A great deal of research has been devoted to nontrivial reasoning in inconsistent knowledge bases. Coherence-based approaches proceed by a consolidation operation which selects several consistent subsets of the knowledge base and an entailment operation which uses classical implication on these subsets in order to conclude. An important advantage of these formalisms is their flexibility: consolidation operations can take into account the priorities of declarations stored in the base, and different entailment operations can be distinguished according to the cautiousness of reasoning. However, one of the main drawbacks of these approaches is their high computational complexity. The purpose of our study is to define a logical framework which handles this difficulty by introducing the concepts of anytime consolidation and anytime entailment. The framework is semantically founded on the notion of resource which captures both the accuracy and the computational cost of anytime operations. Moreover, a stepwise procedure is included for improving approximations. Finally, both sound approximations and complete ones are covered. Based on these properties, we show that an anytime view of coherence-based reasoning is tenable.
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