the A-excluding word W is not contained in L A T (A; P) it obviously follows A v T 8W.P. Thus, in order to represent the value restrictions that are satis ed by A, beside L A T (A; P), additionally, A-excluding words have to be taken into account.Since we are interested in an automata theoretic characterization of subsumption, it is necessary to characterize A-excluding words based on A T .Proposition 26 (Exclusion) Let T denote an acyclic ALN-terminology, and let S denote the minimal subset of such that the following conditions hold: 1.If there is a primitive concept P or con icting number restriction ( l R) and ( r R), l > r, such that W 2 L A T (A; P) \ L A T (A; :P) or W 2 L A T (A; ( l R)) \ L A T (A; ( r R)), then W 2 S. 2. If WR 2 , W 2 , R 2 , and W 2 L A T (A; ( 0 R)), then WR 2 S. 3. If WR 2 S, W 2 , R 2 , and W 2 L A T (A; ( l R)), l 1, then W 2 S. 4. If W 2 S and V 2 , then WV 2 S.
Then S = E(A).
Proof.Consequence of Lemma 35 in 11].Since the proof is lengthy and technical it is omitted here.
In order to simplify the comparison of structural subsumption and the automata theoretic approach, the characterization stated in Proposition 26 di ers from the de nition of E(A) in 12, 11].However, it can easily be veri ed that these descriptions of E(A) are equivalent.
To illustrate Proposition 26 we consider the concept C 0 from Example 25.Obviously, RSS 2 E(A 0 ) since RSS 2 L A T C 0 (A 0 ; Q) \ L A T C 0 (A 0 ; :Q), i.e., any RSS-successor of an individual in A 0 must belong both to Q and :Q, which is impossible.Furthermore, by Proposition 26, 3. RS 2 L A T C 0 (A 0 ; ( 1 S)) implies RS 2 E(A 0 ).This is motivated by the following fact: Every RS-successor of an individual in A 0 also has an RSS-successor.
Since RSS 2 E(A 0 ) means that individuals in A 0 cannot have RSS-successors, this implies that they cannot have RS-successors.Finally, since RS 2 E(A 0 ) we know by Proposition 26, 4 that RSW 2 E(A 0 ) for all W 2 .These words must be contained in E(A 0 ) since A 0 v T C 0 8RS.? implies A 0 v T C 0 8RSW.?.Now, it is not hard to see that E(A 0 ) = RS .
It can be shown that L A T (A; P) E(A) contains exactly those words W such that A v T 8W.P. Intuitively, L A T (A; P) E(A) represents all value restrictions for P that are satis ed by A. For primitive negation :P there is an analogous set.
For number restrictions, beside excluding words, another phenomenon comes into the holds for all l r.Therefore, the value restrictions for ( l R) that are satis ed for A include all words in S r l L A T (A; ( r R)).In fact, it can be shown that S r l L A T (A; ( r R)) E(A) contains all value restrictions for ( l R), l > 0, that are satis ed by A.
For -restrictions S r l L A T (A; ( r R)) E(A) is not su cient for ( l R).If WR 2 E(A), then it follows A v T 8WR.?, Consequently, it holds A v T 8W.( 0 R).More generally, it can be observed that W 2 E(A)R 1 implies A v T 8W.( l R) for all l 0. 4Now, it can be shown that S r l L A T (A; ( r R)) E(A)R 1 represents exactly those value restrictions for ( l R) that are satis ed by A.
Intuitively, A is subsumed by B if the set of value restrictions that has to be satis ed by B are contained in those which are satis ed by A, i.e., A satis es at least those value restrictions that must hold for individuals in B.
Theorem 27 (Characterizing subsumption for ALN) Let T A and T B be acyclic ALNterminologies de ned over the same set of primitive concepts and roles.Furthermore, let A be de ned in T A and B in T B .Then A v T A ;T B B i 1. for all primitive concepts P it holds that L A T B (B; P) L A T A (A; P) E(A), 2. for all primitive negation :P it holds that L A T B (B; :P) L A T A (A; :P) E(A), 3. for all -restrictions ( l R), l > 0, it holds that L A T B (B; ( l R)) S r l L A T A (A; ( r R)) E(A), and 4. for all -restrictions ( l R) it holds that L A T B (B; ( l R)) S r l L A T A (A; ( r R)) E(A)R 1 .
