What s in an Attribute? Consequences for the Least Common Subsumer

Transcription

1 What s in an Attribute? Consequences for the Least Common Subsumer Ralf Küsters LuFG Theoretical Computer Science RWTH Aachen Ahornstraße Aachen Germany kuesters@informatik.rwth-aachen.e Alex Borgia Department of Computer Science Rutgers University Piscataway, NJ USA borgia@cs.rutgers.eu Abstract Functional relationships between objects, calle attributes, are of consierable importance in knowlege representation languages, incluing Description Logics (DLs). A stuy of the literature inicates that papers have mae, often implicitly, ifferent assumptions about the nature of attributes: whether they are always require to have a value, or whether they can be partial functions. The work presente here is the first explicit stuy of this ifference for (sub-)classes of the Classic DL, involving the same-as concept constructor. It is shown that although etermining subsumption between concept escriptions has the same complexity (though requiring ifferent algorithms), the story is ifferent in the case of etermining the least common subsumer (lcs). For attributes interprete as partial functions, the lcs exists an can be compute relatively easily; even in this case our results correct an exten three previous papers about the lcs of DLs. In the case where attributes must have a value, the lcs may not exist, an even if it exists it may be of exponential size. Interestingly, it is possible to ecie in polynomial time if the lcs exists. This research was supporte in part by grant NSF IRI

2 Contents 1 Introuction 3 2 Formal Preliminaries 5 3 Subsumption Description Graphs Translating Concept Descriptions to Description Graphs Translating a Description Graph to a Concept Description Canonical Description Graphs Subsumption Algorithm Sounness an Completeness of the Subsumption Algorithm Computing the lcs in Classic The Prouct of Description Graphs Computing the lcs The Lcs for Same-as an Total Attributes The Existence of the Lcs Characterizing the Existence of an Lcs Deciing the Existence of an Lcs Computing the Lcs Conclusion 39 2

3 1 Introuction Knowlege representation systems base on Description Logics (DL systems) have been the subject of continue attention in Artificial Intelligence, both as a subject of theoretical stuies (e.g., [2, 11, 14]) an in applications (e.g., [1, 17]). More impressively, DLs have foun applications in other areas involving information processing, such as atabases [6, 16], semi-structure ata [12, 13], information integration [15, 9], as well as more general problems such as configuration [28] an software engineering [7, 21]. In Description Logics, one takes an object-centere view, where the worl is moele as iniviuals, connecte by binary relationships (here calle roles), an groupe into classes (calle concepts). In every DL system the concepts of the application omain are escribe by concept escriptions that are built from atomic concepts an roles using the constructors provie by the DL language. For example, using the atomic concepts Moel, Manufacturer, an INTEGER as well as the roles moel, seats, an maeby the concept Car can be escribe as follows: Car := maeby moel maeby seats moel seats seats.integer moel.m oel maeby.m anuf acturer By using same-as equalities (like seats moel seats), we ensure that the manufacturer (number of seats) of a car is the same as the manufacturer (number of seats) of the moel of that car. DLs support a variety of inferences, incluing eciing if a concept subsumes another one. Subsumption algorithms allow one to etermine subconcept-superconcept relationships: C is subsume by D (C D) if an only if all instances of C are also instances of D, i.e., the first escription is always interprete as a subset of the secon escription. The traitional inference problems for DL systems, such as subsumption, inconsistency etection, membership checking, are by now well-investigate. Algorithms an etaile complexity results for realizing such inferences are available for a great variety of DL languages of iffering expressive power (e.g., [23, 3, 29, 22, 25]). In most knowlege representation systems, incluing DLs, functional relationships, here calle attributes (in the literature also calle features), are istinguishe as a subclass of general relationships, at least in part because functional restrictions occur so frequently in practice. In the above example, maeby an seats are meant to be attributes, thus making unnecessary number restrictions like (( 1 maeby) ( 1 maeby)). In aition, istinguishing attributes helps ientify tractable subsets of DL constructors: in classic, coreferences between attribute chains (as in the above examples) can be reasone with efficiently [10], while if we change to roles (e.g., allowe (repairs owneby repairspaifor)), the subsumption problem becomes uneciable [30]. The istinction between roles an attributes in DLs is both theoretically an practically well unerstoo. However, it turns out that there is a istinction to be mae between attributes being interprete as total (total attributes) or partial functions (partial attributes). This istinction is useful in practice, since there is a ifference between a car not having a license plate, an having a license plate whose value is not currently known. In DLs, the latter is moele by having the attribute haslicenseplate, with restriction ( 1 haslicenseplate), while the former is moele with the restriction ( 0 haslicenseplate). If attributes were total functions, the last assertion woul immeiately lea to a contraiction. It turns out that the stuy of the theoretical implications of this istinction has slippe through the cracks of contemporary research. The purpose of this paper is to show explicitly an precisely the effect of allowing attributes to be total or partial. Specifically, we show that for one group 3

