Considerations about the identification of forward- and backward-graded knowledge structures

The application of the basic local independence model (BLIM) to a knowledge structure (Q,K) that satisfies a particular kind of gradation (namely forward- or backward-gradedness) leads the model to be not identifiable. In the present article, we show that many important types of knowledge structures happen to be either forward- or backward-graded. This means that the application of the BLIM to these structures leads to unidentifiable models. No universal remedy for recovering identifiability is presently known. However, we propose a construction that consists in introducing an equally informative item for each item in Q. We conjecture that the BLIM based on the resulting knowledge structure is always identifiable. This conjecture is proven to be true for knowledge structures on small sets of items.