In our example we have L A T C (A; P) = fR; RSg L A T C 0 (A 0 ; P) E(A 0 ) = fRg RS as well as L A T C (A; Q) = fR; Sg L A T C 0 (A 0 ; Q) E(A 0 ) = fS; Rg RS .The other languages for A are empty.Thus by Theorem 27, C 0 is subsumed by C.
Note that although we only consider acyclic terminologies the set of excluding words are either empty or in nite.Thus, in order to use Theorem 27 for deciding subsumption inclusions of in nite languages have to be tested.This is achieved by applying automata theoretic techniques.First we have to show that the set of excluding words is regular.In fact, it turns out 12] that for an ALN-terminology T the set E(A) of A-excluding words is accepted by a certain extension of the powerset automaton of A T .To see this, we need the following De nition 28 (Exclusion set) Let T be an ALN-terminology and A T = ( ; Q; ) the corresponding semi-automaton.The set F 0 Q is called exclusion set w.r.t.A T if there is a non-negative integer n, a word R 1 R n 2 , con icting number-restrictions ( l R) and ( r R), l > r, or a primitive concept P, and for all 1 i n there are integers m i 1 such that for F i := next " (F i 1 ; R i ), 1 i n, it holds that ( m i R i ) 2 F i 1 for all 1 i n and ( l R), ( r R) 2 F n or P; :P 2 F n .Now, an automata theoretic characterization of exclusion can be shown 12]: Lemma 29 Let T be an ALN-terminology, A T the corresponding semi-automaton, and A an atomic concept in T. Then E(A) = fW 2 j there is a pre x V of W such that next A T (fAg; V ) is an exclusion set or there is a pre x V R of W, V 2 , R 2 such that ( 0 R) 2 next A T (fAg; V )g.
Using this Lemma, we can construct a nite automaton accepting E(A): De nition 30 Let P(A T ) = ( ; b Q; b ) denote the powerset automaton of A T = ( ; Q; ) with initial state "-closure(A).We extend P(A T ) to B T = ( ; Q ; "-closure(A); ; fqg) by a new state q which is the nal state of B T .For every exclusion set F Q we add a transition (F; "; q) in B T and for every state F Q in B T and every -restriction ( 0 R) 2 F we add the transition (F; R; q).Finally, we add (q; R; q) for every R 2 .
If, furthermore, every F Q that contains P is a nal state in B T then this automaton accepts the language L A T (A; P) E(A).Analogously, one can de ne nal states in B T C such that the languages on the right-hand side of the inclusions in Theorem 27 for primitive negation, -restrictions, and -restrictions are accepted.Note, that for -restrictions some transitions must be added as well, since we are faced with E(A)R 1 instead of E(A).This extension can easily be achieved.Now, one can test the inclusions in Theorem 27 as pointed out in Section 3.3 using A T D and B T C .

Structural subsumption for ALN
In this section we extend our notion of description graphs to ALN by allowing for negated primitive concepts and number restrictions in the labels of nodes, i.e., an ALN-description graph G = (V; E; v 0 ; l) over a set C of primitive concepts and a set of role names is a nite tree with root v 0 such that E V V and for all v 2 V l(v) is a nite subset of C f:P j P 2 Cg f( n R) j n 2 IN; R 2 g f( n R) j n 2 IN; R 2 g.The semantics of ALN-description graphs is de ned in De nition 31 (Extension of ALN-description graphs) Let I be an interpretation of C and , G = (V; E; v 0 ; l) an ALN-description graph over C and .The extension of a node v 2 V is recursively de ned by x 2 v I i x 2 P I for all P 2 l(v) and x 6 2 P I for all :P 2 l(v) and jfy j (x; y) 2 R I gj n for all ( n R) 2 l(v) and jfy j (x; y) 2 R I gj n for all ( n R) 2 l(v) and for all vRv 0 2 E and y 2 dom(I) with (x; y) 2 R I it holds that y 2 v 0 I .The extension of G is de ned by G I := v I 0 .
The recursive algorithm in Figure 3 for translating FL 0 -concept descriptions into FL 0description graphs can be easily extended to ALN.We are concerned with ALN-concept descriptions of the form C = P 1 u : : : u P n u :Q 1 u : : : u :Q k u ( 1 S 1 ) u : : : u ( l S l ) u ( 1 T 1 ) u : : : u ( r T r )u 8R 1 :C 1 u : : : u 8R m :C m : We replace the de nition of the label of v 0 in Figure 3  Proof.See 13].