4 of DLs relate to the Classic system, although this istinction oes not affect the complexity of computing subsumption (the etails of the algorithm o nee to be change), it oes have a significant impact on the problem of computing the least common subsumer (lcs) of concepts, i.e., the most specific concept escription subsuming a set of given concepts. Since, as etaile below, a number of publishe papers on the above topic mae iffering assumptions about the nature of attributes, but i not highlite these ifferences, the more general lesson we want to impart is that in all knowlege representation schemes an investigations it is very important to be clear about whether attributes are total or partial. Least common subsumer The lcs was first introuce as a new inference problem in DLs in [20]. One motivation for consiering the lcs is to use it as an alternative to isjunction. The iea is to replace isjunctions like C 1 C n by the lcs of C 1,..., C n. In [8, 20], this operation is calle knowlege-base vivification. Although, in general, the lcs is not equivalent to the corresponing isjunction, it is the best approximation of the isjunctive concept within the available language. Using such an approximation is motivate by the fact that, in many cases, aing isjunction woul increase the complexity of reasoning. 1 As propose in [4, 5], the lcs operation can be use to support the bottom-up construction of DL knowlege bases, where, roughly speaking, starting from typical examples an lcs algorithm is use to compute a concept escription that (i) contains all these examples, an (ii) is the most specific escription satisfying property (i). In [4], such an algorithm has been presente for cyclic ALN -concept escriptions; ALN is a sublanguage of Classic allowing for concept conjunction, primitive negation, value restrictions, an number restrictions. Baaer et al. [5] have propose an lcs algorithm for a DL allowing for existential restrictions instea of number restrictions. Originally, the lcs was introuce as an operation in the context of inuctive learning from examples [20], an several papers followe up this lea. The DLs consiere were mostly sublanguages of Classic which allowe for same-as equalities, i.e., expressions like maeby moel maeby. Cohen et al. [20] propose an lcs algorithm for ALN an a language that allows for concept conjunction an same-as, which we will call S. In [18], the algorithm for S was extene to Core- Classic, which aitionally allows for value restrictions (see [19] for experimental results). Finally, Frazier an Pitt [24] presente an lcs algorithm for full Classic. All these algorithms are base on a translation of concept escriptions into so-calle escription graphs, which ha been use in [10] to ecie subsumption. More precisely, the lcs is compute in three steps: First, the concept escriptions are turne into escription graphs. Secon, the lcs is compute as the prouct of the escription graphs. Finally, the prouct graph thus obtaine is turne back into a concept escription, representing the lcs of the given concepts. However, there is a mismatch between the semantics of attributes unerlying the subsumption algorithm on the one han an the lcs algorithms on the other han. In particular, in the work of Borgia an Patel-Schneier [10], attributes are interprete as total functions, whereas a careful examination of the lcs algorithms propose, especially the ones involving same-as, reveals that the lcs is compute for DLs with partial attributes. Furthermore, it turns out that the lcs algorithm presente by Frazier an Pitt [24] oes not hanle properly inconsistency, which can be expresse in full Classic. 1 Observe that if the language alreay allows for isjunction, we have lcs(c 1,..., C n) C 1 C n. In particular, this means that, for such languages, the lcs is not really of interest. 4

5 New results In Section 3, we provie a subsumption algorithm for full Classic with partial attributes, which essential extens ALN by same-as equalities, but also allows for the fills an one-of constructors on (host) iniviuals. This algorithm is a moification of the corresponing algorithm for the total attribute case presente in [10]. Then we present an lcs algorithm for this language along the lines of [18], an formally prove its correctness using the subsumption algorithm specifie before. It turns out that, as in CoreClassic, the lcs always exists, an, for two concept escriptions, it can be compute in time polynomial in their size. Finally, the central new results of this paper examines the problem of computing lcs when attributes are total (Section 5). The surprising result is that in this case the lcs oes not exist in general for the language S the construction in [20] being actually for partial attributes. We o however provie a polynomial-time algorithm for eciing when the lcs exists, an a (necessarily) worst-case exponential-time algorithm for computing the lcs in case it exists. We start by introucing the basic notions necessary for our investigations. 2 Formal Preliminaries Following [10], in this section we formally introuce the (full) Classic language except for the noneclarative test-efine concepts employe in the Classic system to algorithmically approximate representations for ieas that cannot be encoe using the constructors provie by Classic. Concept escriptions in Classic are built up from a collection of concept names, role names, attribute names, an iniviuals. Roles, attributes, an iniviuals are always atomic but escriptions can be built up using constructors. Classic incorporates two ifferent kins of concept escriptions, namely, host concept escriptions an classic concept escriptions. Therefore, we istinguish host concept names an classic (or atomic) concept names as well as host iniviuals an classic iniviuals. Host concept escriptions are use to escribe objects in a concrete omain, e.g., a programming language. A general scheme for incorporating such host objects has been presente in [3]. Host concept escriptions in Classic are relatively simple. They are efine built up from host concepts names an host iniviuals. More precisely, such concepts have the following syntax: Syntax Constructor Name H top concept of the host omain E host concept (name) {I 1... I n } one of C D concept conjunction where I 1,..., I n are host iniviuals an C an D are host concept escriptions. The semantics of host concepts an host iniviuals is preefine an fixe. The extension of a host concept is a subset of H, the so-calle host realm. A host iniviual is interprete as an element in H where ifferent iniviuals are assigne to ifferent elements of the host realm (unique name assumption). The elements in the host realm have no role or attribute successors. However, they can be successors of elements in the classic realm (see below). Finally, we require that (i) all host concept names have an extension that is either of infinite size or is empty; an (ii) that if the extensions of two host concepts overlap, then one must be subsume by the other (i.e., host concept names are isjoint, unless they are subconcepts of each other). an (iii) that the ifference (A B) of the extensions of two host concept names is either infinite or empty. (These 5

