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..., ch. 10]. Although OWL-DL is very expressive, it is a decidable fragment of firstorder logic, and thus cannot express arbitrary axioms: the only axioms it can express are of a certain tree-structure =-=[8]-=-. In contrast, decidable rule-based formalism such as function-free Horn rules do not share this restriction, but lack 3 Throughout this paper, we use “rules” and “clauses” synonymously, following [11...

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...e logically follows from symmetry and transitivity.) (11) (12) (13) (14) a ≈ a x ≈ y ← y ≈ x x ≈ z ← x ≈ y, y ≈ z R(x1, . . .,x ′ i, . . .,xn) ← R(x1, . . .,xi, . . .,xn), xi ≈ x ′ i It is well-known =-=[9]-=- that P ∪ P≈, where ≈ is treated as an “ordinary” predicate, entails the same set of consequences as P, where ≈ is taken to have the usual semantics of equality. 23sNow for a query predicate Q and a p...

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...s. A related logic, SHIQ(D) [13, 12], distinguished from SHOIN(D) mainly by not supporting nominals (or named objects), has been successfully implemented in practical reasoning systems, such as Racer =-=[9]-=- and FaCT [10]. Description logics have been found useful in numerous applications such as information integration [1, ch. 16], software engineering [1, ch. 11], and conceptual modeling [1, ch. 10]. A...

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...afe rules are a proper subset of SWRL, where some expressivity is traded for decidability. Hence, our approach provides an optimal query answering algorithm covering a significant portion of SWRL. In =-=[8]-=- an approach for combining answer set programming with description logics is presented. The interaction between the subsystems is enabled by exchanging only unit (i.e. non-disjunctive) ground conseque...

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... minor as a person whose age is less than or equal to 18 in the following way: Minor ≡ Person ⊓ ∃age. ≤18. Instead of axiomatizing concrete datatypes in logic, SHOIN(D) employs an approach similar to =-=[2]-=-, where the properties of concrete datatypes are encapsulated in so-called concrete domains. A concrete domain is a pair (△D,ΦD), where △D is an interpretation domain and ΦD is a set of concrete domai...

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...formance penalty is paid only for features actually used. Finally, we note that evaluating queries can be further optimized by using the magic sets transformation. In its basic form for Horn programs =-=[5]-=-, magic sets transformation reduces the amount of irrelevant computation 24sby simulating the binding propagation of top-down SLD-resolution. This technique has recently been extended to disjunctive p...

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...logic, SHIQ(D) [13, 12], distinguished from SHOIN(D) mainly by not supporting nominals (or named objects), has been successfully implemented in practical reasoning systems, such as Racer [9] and FaCT =-=[10]-=-. Description logics have been found useful in numerous applications such as information integration [1, ch. 16], software engineering [1, ch. 11], and conceptual modeling [1, ch. 10]. Although OWL-DL...

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...heorem summarizes the properties of our algorithm (we use |=c for cautions entailment in disjunctive datalog, which coincides on ground facts with first-order entailment for positive datalog programs =-=[6]-=-): Theorem 2 ([14]). Let KB be an SHIQ(D) knowledge base, defined over an admissible concrete domain D, such that satisfiability of finite conjunctions over ΦD can be decided in deterministic exponent...

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...B) according to Table 1. Assuming unary coding of numbers, π(KB) can be computed in polynomial time. To transform π(KB) into a set of closures Ξ(KB), we apply the well-known structural transformation =-=[26]-=-. Roughly speaking, the structural transformation introduces a new name for each nonatomic subconcept of KB. For example, in the axiom C ⊑ ∃R.(∀S.C), a new concept Q is introduced for the subconcept ∀...

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...gram. For unary coding of numbers and assuming a bound on the arity of predicates in rules, our algorithm runs in deterministic exponential time, which makes it optimal since SHIQ is ExpTime-complete =-=[21]-=-. The full presentation of the algorithm and a proof of its correctness are technically involved and lengthy. Here, we just provide an overview of the procedure, without going into details. For a comp...

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...vity while still being decidable. For example, SHOIN(D) provides full negation, disjunction, and (a restricted form of) universal and existential quantification of variables. A related logic, SHIQ(D) =-=[13, 12]-=-, distinguished from SHOIN(D) mainly by not supporting nominals (or named objects), has been successfully implemented in practical reasoning systems, such as Racer [9] and FaCT [10]. Description logic...

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... satisfiability of finite conjunctions over concrete predicates. Satisfiability checking of admissible concrete domains can successfully be combined with logical reasoning for many description logics =-=[17]-=-. We use a set of concept names NC, sets of abstract and concrete individuals NIa and NIc , respectively, and sets of abstract and concrete role names NRa and NRc , respectively. An abstract role is a...

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...to deal with equality. It is well-known that this calculus remains complete if ground literals are totally ordered under an arbitrary ordering ≻, and inferences are performed on maximal literals only =-=[3]-=-. This ordering has the useful property that, in each ground disjunction, exactly one literal is maximal. Hence, instead of performing an inference on each literal of a ground fact, it is sufficient t...

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...[8]. In contrast, decidable rule-based formalism such as function-free Horn rules do not share this restriction, but lack 3 Throughout this paper, we use “rules” and “clauses” synonymously, following =-=[11]-=-.s2 Boris Motik, Ulrike Sattler, and Rudi Studer some of the expressive power of OWL-DL: they are restricted to universal quantification and lack negation in their basic form. To overcome the limitati...