Example 25 shows that Theorem 21 does not hold for ALN-concept descriptions C and D. Notice that the description graph G C 0 of C 0 depicted in Figure 8 is already deterministic, i.e., the normalization rule in Figure 5 is not applicable to G C 0 .We will deal with the problems caused by inconsistencies by applying additional normalization rules when computing the canonical description graph 5, 6, 13] (see Figure 9).
As for FL 0 -description graphs we rst obtain a deterministic graph by merging all R-successor nodes of a node v to one new R-successor of v (Rule 1).The rules 2, 3, and 4 cope with nodes labeled by inconsistent sets, i.e., nodes v with fP; :Pg l(v) or f( l S); ( r S)g l(v), l > r.Nodes labeled by inconsistent sets and the edges leading to these nodes are removed.In addition, if there was an edge labeled R from node v to the inconsistent node, the label of v is extended by ( 0 R) (rules 2, 3).This is due to the equivalence 8R:? ( 0 R).For the same reason, we have to remove each subgraph with root v if the label of the R-predecessor of v contains ( 0 R) (rule 4).
If the root v 0 is labeled by an inconsistent set, then the whole concept is inconsistent.
In this case, we remove all nodes except the root and all edges and label v 0 by ?(rule 7).The rules 5 and 6 deal with number restrictions.Using ( n R) v ( m R) i n m and ( n R) v ( m R) i n m, we can reduce all -restrictions and all -restrictions for an R 2 to one -restriction and one -restriction, respectively, in the label of a node v.
Example 33 (Example 24 continued) Consider the description graph G C 0 = (V 0 ; E 0 ; v 0 0 ; l 0 ) of C 0 in Figure 8.Because of fQ; :Qg l 0 (v 0 3 ), the node v 0 3 and the edge v 0 2 Sv 0 3 are removed, and ( 0 S) is added to the label of v 0 2 .Now, v 0 2 is labeled with f( 1 S); ( 0 S)g, which is again inconsistent.Consequently, it is removed, and ( 0 S) is added to the label of the S-predecessor node v 0 1 .The description graph obtained this way is depicted in Figure 10.
As for FL 0 each iterated application of the normalization rules terminates since jGj > jG 0 j if G 0 is obtained from G by applying one of the rules in Figure 9.As mentioned above, each rule is based on an equivalence between concept descriptions, e.g., 8R:C u 8R:D 8R:(C u D).Thus, it is not hard to see, that the rules are sound, i.e., if G 0 is obtained from G, then it is G I = G 0 I for all interpretations I.
In order to distinguish the two normal forms used in the structural approach for FL 0 and ALN, respectively, we refer to the description graph G 0 C that is obtained from G C by applying only the rst normalization rule in Figure 9  De nition 35 (More speci c nodes and paths) Let G = (E; V; v 0 ; l) and G0 = (V 0 ; E 0 ; v 0 0 ; l 0 ) be ALN-description graphs.A node v 2 V is more speci c than a node v 0 2 V 0 i for each primitive concept P 2 l 0 (v 0 ) it is P 2 l(v), for each negated primitive concept :P 2 l 0 (v 0 ) it is :P 2 l(v), for each ( 0 R) 2 l 0 (v 0 ), there exists ( R) 2 l(v) with 0 , and for each ( 0 R) 2 l 0 (v 0 ) there exists ( R) 2 l(v) with 0 .
A rooted path p = v 0 R 1 v 1 : : i for 1 i min(m; n), for all 0 i min(m; n) it is v i more speci c than v 0 i , and if n < m, then ( 0 R 0 n+1 ) 2 l(v n ).
The conditions on more speci c nodes v and v 0 ensure that the conditions given by atomic concepts in the label of v 0 are satis ed by each instance of v.As an example consider the node b v 0 1 in Figure 10 C2l(v) C: More generally, the conjunction of all atomic concepts in l 0 (v 0 ) subsumes the conjunction of all atomic concepts in l(v) if v is more speci c than v 0 .
Due to number restrictions of the form ( 0 R), a path, which is more speci c than a path p 0 , can be shorter than p 0 .To be more precise, let G = (V; E; v 0 ; l) and G 0 = (V 0 ; E 0 ; v 0 0 ; l 0 ) be description graphs.If v is the W-successor node of the root v 0 and ( 0 R) 2 l(v), then each instance x of v 0 has no WR-successor.Thus, all conditions on WR-successors v 0 of v 0  We summarize the results of the comparison.