6 conitions are neee to avoi being able to infer conclusions from the size of host escriptions.) This, for example, allows for host concepts like INTEGER, REAL, COMPLEX, an STRING, but not BOOLEAN. Classic concept escriptions in Classic allow for more complex concepts than host concept escriptions. They are forme accoring to the following syntax: Syntax Constructor Name C top concept of the classic realm E classic/atomic concept name ( n R) at least restriction ( m R) at most restriction R : I fills (on a role) A : I fills (on an attribute) {I 1... I n } one of A 1 A k B 1 B h sams-as equality C D concept conjunction R.C (role) value restriction A.C (attribute) value restriction where E is an atomic concept name; R is a role; A, A i, an B j are attributes; I is the name of a classic or host iniviual; I j are names of classic iniviuals; C an D are classic concept escriptions; F is a host or classic concept escription; an k, h, m, n are non-negative integers. A escription which is either a host concept escription or a classic concept escription is calle (Classic) concept escription. In aition, a concept escription might be the top concept, which will be interprete as the whole omain. The sublanguage S of Classic only allows for same-as an concept conjunction. In the sequel, the set of concept names (host an classic) is calle C, the set of role names R, the set of attributes A, an the set of iniviuals IN D (which consists of the set of host iniviuals IN D H an classic iniviuals IN D C ). All these sets are pairwise isjoint. Furthermore, we will use R 1 R n.c as abbreviation of R 1. R 2 R n.c where ε.c enotes C. As argue in [10], for iniviuals it is reasonable to introuce a non-stanar semantics. The most significant reason is to avoi intractability of subsumption. It has been shown in [10, 27] that subsumption in Classic is NP-complete when iniviuals are interprete as single elements of the omain. To avoi this problem the semantics of iniviuals are efine as follows: instea of mapping classic iniviuals onto single elements of the omain, as we have one it for host iniviuals or as it is one in stanar semantics, classic iniviuals are mappe onto isjoint subsets of the omain, intuitively representing ifferent possible realizations of that (Platonic) iniviual. For a given interpretation I the classic iniviuals inuce the following congruence relation over the omain om(i) of I: two elements in om(i) are sai to be congruent if an only if they belong to the same extension of the same iniviual I IN D C or if they are ientical. The carinality of a set of elements of the omain is then the size of the set moulo this congruence relationship. As usual, the enotational semantics for concept escriptions is recursively built on the extensions of atomic ientifiers (i.e., of concept names, role names, attribute names, an iniviuals) by an interpretation: Definition 1 An interpretation I consists of a omain an an interpretation function I. The omain is isjointly ivie into a classic realm C an the host realm H which is fixe for all interpretations. The interpretation function assigns extensions to atomic ientifiers as follows: 6

7 The extension of an atomic concept name E is some subset E I of the classic realm. The extension of a host concept name H is some preefine subset of H which is fixe for all interpretations an satisfies the conitions state above. The extension of an atomic role name R is some subset R I of C. The extension of an atomic attribute name A is some partial function A I from C to, i.e., if (x, y 1 ) A I an (x, y 2 ) A I then y 1 = y 2. The extension of a classic iniviual I is some non-empty subset I I of C where the interpretations of istinct ientifiers must be isjoint as iscusse above. Host iniviuals are interprete accoring to the unique name assumption (see above). The extension I I of a host iniviual I is an element in H. As alreay mentione, the interpretation of the host iniviuals is fixe for all interpretations. Therefore, we occasionally refer to I I as I. By (R 1 R n ) I for role names or attributes R i we enote the composite of the binary relations R I i. If n = 0 then εi enotes the ientical relation, i.e., ε I := {(, ) C }. For an iniviual, we efine R I () := {e (, e) R I }. If the R i are attributes, we say that (R 1 R n ) I is efine for iff (R 1 R n ) I () ; occasionally, we will refer to the image of by (R 1 R n ) I (). The extension C I of a concept escription C is inuctively efine as follows: I = ; I C = C; I H = H; p : I I = { C x (, x) p I x I I } where p is a role or an attribute; {I 1... I n } I = n k=1 I I k. If the I k s are all host iniviuals then this means {I 1... I n } I = {I 1... I n }; ( n R) I (resp. ( n R) I ) is those objects in C with at least (resp. at most) n non-congruent successors for the role R; (A 1 A k B 1 B h ) I = { C (A 1 A k ) I an (B 1 B h ) I are efine for an A 1 A k I () = B 1 B h I ()}; (C D) I = C I D I ; ( p.c) I = { C p I () C I } where p is a role or an attribute. If we ha not restricte the interpretation of the constructors like value-restriction an number restriction to subsets of C then subsumption relationships as INTEGER R.C or INTEGER ( n R) woul hol. Note that the above efinition supports partial attributes. Since the main point of this paper is to emonstrate the impact of ifferent semantics for attributes, we occasionally restrict the set of interpretations to those that map attributes to total functions. Such interpretations are calle t-interpretations. The main inference service provie by a system base on escription logics is to compute subconcept-superconcept relationships, which are also calle subsumption relationships. Definition 2 A concept escription C is subsume by the concept escription D (C D for short) if an only if for all interpretations I it is C I D I. If we consier only total interpretations, we get t-subsumption: C t D iff C I D I for all t-interpretations I. 7

