We propose an algebraic approach to investigate K -diagnosability of partially observed labeled Petri nets which can be either bounded or unbounded. Namely, a necessary and sufficient condition for K-diagnosability is established based on the resolution of an Integer Linear Programming (ILP) problem. When the system is K -diagnosable, our approach also yields the minimal value K min ? K that ensures K min -diagnosability. The value of K min is calculated directly, using the same ILP formulation, i.e, without testing 1 , . . . , (K min ? 1)-diagnosability. A second K -diagnosability approach, which is derived from the first one, is also developed on a compacted horizon providing a sufficient condition for K-diagnosability. This second technique allows for reducing the system dimensionality yielding a higher computational efficiency and allowing the characterization of the length of the sequences that lead to the fault occurrence, which is necessary to perform the K -diagnosability test of the first approach.