This paper deals with the problem of diagnosability of a fault after the firing of a finite number events (i.e., K-diagnosability). This problem corresponds to diagnosability of a fault within a finite delay in the context of discrete event systems. The main contribution of this paper is a necessary and sufficient condition for K-diagnosability of bounded nets. The proposed approach exploits the mathematical representation of Petri nets and the Integer Linear Programming optimization tool. In particular no specific assumptions are made on the structure of the net induced by the unobservable transitions, since the proposed approach permits to detect also the undiagnosability due to the presence of unobservable cycles. (C) 2012 Elsevier Ltd. All rights reserved.