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The problem of automatic fault diagnosis of systems decomposed into a number of interconnected units is considered by using a simplified version of the diagnostic model introduced by Preparata et al. The model used in this paper is supposed to be a realistic representation of systems where each unit has a considerable computational capability. For any system of n units whose set of testing links is given, necessary and sufficient conditions for t-diagnosability are presented in both cases of one-step diagnosis and diagnosis with repair, and it is shown that the procedure for diagnosis with repair has very small complexity. The problem of optimal assignment of testing links in order to achieve a given diagnosability is also considered and classes of optimal t-diagnosable systems are presented for arbitrary values of t in both cases of one-step diagnosis and diagnosis with repair.