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This paper describes the design and implementation of the Ariel DBMS and its tightly-coupled forward-chaining rule system. The query language of Ariel is a subset of POSTQUEL, extended with a new production-rule sublanguage. Ariel supports traditional relational database query and update operations efficiently, using a System R-like query processing strategy. In addition, the Ariel rule system is tightly coupled with query and update processing. Ariel rules can have conditions based on a mix of selections, joins, events, and transitions. For testing rule conditions, Ariel makes use of a discrimination network composed of a special data structure for testing single-relation selection conditions efficiently, and a modified version of the TREAT algorithm, called A-TREAT, for testing join conditions. The key modification to TREAT (which could also be used in the Rete algorithm) is the use of virtual驴-memory nodes which save storage since they contain only the predicate associated with the memory node instead of copies of data matching the predicate. In addition, the notions of tokens and 驴-memory nodes are generalized to support event and transition conditions. The rule-action executor in Ariel binds the data matching a rule's condition to the action of the rule at rule fire time, and executes the rule action using the query processor.