8 Having efine subsumption, equivalence of concept escriptions is efine in the usual way: C D if an only if C D an D C. Analogously, t-equivalence C t D is specifie. Although, Classic as introuce here oes not contain the bottom concept explicitly, it can be expresse by, e.g., ( 1 R) ( 0 R). We will use as abbreviation of some inconsistent concept escription. Furthermore, primitive negation can be expresse by number restrictions. For an atomic concept E one can replace every occurrence of E by 1R E an the negation E of E by 0R E where R E is a new role name. We also o not allow for number restrictions on attributes. However, ( na) an ( na) C for every n 2. Moreover, ( 0A) A., ( 1A) C, ( 0 A) C, an ( 1 A) A A. Usually, Classic also allows for min n := {m m is a nonnegative integer greater or equal n} an max n := {m m is a nonnegative integer less or equal n} where n is a nonnegative integer. This constructor can be simulate by introucing a host concept name NON-NEGATIVE-INTEGER which is interprete as the nonnegative integers an an infinite set of host iniviuals 0, 1, 2,... Then, for example, min n is equivalent to the concept escription {n, n+1, n+2,...} (infinite one-of) which can be represente finitely. To sum up, the results presente in this paper, can easily be extene to a language that in aition to the constructors introuce for Classic also allows for the bottom concept, primitive negation, number restrictions on attributes, an min n, max n. The least common subsumer of a set of concept escriptions is the most specific concept subsuming all concept escriptions of the set: Definition 3 The concept escription D is the least common subsumer (lcs) of the concept escriptions C 1,..., C n (lcs(c 1,..., C n ) for short) iff i) C i D for all i = 1,..., n an ii) for every D with that property D D. Analogously, we efine lcs t (C 1,..., C n ) using t instea of. Note that the lcs of concept escriptions may not exist, but if it oes, by efinition it is uniquely etermine up to equivalence. In this sense, we may refer to the lcs. In the following two sections, attributes are always interprete as partial functions; only in Section 5 we consier total attributes. 3 Subsumption As propose in [10] subsumption is ecie by a multi-part process. First, escriptions are turne into escription graphs. Next, escription graphs are put into canonical form, where certain inferences are explicate an other reunancies are reuce by combining noes an eges in the graph. Finally, subsumption is etermine between a escription an a canonical escription graph. Since in [10] attributes are interprete as total functions, we nee to ajust the steps liste above to ecie subsumption in the case of partial attributes. However, we have trie to reuce the changes necessary. For this reason, roughly speaking, attributes are treate as roles unless they form part of a same-as equality. (Note that attributes participating in a same-as construct must have values!) To some extent, this will allow us to aopt the semantics of the original escription graphs, which is crucial for proofs. However, the two ifferent occurrences of attributes, namely, in a same-as equality vs. a role in a value-restriction, require us to moify an exten the efinition of escription graphs, the normalization rules, an the subsumption algorithm itself. Furthermore, the subsumption algorithm propose in [10] was still incomplete because some normalization rules were missing, e.g., for ealing with singleton sets of host iniviuals. In the following, we will present the steps of the subsumption algorithm in etail. We start with the efinition of escription graphs. 8

9 maeby C maeby G(INTEGER) G(Moel) seats seats [0,1] moel [0,1] { C } moel maeby [0, 1] moel C seats G(Manufacturer) Figure 1: The escription graph for Car where the noe in the mile is the root of the graph 3.1 Description Graphs Intuitively, escription graphs reflect the syntactic structure of concept escriptions. A escription graph is a labele, irecte multigraph, with a istinguishe noe. Roughly speaking, the eges (a-eges) of the graph capture the constraints expresse by same-as equalities. The noes are labele with a set of so-calle r-eges which correspon to value restrictions. These r-eges lea to escription graphs again which represent the concept escriptions of the corresponing value restrictions. Unlike the graphs propose in [10], the value restrictions represente by noes not only contain restrictions on roles, as in [10], but also on attributes. We shall comment on the avantage of this moification in orer to eal with attributes as partial functions instea of total functions. Before efining escription graphs formally, we will look at our example concept Car. The escription graph epicte in Figure 1 correspons to C. We use G(M anuf acturer), G(M oel), G(INTEGER) to enote the escriptions graphs for the atomic concept names Manufacturer an Moel as well as the host concept name INTEGER. In this case, these escription graphs are very simple; they merely consist of one noe labele with the corresponing concept name. In general, the escription graphs in r-eges are more complicate since in a value restriction like R.C, C is an arbitrary concept escription. For the sake of simplicity, we have omitte the components of the noes an eges corresponing to the one of constructor an fills. Although, the concept Car oes not have number restrictions, the corresponing graph has the restriction [0, 1] on the r-eges since in our example these eges are restrictions on attributes, which have at most one irect successor. By aing these aitional informations for attributes it is possible to treat attributes like roles, unless no successors by fills or a same-as equality are require. In our example, there are same-as restrictions on the attributes. As we will see later, it is necessary to lift the r-eges to a-eges in orer to get a complete subsumption algorithm. This normalization operation was not necessary for the graphs efine in [10] since attributes were not allowe in r-eges. Formally, escription graphs are efine as follows: Definition 4 A escription graph G is a tuple (N, E, r, l), consisting of a finite set N of noes; a finite set E of eges (a-eges); a istinguishe noe r in N (root of the graph); an a function l from N into the set of labels of noes. We will occasionally use the notation G.N oes, G.Eges, an G.root to access the components N, E an r of the graph G. 9

10 An a-ege is a tuple of the form (n 1, n 2, A, F) where n 1, n 2 are noes, A is an attribute name, an F is a set of iniviuals (the fillers of the noe). A label of a noe is efine to be or a tuple of the form (C, D, H), consisting of a finite set C of concept names (the atoms of the noe), a finite set D of classic iniviuals or (the om of the noe), an a finite set H of tuples (the r-eges of the noe). Concept names in a escription graph are atomic concept names, host concept names,, C, or H. We will occasionally use the notation n.atoms, n.dom, an n.reges to access the components C, D, an H of the noe n. An r-ege is a tuple, (R, m, M, F, G ), of a role or an attribute name, R; a min, m, which is a non-negative integer; a max, M, which is a non-negative integer or ; a finite set F of iniviuals (the fillers of the r-ege); an a (recursively neste) escription graph G. The graph G will often be calle the restriction graph of the noe for the role R. We assume that the noes of G are istinct from all noes of G an from all other neste escription graphs of G. If R is an attribute then we require m = 0, M {0, 1}, an F =. A path p in G from the noe n 0 to n m is a sequence of a-eges of the form (n 0, n 1, A 1, F 1 ), (n 1, n 2, A 2, F 2 )..., (n m where m 0 (for m = 0 the path p is empty); w = A 1 A m is calle attribute-label of p (the empty path has attribute-label ε). For n N we efine G n to be the escription graph (N, E, n, l). In orer to merge escription graphs we nee the notion of recursive set of noes of a escription graph G: The recursive set of noes of G is the union of the noes of G an the recursive set of noes of all escription graphs in the r-eges of noes in G. Just as for concept escriptions, the semantics of escription graphs is efine by means of an interpretation I. We introuce a function Υ which assigns an iniviual of the omain of I to every noe of the graph. This ensures that all same-as equalities are satisfie. Definition 5 Let G = (N, E, r,l) be a escription graph an let I be an interpretation. An element,, of is in G I, iff there is some function, Υ, from N into such that 1. = Υ(r); 2. for all n N it is Υ(n) n I ; 3. for all (n 1, n 2, A, F) E we have (Υ(n 1 ), Υ(n 2 )) A I, an for all f F, Υ(n 2 ) f I. The extension of a noe n with label is the empty set. An element,, of is in n I, where n = (C, D, H), iff 1. for all B C, we have B I ; 2. If D is not then there exists f D such that f I. 3. for all (R, m, M, F, G ) H, (a) there are between m an M elements,, of the omain such that (, ) R I ; (b) for all f F there is a omain element,, such that (, ) R I an f I ; an (c) G I for all such that (, ) R I The semantics of the graphs in [10] has been efine in the same way. However, in their paper not only same-as equalities have been expresse by a-eges but also value restrictions on attributes. But then, in the context of partial functions, we coul not efine the semantics of escription graphs by means of the function Υ since iniviuals nee not to have successors for attributes. For that 10