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... ones. Importantly, we do not restrict the component languages, but only reduce the interface between them. Generalizing the approaches of other decidable combinations of rules and description logics =-=[16, 5]-=-, in DL-safe rules, concepts and roles are allowed to occur in both rule bodies and heads as unary, respectively binary predicates in atoms, but each variable of a rule is required to occur in some bo...

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... ones. Importantly, we do not restrict the component languages, but only reduce the interface between them. Generalizing the approaches of other decidable combinations of rules and description logics =-=[16, 5]-=-, in DL-safe rules, concepts and roles are allowed to occur in both rule bodies and heads as unary, respectively binary predicates in atoms, but each variable of a rule is required to occur in some bo...

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...idable. Moreover, we present an algorithm for query answering in the extension of SHIQ with DL-safe rules which is based on a novel technique for reducing SHIQ knowledge bases to disjunctive programs =-=[20, 18]-=-. This yields a query answering algorithm which follows the principle of “graceful degradation”: the user “pays” only for the features she actually uses. Although a full evaluation is not yet finished...

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...s because we can restrict our attention to certain “nice” models. For numerous DLs, we can restrict our attention to tree models, i.e. to models where the underlying relational structure forms a tree =-=[22]-=-. This is so because every satisfiable knowledge base has such a tree model (to be precise, for some DLs we consider tree-like abstractions of Legend: Person � Grandchild father equivalent nodes peter...

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...hm for Reducing SHIQ(D) to Datalog Programs Disjunctive Program DD(KB)sQuery Answering for OWL-DL with Rules 11 π(Ω(KB)) into clausal form using structural transformation to avoid exponential blow-up =-=[19]-=-. We call the result Ξ(KB). Saturation by Basic Superposition. We next saturate the RBox and TBox clauses of Ξ(KB) by basic superposition [4] — a clausal calculus optimized for theorem proving with eq...

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...me way as this is done for SHIQ in [14] and (ii) the resulting logic is a syntactic variant of the two variable fragment of first-order logic with counting quantifiers, which is known to be decidable =-=[7]-=-. ⊓⊔ We strongly believe that Theorem 1 also holds for SHOIN(D): (i) the decidability proof of SHOIN should be easily adaptable to SHOIN(D), and (ii) the same non-deterministic reduction of ground DL-...

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...ction-free formalism compatible with the one from [15]. 2.3 Basic Superposition Our algorithms from Section 5 are based on basic superposition — a calculus optimized for theorem proving with equality =-=[4, 24]-=-, which we outline in the rest of this section. We assume the standard notions of first-order clauses with equality: all existential quantifiers have been eliminated using Skolemization; all remaining...

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...ng structural transformation to avoid exponential blow-up [19]. We call the result Ξ(KB). Saturation by Basic Superposition. We next saturate the RBox and TBox clauses of Ξ(KB) by basic superposition =-=[4]-=- — a clausal calculus optimized for theorem proving with equality. In this key step of the reduction, we compute all nonground consequences of KB. We can prove that saturation terminates because appli...

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...inimal models and general first-order models fundamentally changes the computational properties 11sof concept subsumption: equivalence of general programs under minimal model semantics is undecidable =-=[27]-=- whereas, under first-order semantics, it is decidable and can be reduced to satisfiability checking using standard transformations. To summarize, the difference between first-order and minimal model ...

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...a]) Conditions: (i): In any term f(t), the inner term t occurs marked. (ii): In all positive equality literals with at least one function symbol, both sides are marked. Decomposition. As discussed in =-=[21, 18]-=-, if KB contains number restrictions on roles that have subroles, saturating Ξ(KB) by BS need not terminate. To remedy that, we introduce decomposition — an additional inference rule which transforms ...

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...er, we present an algorithm for query answering in the extension of SHIQ(D) with DL-safe rules which is based on a novel technique for reducing SHIQ(D) knowledge bases to disjunctive datalog programs =-=[15, 14]-=-. This yields a query answering algorithm which follows the principle of “graceful degradation”: the user “pays” only for the features she actually uses. Although a full evaluation is not yet finished...

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...sets transformation reduces the amount of irrelevant computation 24sby simulating the binding propagation of top-down SLD-resolution. This technique has recently been extended to disjunctive programs =-=[11]-=-, and has shown significant benefits in practice. Magic sets transformation is not tied to the query evaluation method, and therefore can be used in conjunction with query answering by the RQ calculus...

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...g algorithm which follows the principle of “graceful degradation”: the user “pays” only for the features she actually uses. Although a full evaluation is not yet finished, our partial evaluation from =-=[18]-=- is very promising, and we believe that this algorithm can be efficiently realized in practice. Please note that we are primarily concerned with the semantic and decidability aspects of hybrid reasoni...

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...antics; for example, the fact that Oedipus is a child cannot be derived from the fact that he is a GoodChild or a BadChild ′ , c.f. our example at the end of Section 4.3. The approaches from [12] and =-=[30]-=- for reducing certain fragments of description logics to logic programming can easily be extended with rules, by simply appending the rules to the result of the transformation. However, the descriptio...

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...s because we can restrict our attention to certain “nice” models. For numerous DLs, we can restrict our attention to tree models, i.e. to models where the underlying relational structure forms a tree =-=[29]-=-. This is so because every satisfiable knowledge base has such a tree model (to be precise, for some DLs we consider tree-like abstractions of non-tree models). Even if such a tree model is infinite, ...