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...nal closure is no harder than global entailment checking. Furthermore, the algorithm can readily be extended to implement refinements presented by presumptive reasoning [18], and inheritance networks =-=[7]-=-. We are currently investigating further optimizations of the algorithm via suitable definitions of modularity for defeasible knowledge bases [15,16]. 9 Concluding Remarks The contributions of this pa...

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...ed, and different orders determine different sets of consequences. The authors show that, for each order s, the consequence relation ⊩s is rational and can be computed in PSPACE. In a subsequent work =-=[6]-=-, the authors introduce an approach based on the combination of rational closure and Defeasible Inheritance Networks (INs). The logic ALC R T we consider as our base language is equivalent to the logi...

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...r to extend the basic formalism of DLs with nonmonotonic reasoning features. The traditional approach is to handle defeasible inheritance by integrating some kind of nonmonotonic reasoning mechanisms =-=[5, 2, 10, 3, 4]-=-. A simple but powerful nonmonotonic extension of DLs is proposed in [6]. In this approach, “typical” or “normal” properties can be directly specified by means of a “typicality” operator T enriching t...

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...ng the evolution from frame systems to description logics (DLs), nonmonotonic inferences and constructs (such as those supported by the LOOM system in the 1990s) have disappeared from the mainstream. However, a range of knowledge engineering requirements kept interest in nonmonotonic DLs alive, see e.g. [21,23,5] for more details. In fact, along the years all of the major nonmonotonic semantics have been adapted to DLs, including the integration of default rules and DLs [2,12,20,16,17], circumscription [7,22], and variations of autoepistemic logics, preferential semantics and rational closure [8,11,14,15,10]. In this paper, we focus on circumscription, which was first applied in the DL context by Gerd Brewka to whom this volume is dedicated [7]. The general idea of circumscription is to select a subclass of the classical models of the knowledge base by minimizing the extension of some selected predicates that represent abnormal situations. During minimization, the interpretation of the other predicates can be fixed or vary freely. To achieve a faithful modeling, in addition it is often necessary to allow a preference order on the minimized predicates, that is, if P1 is preferred to P2, then we al...

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...ows: K (rc C |„D iff rKpCq ă rKpC ^ D) or rKpCq “ 8. We let RCpKq “ tC |„D |K (rc C |„Du. We extend the notion of coherence to conditional knowledge bases: C is coherent for K iff K *rc C $ K iff K *rc C |„K. K is inconsistent iff K (rc J $ K. Two knowledge bases are rank equivalent iff their Rational Closures are equal. We chose to use RC in this paper for a number of reasons. Most importantly, below we are going to investigate a form of belief contraction of conditionals of the form C |„ K. It turns out that all the prominent entailment relations proposed in the preferential framework, e.g. [11, 13, 12, 4, 3], are equivalent for such conditionals (C |„ K is a consequence of K in one of them iff it is a consequence in all of them). So, if we model a contraction operator for RC, we can immediately apply the same operator to all the other entailment relations in this family. Note that only one half of the Deduction Theorem holds for defeasible conditionals in the context of RC, but the other direction doesn’t. Proposition 1 If K (rc C ^ D |„E then K (rc C |„ D _ E. However, it may be that K *rc C ^D |„E when K (rc C |„ D _ E. So, C |„ D _ E can be viewed as a weakening of C ^D |„E in the context of R...

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...ed, and different orders determine different sets of consequences. The authors show that, for each order s, the consequence relation ⊩s is rational and can be computed in PSPACE. In a subsequent work =-=[5]-=-, the authors introduce an approach based on the combination of rational closure and Defeasible Inheritance Networks (INs). 5 Conclusions In the first part of the paper we have provided a semantic rec...