There exists an isomorphism between the automaton A T C and the description graph G C of an ALN-concept description C.
We have de ned a bijective mapping from the FL 0 -canonical description graph G 0 C to the deterministic automaton B T C .
There is a 1-1-correspondence between the additional normalization rules for computing ALN-canonical description graphs and the characterization of A-excluding words by terms of the automaton B T C .We have shown that looking for a rooted path in G D without a more speci c rooted path in b G C corresponds to testing the set inclusions for all atomic concepts occuring in C and D in parallel.As a consequence of these results structural subsumption algorithms based on description graphs can be seen as a special implementation of the inclusion tests required by the automata theoretic characterization of subsumption for ALN-concept descriptions.

Conclusion and future work
We have shown that structural subsumption algorithms are special implementations of the language inclusion tests required by the automata-theoretic characterization of subsumption.This provides a more abstract understanding of how structural subsumption algorithms work.
More precisely, we have pointed out that there exists an isomorphic relation between The size of the automaton A T C corresponding to a concept description C is also linear in the size of C. Due to the weak tree structure of A T C , the powerset automaton of C is also linear in the size of C. The inclusion of regular languages can be decided in time polynomial in the size of the automata de ning these languages.Consequently, one can decide subsumption of concept descriptions within the automata theoretic approach in time polynomial in the size of the concept descriptions.
In future work, we will extend the comparison to cyclic terminologies, by comparing the automata-theoretic characterization of subsumption with the structural subsumption algorithm for cyclic terminologies realized in K- Rep 7].
The comparison between the structural and the automata-theoretic approach can also be extended to other inference tasks such as computing the least common subsumer (lcs) of ALN-concept descriptions.Again, the algorithm for computing the lcs based on description graphs 6] can be seen as a special implementation of the automata theoretic characterization of the lcs 3, 4].An advantage of the automata-theoretic approach is that it easily carries over to computing the lcs for concepts de ned by cyclic terminologies 3, 4].
(C) + 1, depth(( n R)) := depth(( n R)) := 1, depth(C 1 u C 2 ) := maxfdepth(C 1 ); depth(C 2 )g.De nition 2 (Subsumption) Let C; D be ALN-concept descriptions.D subsumes C (for short C v D) i C I D I for all interpretations I. C is equivalent to D (for short C D) i C v D and D v C, i.e., C I = D I for all interpretations I.

Figure 1 :
Figure 1: The semi-automata corresponding to T C and T D .

Example 9 (
Example 8 continued) Consider the semi-automata A T C and A T D in Figure 1 of the concept descriptions C and D from Example 8.In this example, we have L A T C (A; P) = fR; RSg, L A T C (A; Q) = fR; Sg, L A T D (B; P) = fRSRg, and L A T D (B; Q) = fSg.

;Figure 2 :
Figure 2: The powerset automaton of A T C

4
Structural subsumption algorithms based on description graphsIn this section we present a characterization of subsumption of FL 0 -concept descriptions based on structural subsumption inClassic 5,6].The idea behind is as follows: given two FL 0 -concept descriptions C and D, we translate the concept descriptions into equivalent description graphs G C and G D .A normalization of G C yields the canonical description graph b G C of C. Thereafter, we can decide C v D by some kind of structural comparison of b G C and G D .4.1 Description GraphsDescription graphs were introduced in 5, 6] for deciding subsumption of concept descriptions in Classic.Since FL 0 is a sublanguage of Classic, we rst con ne the notion of description graphs given in 5].Description graphs are rooted directed acyclic graphs whose nodes are labeled by sets of primitive concepts and whose edges are labeled by roles.Concept descriptions can be turned into description graphs by a straightforward translation of the syntactic structure of the descriptions.It will turn out, that the description graphs corresponding to FL 0 -concept descriptions are trees.