11 purpose, value restrictions of attributes are always translate into r-eges. The next section will present the translation of concept escription into escription graphs in etail. Having efine the semantics of escription graphs, subsumption an equivalence between escription graphs (e.g., H G) as well as concept escriptions an escription graphs (e.g., C G) is efine in the same way as subsumption an equivalence between concept escriptions. 3.2 Translating Concept Descriptions to Description Graphs A Classic escription is turne into a escription graph by a recursive process, working from the insie out. In this process, noes an escription graphs are often merge. Definition 6 The merge of two noes, n 1 n 2, is a new noe n with the following label: if n 1 or n 2 has label then the label of n is. Otherwise if both labels are not equal to then the atoms of n are the union of the atoms of n 1 an n 2 ; the om is the om of n 2, if the om of n 1 is an vice versa; otherwise, if both the om of n 1 an n 2 are not equal to then om of n is the intersection of the om of n 1 an n 2 ; the set of r-eges is the union of the r-eges of the two noes. Definition 7 Let G 1 an G 2 be two escription graphs with isjoint recursive sets of noes (see above). Then the merge of G 1 an G 2, G := G 1 G 2, is efine as follows: The noes of G are the union of the noes of G 1 an G 2 without the roots of G 1 an G 2 but with an aitional noe r. The a-eges of the merge graphs are the union of the a-eges of G 1 an G 2, except that eges touching on the roots of G 1 an G 2 are moifie to touch r, i.e., in all a-eges of G 1 an G 2 the roots are replace by r. The new noe r is the root of G an its label is efine by the merge of the two root noes of G 1 an G 2. The rules for translating a escription C in Classic into a escription graph G(C) are as follows: 1. ( C or H ) is turne into a escription graphs with one noe r an no a-eges. The only atom of r is ( C or H ); the om of r is ; an the set of r-eges is empty. 2. is turne into a escription graph with one noe r an no a-eges. The label of r is. 3. A concept name is turne into a escription graph with one noe an no a-eges. The atoms of the noe contain only the concept name; om is ; an the noe has no r-eges. 4. A escription of the form ( n R) is turne into a escription graph with one noe an no a- eges. The noe has as its atoms C ; om is ; an it has a single r-ege (R, n,,, G( )). s 5. A escription of the form ( n R) is turne into a escription graph with one noe an no a-eges. The noe has as its atom C ; om is, an it has a single r-ege (R, 0, n,,g( )). 6. A escription of the form R : I is turne into a escription graph with one noe an no a- eges. The noe has as its atom C, as om, an it has a single r-ege (R, 0,, {I}, G( C )) if I is a classic iniviual, an (R, 0,, {I}, G( H )) otherwise. 7. A escription of the form A : I is turne into a escription graph with two noes r, n, an the a-eges (r, n, A, {I}) where r enotes the root of the graph. The atom of r is C ; om is ; an r has no r-eges. The atom of n is C if I is a classic iniviual an H otherwise; om is ; an n has no r-eges. 11

12 8. A escription of the form {I 1... I n } is turne into a escription graph with one noe. The noe has as om the set containing I 1 through I n, an no r-eges. The only atom of the noe is H if all of the iniviuals are host values, an C if all of the iniviuals are classic iniviual names. (Note that classic an host iniviuals can not be together in one set.) 9. A escription of the form A 1 A n B 1 B m is turne into a graph with the pairwise istinct noes a 1,..., a n 1, b 1,..., b m 1, the root a 0 = b 0 = r, an an aitional noe a n = b m = e; the set of a-eges consists of (a 0, a 1, A 1, ), (a 1, a 2, A 2, ),...,(, a n 1, a n, A n, ) an (b 0, b 1, B 1, ), (b 1, b 2, B 2, ),..., (b m 1, b m, B m, ), i.e., two isjoint paths from r to e. (Note that for n = 0 the first path is the empty path from r to r an for m = 0 the secon path is the empty path from r to r.) All noes except e have C as there only atom. If r e then the atom of e is. Finally, the omain of all noes is, an there are no r-eges. 10. A escription of the form R.C, where R is a role, is turne into a escription graph with one noe an no a-eges. The noe has the atom { C }, its om is, an it has a single r-ege (R, 0,,, G(C)). 11. A escription of the form A.C, where A is an attribute, is turne into a escription graph with one noe an no a-eges. The noe has the atom { C }, its om is, an it has a single r-ege (A, 0, 1,,G(C)) To turn a escription of the form C D into a escription graph, construct G(C) an G(D) an merge them. Note that this translation is well-efine since it ensure that for every r-ege containing an attribute min is 0 an max = 1 {0, 1}. Figure 1 shows the escription graph for the concept escription Car of our example. Now we want to show that this process preserves extensions. As we use the merge operations we first show that they work correctly. Lemma 1 If n 1 an n 2 are noes then (n 1 n 2 ) I = n I 1 n I 2. If D 1 an D 2 are escription graphs then (D 1 D 2 ) I = D1 I DI 2. Proof: It is easy to see that the claim is true for noes if one of them has label. Otherwise, if both labels are not then atoms an r-eges of the merge noe are obtaine by unioning the components of the respective noes; the om of the new noe is the intersection of om of n 1 an n 2 if both are not, otherwise the om of the noe which is not or it is if both noes have om. This implies that the interpretation of each component of the new noe is the intersection of the interpretation of the corresponing component of n 1 an n 2. Furthermore, since the interpretation of a noe is the intersection of the interpretations of its components, the result is obviously true for noes. For merging graphs, the only ifference is that the root noes are replace by their merger in all eges an that the new root is the merger of the roots of the merge graphs. But then an element of (D 1 D 2 ) I is clearly an element of both D1 I an DI 2. Conversely, since we take the isjoint union of the other noes in the two graphs, the mapping functions Υ 1 an Υ 2 in Definition 5 can simply be unione, so that an element of both D1 I an DI 2 is an element of the merge root noe, an hence of (D 1 D 2 ) I. 2 In [10], the concept escription A.C woul be turne into an a-ege. As alreay mentione, this woul cause problems for attributes interprete as partial functions when efining the semantics by means of Υ as specifie in Definition 5. 12

