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A dialogue model is an abstract model that is used to describe the structure of the dialogue between a user and an interactive computer system. Dialogue models form the basis of the notations that are used in user interface management systems (UIMS). In this paper three classes of dialogue models are investigated. These classes are transition networks, grammars, and events. Formal definitions of all three models are presented, along with algorithms for converting the notations into an executable form. It is shown that the event model has the greatest descriptive power. Efficient algorithms for converting from the transition diagram and grammar models to the event model are presented. The implications of these results for the design and implementation of UIMSs are also discussed.