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In this paper, we present our research on defining a correct semantics for a class of update rule (UR) programs, and discuss implemanting these programs in a DBMS environment. Update rules execute by updating relations in a database which may cause the further execution of rules. A correct semantics must guarantee that the execution of the rules will terminate and that it will produce a minimal updated database. The class of UR programs is syntactically identified, based upon a concept that is similar to stratification. We extend that strict definition of stratification and allow a relaxed criterion for partitioning of the rules in the UR program. This relaxation allows a limited degree of nondeterminism in rule execution. We define an execution semantics based upon a monotonic fixpoint operator TUR, resulting in a set of fixpoints for UR. The monotionicity of the operator is maintained nby explicitly representing the effect of asserting and retracting tuples in the database. A declarative semantics for the update rule program is obtained by associating a normal logic program UR to represent the UR program. We use the stable model semantics which characterize a normal logic program by a set of minimal models which are called stable models. We show the equivalence between the set of fixpoints for UR and the set of stable models for UR. We briefly discuss implementing the fixpoint semantics of the UR program in a DBMS environment. Relations that can be updated by the rules are updatable relations and they are extended with two flags. An update rule is represented by a database query, which queries the updatable relations as well as database relaions, i.e., those relations which are not update by rules. We describe an algorithm to process the queries and compute a fixpoint in the DBMS environment and obtain a final database.