13 Theorem 1 A concept escription C an its corresponing escription graph G(C) are equivalent, i.e., C G(C). Proof: The proof is by structural inuction on concept escriptions. The extension of concept names, fills, one of, number restrictions, an -restrictions on roles an attributes can be easily seen to agree with the extension of escription graphs forme from them. Lemma 1 shows that conjunction is properly hanle. For same-as equalities A 1 A n B 1 B m the construction forms a escription graph with two isjoint paths from the istinguishe noe r to an en noe e, one labele by the A i s, through noes a i, an the other labele by the B j s, through noes b j. If (A 1 A n B 1 B m ) I then efining Υ(a i ) := (A 1 A i ) I (), an Υ(b j ) := (B 1 B j ) I (), yiels the mapping require by Definition 5. The converse is satisfie by the requirement in Definition 5 that for each a-ege (n 1, n 2, A, F) E, we have (Υ(n 1 ), Υ(n 2 )) A I. 3.3 Translating a Description Graph to a Concept Description Although, we o not nee the converse translation from escription graphs to concept escriptions for characterizing subsumption, the translation is presente here alreay in orer to show that concept escriptions an escription graphs are equivalent representations. Later on, when iscussing the lcs, we will actually nee to translate escription graphs into concept escriptions. In the sequel, let G = (N, E, r,l) be a escription graph. W.l.o.g. we assume that G an (recursively) all escription graphs neste in G are connecte. A escription graph is sai to be connecte if all noes of the graph can be reache from the root of the graph by a irecte path an if all neste graphs are connecte. Note that the semantics of a graph is not change, if noes that are not connecte via a path with the root are elete. We now (recursively) specify C G which correspons to the escription graph G using the concept escription C n corresponing to the label of the noe n in G. Let n be a noe in G. If the label of n is, then C n :=. Now, let (S, D, H) be the label of n. Then, C n is a conjunction consisting of the following conjuncts: 1. atomic concepts: If S =, then ; otherwise E S E. 2. one-of: If D =, then ; if D =, then ; otherwise D. 3. r-eges: If H =, then ; otherwise (R,l,k,F,G ) H [ R.C G ( l R) ( k R) f F (R : f) If R is an attribute, then we know l = 0, k {0, 1}, an F =. In this case, we efine ( lr) to be. Furthermore, we efine ( k R) to be if k = 1 an R. otherwise. Using C n, C G is efine as follows: 1. Same-as: Let T be a spanning tree of G. (Note that because G is connecte, T contains all noes of G.) For every leave of T, C G has a same-as equality v v where v is the attributelabel of the roote path in T to the leave. Furthermore, for every a-ege (n 1, n 2, A, F) not containe in T we have v w in the conjunction where v an w are efine as follows: w is the attribute-label of the roote path in T to n 2 ; v is the attribute-label of the path in T to n 1 concatenate with the a-ege (n 1, n 2, A, F) from n 1 to n ].

14 2. noes an a-eges: For every noe n in T we have [ ( ) ] al(n). C n (A : f) (n,m,a,f) E f F where al(n) enotes the attribute-label of the roote path in T. We can show: Lemma 2 G C G. Proof iea: Let n be a noe in G. Then, it is easy to see that if C n is translate back into a escription graph G where accoring to part 3 of the efinition of C n the r-eges for R.C G ( l R) ( k R) (R : f) are merge to one r-ege, then G is isomorphic to the subgraph of f F G only consisting of the noe n. The conjunction consisting of the conjuncts al(n).c n where n is a noe in T can be translate back into a escription graph where certain a-eges are merge such that the resulting graph is a tree isomorphic to T except for the fillers of the a-eges. Now aing the remaing same-as equalities yiels a escription graph G which is isomorphic to G except for fillers on a-eges. Finally, [ it is not har to verify that the escription graph for the conjunction consisting of ( ) ] al(n). (A : f) on the one han, where, again, n is a noe in T, an the graph (n,m,a,f) E f F G on the other han can be merge in such a way that the resulting escription graph (which by construction is equivalent to C G ) is isomorphic to G. 3.4 Canonical Description Graphs In the following we occasionally refer to marking a noe incoherent ; this means that the label of this noe is change to. Marking a escription graph as incoherent means that the escription graph is replace by the graph G( ) corresponing to. One important property of canonical escription graphs is that they are eterministic, i.e., every noe has at most one outgoing ege (r-ege or a-ege) with the same attribute or role name. Therefore, as in [10], to turn a escription graph into a canonical graph we nee to merge a-eges an r-eges. In aition, because in the graphs efine here, attributes can occur in r-eges an in a-eges it might be necessary to lift r-eges to a-eges. Finally, because of the one of constructor on host values, it might be necessary to merge noes. To merge two a-eges (n, n 1, A, F 1 ) an (n, n 2, A, F 2 ) in a escription graph G, replace them with a single new ege (n, n, A, F 1 F 2 ) where n is the result of merging n 1 an n 2. (If n 1 = n 2 then n = n 1.) In aition, replace n 1 an n 2 by n in all other a-eges of G. To merge two r-eges (R, l 1, r 1, F 1, G 1 ), (R, l 2, r 2, F 2, G 2 ) replace them by (R, max(l 1, l 2 ), min(r 1, r 2 ), F 1 F 2, G 1 G 2 ). To lift up an r-ege (A, l, r,f A, G A ) of a noe n in concept graph G when G has ege (n, n 1, A, F), remove it from n.reges, an augment G by aing G A.Noes to G.Noes, G A.Eges to G.Eges, as well as aing (n, G A.Root, A, F A ) to G.Eges. To merge two noes n 1, n 2 in a concept graph G let n be the result of merging n 1, n 2 an replace n 1 an n 2 by n in all a-eges of G. Description graphs are transforme into canonical form by repeating the following normalization rules whenever possible for the escription graph an all its escenants. In the sequel, the carinality of the om of a noe is efine as follows: has carinality an if the omain is a subset of IN D the carinality is the number of elements in this subset. 14

