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A commonplace assumption in the fault diagnosis of discrete event systems (DESs) is that of modeling faulty events with unobservable transitions, i.e., transitions whose occurrence does not produce any observable label. The diagnostic system must thus infer the occurrence of a fault from the observed behavior corresponding to the firing of nonfaulty transitions. The presence of nonfaulty unobservable transitions is a source of additional complexity in the diagnostic procedure. In this paper, we assume that fault events can also be modeled by observable transitions, i.e., transitions whose occurrence produces an observable label. This does not mean, however, that the occurrence of such a transition can be unambiguously detected: In fact, the same label may be shared with other fault transitions (e.g., belonging to different fault classes) or with other nonfaulty transitions. We generalize to this new setting our previous results on the diagnosis of DESs using Petri nets based on the notions of minimal explanations and basis markings. The presented procedure does not require the enumeration of the complete reachability set but only of the subset of basis markings, thus reducing the computational complexity of solving a diagnosis problem.