This paper describes the problem of doing description logic (DL) reasoning with partially closed world. The issue was addressed by extending the syntax of DL SROIQ with an NBox, which specifies the predicates to close, extending the semantics with the idea of negation as failure, reducing the closed world reasoning to incremental reasoning on classical DL ontologies, and applying the syntactic approximation technology to improve the reasoning performance. Compared with the existing DBox approach, which corresponds to the relation database, the NBox approach supports deduction on closed concepts and roles. Also, the approximate reasoning can reduce reasoning complexity from N2EXPTIME-complete to PTIME-complete while preserving the correctness of reasoning for ontologies with certain properties.