15 1. If any noe in a escription graph is marke incoherent, mark the escription graph as incoherent. (Reason: Even if the noe is not a root, attributes corresponing to a-eges must always have a value (since they participate in same-as equalities), an this value cannot belong to the empty set.) 2. If an a-ege of a noe has more than one filler, then mark the noe incoherent. (Reason: Attributes can only have at most one filler.) 3. If an a-ege of a noe n points to a noe n where the om is not an the filler of the a-ege is not inclue the om of n, mark n incoherent. (Reason: Same as 2.) 4. If the om of a noe is empty, mark the noe incoherent. (Reason: {} ) 5. If some r-ege in a noe has its min greater than its max, mark the noe incoherent. (Reason: ( 2 R) ( 1 R) ) 6. If for an r-ege om of the istinguishe noe of the restriction graph 3 is not an the fillers of this r-ege are not a subset of the om, mark the noe of the r-ege incoherent. (Reason: R : I R.{I 1,..., I n } if I {I 1,..., I n }) 7. If some noe has both H an C in its atoms, mark the noe incoherent. If some noe has in its atoms a pair of host concepts that are not relate by the pre-efine subsumption relationship, mark the noe incoherent. (Reason: The intersection of the atoms will be empty.) 8. If the fillers of an r-ege are host iniviuals an they are not containe in the extension of all atoms of the root of the restriction graph for the r-ege then mark the noe as incoherent. (Reason: R : I R.C H if I is not an element of the extension of the host concept C H.) 9. If the fillers of an a-ege from n to n are host iniviuals an they are not containe in the extension of all atoms of the noe the a-ege is pointing to then mark n as incoherent. (Reason: see 8.) 10. If some noe has in its atoms a pre-efine host concept, a H to its atoms. If some noe has an atomic concept name in its atoms, a C to its atoms. For each pre-efine host concept in the atoms of the noe, a all the more-general pre-efine host concepts to its atoms. 11. If a host iniviual in the om of a noe is not in all the atoms of the noe, remove it from the om. (Reason: {I 1, I 2 } INTEGER {I 1 } INTEGER if I 1 is an integer but I 2 is not an integer.) 12. If om of a noe is not an om contains host iniviuals then a to the atoms of the noe all host concepts which extensions contain all the host iniviuals in om. (Reason: {I 1, I 2 } INTEGER {I 1, I 2 } INTEGER REAL if I 1 an I 2 both belong to INTEGER since INTEGER REAL.) 13. If some r-ege in a noe has its escription graph marke incoherent, change its max to 0. (Reason: ( 0 R) R..) 14. If some r-ege in a noe has a max of 0, mark its restriction graph as incoherent. (Reason: See 13) 3 Recall that this is the graph containe in an r-ege. 15

16 15. If the min on an r-ege is less than the carinality of fillers on it, let the min be this carinality. (Reason: R : I 1 R : I 2 R : I 3 ( 3 R)) 16. If the max on an r-ege is greater than the carinality of the om on the istinguishe noe of the restriction graph, make the max of this ege be the carinality of the om. (Reason: R.{I 1, I 2, I 3 } ( 3 R)) 17. If the min on an r-ege is greater than or equal to the carinality of the om on the istinguishe noe of the restriction graph, let the fillers of the ege be the union of its fillers an the om above. (If min is greater than the carinality, then steps 5 an 16 etect the inconsistency.) (Reason: R.{I 1, I 2, I 3 } ( 3 R) R : I 1 R : I 2 R : I 3 ; R.{I 1, I 2, I 3 } ( 4 R) ) 18. If the max on an r-ege is less than or equal to the carinality of fillers on the ege, let the om on the istinguishe noe of the restriction graph be the intersection of the om an the fillers. (If max is less than the carinality, steps 5 an 15 etect the inconsistency.) (Reason: R : I 1 R : I 2 R : I 3 ( 3 R) R.{I 1, I 2, I 3 }; R : I 1 R : I 2 R : I 3 ( 2 R) ) 19. If an a-ege has a filler an the noe at its en has as its om or the filler is an element of the om, make the om be the filler. (Reason: A : I A : I {I}) 20. If a noe has only one element in its om, a this element to the fillers for all the a-eges pointing to it. (Reason: {I} A : I {I}) 21. If there are two noes n 1, n 2 where the om in each of these noes is {I} for some host iniviual I, merge these noes in G. 4 (Reason: A B C D A.{I} C.{I} A C if I is a host iniviual) 22. If some noe has two r-eges labele with the same role, merge the two eges, as escribe above. (Reason: R.C R.D R.(C D)) 23. If some escription graph has two a-eges from the same noe labele with the same attribute, merge the two eges, as escribe above. (Reason: A.C A.D A.(C D)) 24. If some noe in a graph has both an a-ege an an r-ege for the same attribute, then lift up the r-ege, as escribe above. (Reason: r-eges participating in same as restrictions nee to be a-eges.) These normalization rules are well-efine: Starting with a escription graph the outcome of applying a normalization rule is always a escription graph. In particular, min of an r-ege with attribute is always 0 an max {0, 1}. Furthermore, no rules can force such an r-ege to have fillers. We nee to show that the transformations to canonical form o not change the extension of the graph. The main ifficulty is in showing that the merging processes an the lifting o not change the extension. Lemma 3 Let G = (N, E, r,l) be a escription graph with two mergeable a-eges an let G = (N, E, r, l ) be the result of merging these two a-eges. Then, G G. Proof: Let the two eges be (n, n 1, A, F 1 ) an (n, n 2, A, F 2 ) an the new noe n be n 1 n 2. Choose G I, an let Υ be a function from N into the omain of I satisfying the conitions for extensions (Definition 5) such that Υ(r) =. Then Υ(n 1 ) = Υ(n 2 ) because both are equal 4 This rule was missing in [10]. 16

