So far no great importance has been attached to labelling the links in a graph, although it is clearly possible to do so. But often labelled links are supposed to belong to different classes, implying that the links should be distinguished accordingly. In turn the classes would have individual connectivity matrices. Instead of just tracing paths through the diagram, it could be required that the labels follow a certain sequence, with the attendant question of whether the required path actually exists; a question that could be resolved by multiplying the corresponding connectivity matrices in the prescribed order.
Frequently the question is more general --- does the required path exist anywhere at all in the diagram? One approach would be to list all the paths, so as to search the list for the presence of the candidate. Something of the nature can hardly be avoided, so it is a matter of organizing the technique, which is usually known as Moore's subset construction [17], which was introduced in 1956 by Edward F. Moore for a different reason.