# Combining Answer Set Programming with Description Logics for the Semantic Web

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3 We use dom(d) to denote d D dom(d). A datatype is either an element of D or a subset of dom(d) (called datatype oneof). Let A, R A, R D, and I be nonempty finite and pairwise disjoint sets of atomic concepts, abstract roles, datatype roles, and individuals, respectively. We use R A to denote the set of all inverses R of abstract roles R R A. A role is an element of R A R A R D. Concepts are inductively defined as follows. Every C A is a concept, and if o 1, o 2, I, then {o 1, o 2, } is a concept (called oneof). If C and D are concepts and if R R A R A, then (C D), (C D), and C are concepts (called conjunction, disjunction, and negation, respectively), as well as R.C, R.C, nr, and nr (called exists, value, atleast, and atmost restriction, respectively) for an integer n 0. If d D and U R D, then U.d, U.d, nu, and nu are concepts (called datatype exists, value, atleast, and atmost restriction, respectively) for an integer n 0. We write and to abbreviate C C and C C, respectively, and we eliminate parentheses as usual. An axiom is an expression of one of the following forms: (1) C D, where C and D are concepts (concept inclusion); (2) R S, where either R, S R A or R, S R D (role inclusion); (3) Trans(R), where R R A (transitivity); (4) C(a), where C is a concept and a I (concept membership); (5) R(a, b) (resp., U(a, v)), where R R A (resp., U R D ) and a, b I (resp., a I and v dom(d)) (role membership axiom); and (6) a = b (resp., a b), where a, b I (equality (resp., inequality)). A knowledge base L is a finite set of axioms. (For decidability, number restrictions in L are restricted to simple abstract roles (Horrocks et al. 1999)). The syntax of SHIF(D) is as the above syntax of SHOIN (D), but without the oneof constructor and with the atleast and atmost constructors limited to 0 and 1. Semantics. An interpretation I = (, I) with respect to D consists of a nonempty (abstract) domain disjoint from dom(d), and a mapping I that assigns to each C A a subset of, to each o I an element of, to each r R A a subset of, and to each U R D a subset of dom(d). The mapping I is extended to all concepts and roles as usual (Eiter et al. 2003). The satisfaction of a description logic axiom F in an interpretation I = (, I), denoted I = F, is defined as follows: (1) I = C D iff C I D I ; (2) I = R S iff R I S I ; (3) I = Trans(R) iff R I is transitive; (4) I = C(a) iff a I C I ; (5) I = R(a, b) iff (a I, b I ) R I ; (6) I = U(a, v) iff (a I, v) U I ; (7) I = a = b iff a I = b I ; and (8) I = a b iff a I b I. The interpretation I satisfies the axiom F, or I is a model of F, iff I = F. I satisfies a knowledge base L, or I is a model of L, denoted I = L, iff I = F for all F L. We say that L is satisfiable (resp., unsatisfiable) iff L has a (resp., no) model. An axiom F is a logical consequence of L, denoted L = F, iff every model of L satisfies F. A negated axiom F is a logical consequence of L, denoted L = F, iff every model of L does not satisfy F. Description Logic Programs In this section, we introduce description logic programs (or simply dl-programs), which are a novel combination of normal programs and description logic knowledge bases. Syntax Informally, a dl-program consists of a description logic knowledge base L and a generalized normal program P, which may contain queries to L. Roughly, in such a query, it is asked whether a certain description logic axiom or its negation logically follows from L or not. A dl-query Q(t) is either (a) a concept inclusion axiom F or its negation F ; or (b) of the forms C(t) or C(t), where C is a concept and t is a term; or (c) of the forms R(t 1, t 2 ) or R(t 1, t 2 ), where R is a role and t 1, t 2 are terms. A dl-atom has the form DL[S 1 op 1 p 1,, S m op m p m ; Q](t), m 0, (2) where each S i is either a concept or a role, op i {,, }, p i is a unary resp. binary predicate symbol, and Q(t) is a dlquery. We call p 1,, p m its input predicate symbols. Intuitively, op i = (resp., op i = ) increases S i (resp., S i ) by the extension of p i, while op i = constrains S i to p i. A dlrule r has the form (1), where any literal b 1,, b m B(r) may be a dl-atom. We denote by B + (r) (resp., B (r)) the set of all dl-atoms in B + (r) (resp., B (r)). A dl-program KB = (L, P ) consists of a description logic knowledge base L and a finite set of dl-rules P. We use the following example to illustrate our main ideas. Example 1 (Reviewer Selection) Suppose we want to assign reviewers to papers, based on certain information about the papers and the available persons, using a description logic knowledge base L S (partially given in the appendix), which contains knowledge about scientific publications. We assume not to be aware of the entire structure and contents of L S, but of the following aspects. L S classifies papers into research areas, depending on keyword information. The research areas are stored in a concept Area. The roles keyword and inarea associate with each paper its relevant keywords and the areas it is classified into (obtained, e.g., by reification of the classes). Furthermore, a role expert relates persons to their areas of expertise, and a concept Referee contains all referees. Finally, a role hasmember associates with a cluster of similar keywords all its members. Consider then the dl-program KB S =(L S, P S ), where P S contains in particular the following dl-rules: (1) paper(p 1); kw(p 1, Semantic Web); (2) paper(p 2); kw(p 2, Bioinformatics); kw(p 2, Answer Set Programming); (3) kw(p, K 2) kw(p, K 1), DL[hasMember](S, K 1), DL[hasMember](S, K 2); (4) paperarea(p, A) DL[keyword kw; inarea](p, A); (5) cand(x, P ) paperarea(p, A), DL[Referee](X), DL[expert](X, A); (6) assign(x, P ) cand(x, P ), not assign(x, P ); (7) assign(y, P ) cand(y, P ), assign(x, P ), X Y ; (8) a(p ) assign(x, P ); (9) error(p ) paper(p ), not a(p ).

