Knowledge-based systems in the Semantic Web era can make use of the power of the Semantic Web languages and technologies, in particular those related to ontologies. Recent research has shown that user-defined data types are very useful for Semantic Web and ontology applications. The W3C Semantic Web Best Practices and Development Working Group has set up a task force to address this issue. Very recently, OWL-Eu and OWL-E, two decidable extensions of the W3C standard ontology language OWL DL, have been proposed to support customized data types and customized data type predicates, respectively. In this paper, we propose a flexible reasoning architecture for these two expressive Semantic Web ontology languages and describe our prototype implementation of the reasoning architecture, based on the well-known FaCT DL reasoner, which witnesses the two key flexibility features of our proposed architecture: 1) It allows users to define their own data types and data type predicates based on built-in ones and 2) new data type reasoners can be added into the architecture without having to change the concept reasoner.