On the necessary and sufficient conditions for delineating forward- and backward-graded knowledge structures from skill maps

In recent years two classes of knowledge structures received particular attention. They are, respectively, forward-graded and backward-graded knowledge structures. Their importance is related to the fact that the basic local independence model applied to such structures turns out to be locally unidentifiable. Spoto et al. (2012) established some sufficient conditions for a skill multimap to delineate forward- and/or backward graded knowledge structures. Drawing upon some of the theoretical results contained in that article, the present theoretical note aims at providing necessary and sufficient conditions for a skill map to delineate a forward- and/or backward graded knowledge structure.