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Discusses ad hoc frame-oriented CAI; information structure oriented CAI; intelligent tutoring systems; mixed initiative; intersection search; deep insight; strategy graphs; models of inference systems; truth maintenance systems; ideal and 'buggy' rules; the system configuration; the semantic network for the knowledge representation of a geometrical figure; the atomic knowledge for representing such a figure; the meanings of the link labels in a semantic network; the representation of a geometrical theorem; the knowledge base representing the text structure of a geometrical problem; the process for identifying partial trees; the function working within the dialogue process; the rules of the dialogue procedure; an example of a dialogue; and the relationship between working memories in the production system for the geometrical proof.