Figure 5 :
Figure 5: The normalization rule for FL 0 -description graphs

Figure 6 :
Figure 6: The canonical description graph corresponding to C.
we are equipped to characterize subsumption of FL 0 -concept descriptions by a structural comparison of description graphs.C is subsumed by D i the conditions to Wsuccessors of instances of D are subsets of the conditions to W-successors of instances of C for each W 2 .Intuitively speaking, these conditions to W-successors are represented by the labels of W-successor nodes in the corresponding description graphs b G C and G D , respectively.If the label of a W-successor node of the root contains the primitive concept P, then for each instance x of C all W-successors of x must be in the extension of P. Therefore, we can decide C v D by testing wether the label of each W-successor node in G D is a subset of the label of the W-successor node in b G C 3 for each W 2 .Formally, we can prove Theorem 21 (Structural subsumption for FL 0 ) Let C; D be FL 0 -concept descriptions, b G C the canonical description graph of C and G D the description graph of D. Then C v D i for each rooted path p in G D there exists a more speci c rooted path b p in b G C .Proof.See 13].Example 22 (Example 18 continued) Consider the FL 0 -concept descriptions C and D from Example 18, G D in Figure 4 and b G C in Figure 6.The path with label RS in b G C is more speci c than the path with label RS in G D .However, for the path with label RSR in G D there does not exist a more speci c path in b G C .Consequently, the structural subsumption test recognizes that C is not subsumed by D. An algorithm deciding C v D by Theorem 21 considers the (canonical) description graphs b G C and G D and tests wether there exists a more speci c rooted path b p in b G C for each rooted path p in G D .In other words, for each W-successor node v of v 0 in G D = (V; E; v 0 ; l) we test (1) wether there exists a W-successor node b v b l) or not and (2) if b v is the W-successor of b v 0 in b G C , wether l(v) b l(b v).If (1) or (2) does not hold, then C 6 v D; otherwise C v D. A more formal algorithm can be found in 13].
and Figure 6 reveals, that the powerset automaton P(A T C ) and the canonical description graph b G C are also essentially identical.To be more precise, b

2 V v 0 v 1 v 2 v 3 v 4 v 5 '
of the concept description C from Example 8.The mapping ' : V !D T C is given by v (v) 2 D T C A A 1 A 2 A 3 A 31 A 4 In the next step, we formalize the relation between the canonical description graph b G C of C and the power set automaton P(A T C ) of C. As already mentioned, there is no node in the canonical description graph of a concept description C corresponding to the sink state ; in the powerset automaton of C. The automaton directly corresponding to the canonical description graph of C can be obtained from the powerset automaton by eliminating the sink and all edges leading to the sink.In the sequel, this deterministic automaton of C is denoted by B T C .Analogous to De nition 11, B T C can be de ned by B T C := ( ; Q 0 ; 0 ) with Q 0 := fF Q j F 6 = ;; next A T C ("-closure(A); W) = F for a word W 2 g and for I 2 Q 0 , R 2 0 (I; R) := ( next A T C (I; R) 2 Q 0 ; next A T C (I; R) 6 = ; generating the canonical description graph b G C of C recursively is the same as de ning the deterministic automaton B T C of C recursively.As a consequence of the above observations we can recursively de ne a bijective mapping b ' from the set of nodes b V of the canonical description graph b G C = ( b V ; b E; v 0 ; b l) of C to the set of states Q 0 of the deterministic automaton B T C of C, such that b '(v 0 ) = "-closure(A), b '(v) = 0 ("-closure(A); W) if v is a W-successor of v 0 in b G C , b l(v) = b '(v) n D T C for all v 2 b V , and vRw 2 b be concept descriptions, b G C and G D the corresponding (canonical) description graphs, A T D the automaton of D with de ned concept B of D, and P(A T C ) the powerset automaton of C with de ned concept A of C. Assume that C 6 v D. By Theorem 21, the structural subsumption algorithm detects non-subsumption by nding a rooted path p in G D with label W such that there does not exist a more speci c rooted path b p in b G C .There are two possible reasons why this more speci c path does not exist in b G C : by l(v 0 ) := fP 1 ; : : : ; P n ; :Q 1 : : : ; :Q k ; ( 1 S 1 ); : : : ; ( l S l ); ( 1 T 1 ); : : : ; ( r T r )g: The algorithm obtained by this modi cation yields the ALN-description graph G C of an ALN-concept description C. Analogously to FL 0 , the translation is sound.Lemma 32 (Equivalence of concepts and description graphs for ALN) Let C be an arbitrary ALN-concept description and G C the description graph of C. Then for all interpretations I it holds that C I = G I C .
