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1 Introduction In computer science and artificial intelligence,an ontology is a model of (some aspect of) the world that introduces the vocabulary of a particular domain,and specifies the meaning (semantics) of terms.Today’s ontologies are based on the description logics (DLs).With the help of DL reasoners,implicit knowledge can be inferred from explicit knowledge in an ontology.Different from the relational database or logic programming,ontologies impose the open world assumption (OWA) instead of the closed world assumption (CWA).In CWA,a proposition will be regarded as false,if there is no evidence for it being true.While in OWA,a proposition will be regarded as false if and only if there is clear.evidence for it being false.These two assumptions seem to contradict with each other.However,in real world applications,it is usually desired to combine OWA and CWA.For example,a restaurant should not close the class ‘Vegetarian Customer’ because whether a new customer is vegetarian or not is unknown to the restaurant unless explicitly asserted by the customer.Nevertheless,it can close the class ‘Vegetarian Food Menu’ because it has the complete knowledge about its own menus.This raises the challenge of doing local closed world reasoning (LCWR) with ontologies.In the following sections of this extended abstract,we review the state of the art regarding this topic.We also propose several emerging topics in this research area and discuss possible technical directions.