4 Intuitively, rules (1) and (2) specify the keyword information of two papers, p 1 and p 2, which should be assigned to reviewers. Rule (3) augments, by choice of the designer, the keyword information with similar ones. Rule (4) queries the augmented L S to retrieve the areas that each paper is classified into, and rule (5) singles out review candidates based on this information from experts among the reviewers according to L S. Rules (6) and (7) pick one of the candidate reviewers for a paper (multiple reviewers can be selected similarly). Finally, rules (8) and (9) check if each paper is assigned; if not, an error is flagged. Note that, in view of rules (3) (5), information flows in both directions between the knowledge encoded in L S and the one encoded in P S. To illustrate the use of, a predicate poss Referees may be defined in the dl-program, and Referee poss Referees may be added in the first dl-atom of (5), which thus constrains the set of referees. The dl-rule below shows in particular how dl-rules can be used to encode certain qualified number restrictions, which are not available in SHOIN (D). It defines an expert as an author of at least three papers of the same area: expert(x, A) DL[isAuthorOf ](X, P 1), DL[isAuthorOf ](X, P 2), DL[isAuthorOf ](X, P 3), DL[inArea](P 1, A), DL[inArea](P 2, A), DL[inArea](P 3, A), P 1 P 2, P 2 P 3, P 3 P 1. Semantics We first define Herbrand interpretations and the truth of dl-programs in Herbrand interpretations. In the sequel, let KB = (L, P ) be a dl-program. The Herbrand base of P, denoted HB P, is the set of all ground literals with a standard predicate symbol that occurs in P and constant symbols in Φ. An interpretation I relative to P is a consistent subset of HB P. We say I is a model of l HB P under L, denoted I = L l, iff l I, and of a ground dl-atom a = DL[S 1 op 1 p 1,, S m op m p m ; Q](c) under L, denoted I = L a, iff L m i=1 A i(i) = Q(c), where A i (I) = {S i (e) p i (e) I}, for op i = ; A i (I) = { S i (e) p i (e) I}, for op i = ; A i (I) = { S i (e) p i (e) I does not hold}, for op i =. We say that I is a model of a ground dl-rule r iff I = L H(r) whenever I = L l for all l B + (r) and I = L l for all l B (r), and of a dl-program KB = (L, P ), denoted I = KB, iff I = L r for all r ground(p ). We say KB is satisfiable (resp., unsatisfiable) iff it has some (resp., no) model. Least Model Semantics of Positive dl-programs. We now define positive dl-programs, which are not -free dlprograms that involve only monotonic dl-atoms. Like ordinary positive programs, every positive dl-program that is satisfiable has a unique least model, which naturally characterizes its semantics. A ground dl-atom a is monotonic relative to KB = (L, P ) iff I I HB P implies that if I = L a then I = L a. A dl-program KB = (L, P ) is positive iff (i) P is not -free, and (ii) every ground dl-atom that occurs in ground(p ) is monotonic relative to KB. Observe that a dl-atom containing may fail to be monotonic, since an increasing set of p i (e) in P results in a reduction of S i (e) in L, whereas dl-atoms containing and only are always monotonic. For ordinary positive programs P, it is well-known that the intersection of two models of P is also a model of P. The following theorem shows that a similar result holds for positive dl-programs KB. Theorem 1 Let KB = (L, P ) be a positive dl-program. If the interpretations I 1, I 2 HB P are models of KB, then I 1 I 2 is also a model of KB. Proof. Suppose that I 1, I 2 HB P are models of KB. We show that I = I 1 I 2 is also a model of KB, i.e., I = L r for all r ground(p ). Consider any r ground(p ), and assume that I = L l for all l B + (r) = B(r). That is, I = L l for all classical literals l B(r) and I = L a for all dlatoms a B(r). Hence, I i = L l for all classical literals l B(r), for every i {1, 2}. Moreover, I i = L a for all dl-atoms a B(r), for every i {1, 2}, since every dl-atom in ground(p ) is monotonic relative to KB. Since I 1 and I 2 are models of KB, it follows that I i = L H(r), for every i {1, 2}, and thus I = L H(r). This shows that I = L r. Hence, I is a model of KB. As an immediate corollary of this result, every satisfiable positive dl-program KB has a unique least model, denoted M KB, which is contained in every model of KB. Corollary 2 Let KB = (L, P ) be a positive dl-program. If KB is satisfiable, then there exists a unique model I HB P of KB such that I J for all models J HB P of KB. Example 2 Consider the dl-program comprising rules (1) (5) from Example 1. Clearly, this program is not -free. Moreover, since the dl-atoms do not contain occurrences of, they are all monotonic. Hence, the dl-program is positive. As well, its unique least model contains all review candidates for the given papers p 1 and p 2. Iterative Least Model Semantics of Stratified dl-programs. We next define stratified dl-programs, which are intuitively composed of hierarchic layers of positive dlprograms linked via default negation. Like for ordinary stratified programs, a canonical minimal model can be singled out by a number of iterative least models, which naturally describes the semantics, provided some model exists. We can accommodate this with possibly non-monotonic dl-atoms by treating them similarly as NAF-literals. This is particularly useful, if we do not know a priori whether some dl-atoms are monotonic, and determining this might be costly; recall, however, as noted above, that the absence of in (2) is a simple syntactic criterion which implies monotonicity of a dl-atom (cf. also Example 2).

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