as the FL 0 -canonical description graph.The description graph b G C that is obtained from G C by applying all normalization rules in Figure 9 as long as possible is called ALN-canonical description graph.De nition 34 (ALN-canonical description graphs) Let C be an ALN-concept description and G C the description graph of C. The ALN-canonical description graph of G C is de ned as the description graph b G C that is obtained from G C by an iterated application of the normalization rules in Figure 9 such that no more rule is applicable to b G C .Notice that the size of the description graph G C as well as the size of the ALNcanonical description graph b G C of an ALN-concept description is linear in the size of C. Before we can formalize structural subsumption for ALN-concept descriptions, we have to generalize the notion of more speci c paths 6, 13].
and a node v labeled with l(v) = fP; ( 1 S)g.Obviously, Lemma 39 Let C; D be ALN-concept descriptions such that C 6 v D. There exists a rooted path p in G D with label W such that there exists no more speci c rooted path b p in b G C and W satis es the three conditions (a); : : : ; (c) of at least one of the four points 1:{4: in Proposition 38.Proof.By Theorem 36, there exists a rooted path p in G D such that there exists no more speci c rooted path in b G C .We have to consider several cases.1.b v 0 is not more speci c than v 0 .By De nition 35, one of the following cases holds.(a)There exists a primitive concept P such that P 2 l(v 0 ) and P 6 2 b l(b v 0 ).Then it is " 2 L A T D (B; P) and P 6 2 "-closure(A) in B T C .Since C 6 ?, "-closure(A)is not an exclusion set, and hence there exists no transition ("-closure(A); "; q) in B T C .So, W satis es the conditions (a){(c) of Proposition 38, 1.(b) There exists a primitive concept P such that :P 2 l(v 0 ) and :P 6 2 b l(b v 0 ).Analogous to (a), the three conditions in Proposition 38, 2. are satis ed by W.
(c) There exists ( 0 S) 2 l(v 0 ), and there exists no ( S) 2 b l(b v 0 ) with0 .By our assumption on D, it is 0 > 0. Then it is S r l f( l R)g\"-closure(A) = ;, thus the conditions in Proposition 38, 3. are satis ed.(d) There exists ( 0 S) 2 l(v 0 ) and there exists no ( S) 2 b l(b v 0 ) with 0 .Analogous to (c), the conditions in Proposition 38, 4. are satis ed. 2. Let 0 m < n be the maximal index such that there exists b p each b v i is more speci c than v i , 0 i m.The path b p is uniquely determined by R 1 : : : R m , because b G C is deterministic.Since b p is not more speci c than p, it is ( 0 R m+1 ) 6 2 b l(b v m ) (see De nition 35).(a) There exists an R m+1 -successor b v m+1 of b v m in b G C such that b v m+1 is not more speci c than v m+1 .It follows l(v m+1 ) 6 = ;.Similar to case 1. we can show that the conditions (a) and (b) of one of the points 1.{4. in Proposition 38 are satis ed by W = R 1 : : : R m+1 .Since b G C is canonical, it is ( 0 R i+1 ) 6 2 b l(b v i ) for 0 i < m. ( 0 R i+1 ) 6 2 b l(b v i ) for 0 i m implies that there is no path with label R 1 : : : R m from "-closure(A) to the accepting sink state q in B T C .So, W satis es (a){(c) for one of the points 1.{4. in Proposition 38.(b) There exists no R m+1 -successor b v of b v m in b G C .Without loss of generality, we may assume that the last node v in p has a non-empty label set (otherwise, p can be extended appropriately; see Section 5).Consequently, R 1 : : : R n satis es the conditions (a) and (b) of at least one of the four points 1. {4. of Proposition 38.As before, there exists no path with label R 1 : : : R m form "-closure(A) to the accepting sink state q in B T C .Because of b v m R m+1 b v 2 b E i 0 ( b '(b v m ); R m ) = b '(b v), there exists no R m+1 -successor of b '(b v m ) in the deterministic automaton B T C of C. By construction, it follows that the rejecting sink state ; is the unique state reached by R 1 : : : R n in B T C .In particular, there exists no path with label R 1 : : : R n from "-closure(A) to q in B T C .Thus, R 1 : : : R n satis es the conditions (a){(c) for at least one of the points 1. {4. in Proposition 38.
the description graph G C of a concept description C and the automaton A T C representing the acyclic terminology T C of C. Furthermore, we introduced a bijective mapping from the canonical description graph b G C of C to the deterministic automaton B T C .We have seen that the canonical description graph b G C of C and the description graph of D is linear in the size of C and D, respectively.Hence, wether D subsumes C can be decided in time polynomial in the size of C and D.

Table 1 :
Semantics of concept descriptions