17 to A I (Υ(n)). Let Υ be the same as Υ except that Υ (n ) = Υ(n 1 ) = Υ(n 2 ). Then Υ satisfies Definition 5, part 3, for G, because we replace n 1 an n 2 by n everywhere, an the conitions for fillers are satisfie for Υ. Moreover, Υ (n ) = Υ(n 1 ) n I 1 ni 2, which by Lemma 1 equals (n 1 n 2 ) I ; so part 2 is satisfie too, since n = n 1 n 2. Finally, if the root is moifie by the merger, i.e., n 1 or n 2 is r, say n 1, then = Υ(n 1 ) = Υ (n ), so part 1 of the efinition is also satisfie. Conversely, given G I, let Υ be the function stipulate by Definition 5 such that Υ (r ) =. Let Υ be the same as Υ except that Υ(n 1 ) = Υ(n ) an Υ(n 2 ) = Υ (n ). Then the above argument can be traverse in reverse to verify that Υ satisfies Definition 5, such that G I. Lemma 4 Let n be a noe with two mergeable r-eges an let n be the noe with these eges merge. Then n I = n I for every interpretation I. Proof: Let the two r-eges be (R, m 1, M 1, F 1, G 1, ) an (R, m 2, M 2, F 2, G 2 ). Let n I. Then there are between m 1 (m 2 ) an M 1 (M 2 ) elements of the omain such that (, ) R I. Therefore there are between the maximum of m 1 an m 2 an the minimum of M 1 an M 2 elements of the omain such that (, ) R I. Furthermore, for all f F 1 (f F 2 ) there is a such that (, ) R I an f I. Thus, there are fillers of for all f F 1 F 2. Also, all such that (, ) R I are in G I 1 (G I 2). Therefore, all such that (, ) R I are in G I 1 G I 2, which equals (G 1 G 2 ) I by Lemma 1. Thus, n I. Let n I. Then there are between the maximum of m 1 an m 2 an the minimum of M 1 an M 2 elements of the omain such that (, ) R I. Therefore there are between m 1 (m 2 ) an M 1 (M 2 ) elements of the omain such that (, ) R I. Furthermore, for all f F 1 F 2 there is a such that (, ) R I an f I. Thus, for all f F 1 (f F 2 ) there is a such that (, ) R I an f I. Also, all such that (, ) R I are in (G 1 G 2 ) I = G I 1 GI 2. Therefore, all such that (, ) R I are in G I 1 (G I 2). Hence, n I. Lemma 5 Let G = (N, E, r,l) be a escription graph with noe n an a-ege (n, n, A, F). Suppose n has an associate r-ege (A, l, r,f A, G A ) an let G = (N, E, r, l ) be the result of lifting up the r-ege. Then, G G. Proof: Obviously, it is sufficient to show that G I n = G I n since only the label of n is change in G an only n obtains an aitional a-ege which point to the graph G A not connecte to the rest of G. W.l.o.g. we therefore can assume that n is the root of G, i.e., n = r. Let G I. Thus, there is a function Υ from N into as specifie in Definition 5 an an iniviual e such that = Υ(n), e = Υ(n ), an (, e) A I. This implies e G I A. Hence, there exists a function Υ from G A.Noes into for G A an e satisfying the conitions in Definition 5. Since the sets of noes of G an G A are isjoint, we can efine Υ to be the union of Υ an Υ, i.e., Υ (m) := Υ(m) for all noes m in G an Υ (m) := Υ (m) for all noes m in G A. Since by construction for the aitional a-ege (n, G A.Root, A, F A ) E we have (Υ (n), Υ (G A.Root)) A I an the conitions on the fillers F A are satisfie, it follows that all conitions in Definition 5 are satisfie for an G, an thus, G I. Now let G I. Thus, there is a function Υ from N into accoring to Definition 5. Let e := Υ (G A.Root) = Υ (n ). Let G be the escription graph we obtain from G by eleting the noes corresponing to G A, which is the same graph as G without the r-ege (A, l, r, F A, G A ). If we restrict Υ to the noes of G, then it follows G I. Furthermore, restricting Υ to the noes of G A yiels e G I A. Since e is the only A-successor of, we can conclue GI. Lemma 6 Let G = (N, E, r,l) be a escription graph with noes n 1 an n 2 such that the om of these noes is {I} where I is a host iniviual. Let G = (N, E, r, l ) be the result of merging n 1 an n 2 in G. Then, G G. 17