Recently, there has been a lot of interest in the integration of Description Logics and rules on the Semantic Web.We define guarded hybrid knowledge bases (or g-hybrid knowledge bases) as knowledge bases that consist of a Description Logic knowledge base and a guarded logic program, similar to the DL+log knowledge bases from (Rosati 2006). G-hybrid knowledge bases enable an integration of Description Logics and Logic Programming where, unlike in other approaches, variables in the rules of a guarded program do not need to appear in positive non-DL atoms of the body, i.e. DL atoms can act as guards as well. Decidability of satisfiability checking of g-hybrid knowledge bases is shown for the particular DL DLRO, which is close to OWL DL, by a reduction to guarded programs under the open answer set semantics. Moreover, we show 2-EXPTIME-completeness for satisfiability checking of such g-hybrid knowledge bases. Finally, we discuss advantages and disadvantages of our approach compared with DL+log knowledge bases.
The integration of Description Logics with rules has recently received a lot of attention in the context of the Semantic Web (Rosati 2005a;Rosati 2006;Eiter et al. 2004;Motik et al. 2004;Horrocks and Patel-Schneider 2004b;Motik and Rosati 2007;de Bruijn et al. 2007). R-hybrid knowledge bases (Rosati 2005a), and its extension DL+log (Rosati 2006), is an elegant formalism based on combined models for Description Logic knowledge bases and nonmonotonic logic programs. We propose a variant of r-hybrid knowledge bases, called g-hybrid knowledge bases, which do not require standard names or a special safeness restriction on rules, but instead require the program to be guarded. We show several computational properties by a reduction to guarded open answer set programming (Heymans et al. 2005a;Heymans et al. 2006b).
Open answer set programming (OASP) (Heymans et al. 2005a;Heymans et al. 2006b) combines the logic programming and first-order logic paradigms. From the logic programming paradigm it inherits a rule-based presentation and a nonmonotonic semantics by means of negation as failure. In contrast with usual logic programming semantics, such as the answer set semantics (Gelfond and Lifschitz 1988), OASP allows for domains consisting of other objects than those present in the logic program at hand. Such open domains are inspired by first-order logic based languages such as Description Logics (DLs) (Baader et al. 2003) and make OASP a viable candidate for conceptual reasoning. Due to its rule-based presentation and its support for nonmonotonic reasoning and open domains, OASP can be used to reason with both rule-based and conceptual knowledge on the Semantic Web, as illustrated in (Heymans et al. 2005b).
A major challenge for OASP is to control undecidability of satisfiability checking, a challenge it shares with DL-based languages. In (Heymans et al. 2005a;Heymans et al. 2006b), we identify a decidable class of programs, the so-called guarded programs, for which decidability of satisfiability checking is obtained by a translation to guarded fixed point logic (Grädel and Walukiewicz 1999). In (Heymans et al. 2006), we show the expressiveness of such guarded programs by simulating a DL with n-ary roles and nominals. In particular, we extend the DL DLR (Calvanese et al. 1997) with both concept nominals {o} and role nominals {(o 1 , . . . , o n )}, resulting in DLRO. We denote the DL DLRO without number restrictions as DLRO -{≤} . Satisfiability checking of concept expressions w.r.t. DLRO -{≤} knowledge bases can be reduced to checking satisfiability of guarded programs (Heymans et al. 2006b).
A g-hybrid knowledge base consists of a Description Logic knowledge base and a guarded program. The DL+log knowledge bases from (Rosati 2006) are weakly safe, which means that the interaction between the program and the DL knowledge base is restricted by requiring that variables which appear in non-DL atoms, appear in positive non-DL atoms in the body, where DL atoms are atoms involving a concept or role symbol from the DL knowledge base. G-hybrid knowledge bases do not require such a restriction; instead, variables must appear in a guard of the rule, but this guard can be a DL atom as well. In this paper, we show decidability of g-hybrid knowledge bases for DLRO -{≤} knowledge bases by a reduction to guarded programs, and show that satisfiability checking of g-hybrid knowledge bases is 2-EXPTIME-complete. The DL DLRO -{≤} is close to SHOIN , the Description Logic underlying OWL DL (Horrocks and Patel-Schneider 2004a). Compared with SHOIN , DLRO -{≤} does not include transitive roles and number restrictions, but does include n-ary roles and complex role expressions.
To see why a combination of rules and ontologies, as proposed in g-hybrid knowledge bases, is useful, and why the safeness conditions considered so far in the literature are not appropriate in all scenarios, consider the Description Logic ontology FraternityMember ⊑ Drinker ⊓ ∃hasDrinkingBuddy.FraternityMember which says that fraternity members are drinkers who have drinking buddies, which are also fraternity members. Now consider the logic program problemDrinker (X )
← Drinker (X ), not socialDrinker (X ) socialDrinker (X )
← Drinker (X), not problemDrinker (Y ), hasDrinkingBuddy(X , Y ) FraternityMember (John) ← which says that drinkers are by default problem drinkers, unless it is known that they are social drinkers; drinkers with drinking buddies who are not problem drinkers are social drinkers; and John is a fraternity member. From the combination of the ontology and the logic program, one would expect to derive that John is a social drinker, and not a problem drinker. This logic program cannot be expressed using r-hybrid knowledge bases, or DL+log, because the rules in the program are not weakly safe . However, the logic program is guarded, and thus part of a valid g-hybrid knowledge base, which has the expected consequences.
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