Knowledge spaces with graded knowledge states

Knowledge spaces represent a framework for assessment of knowledge for which solid theoretical foundations, methodology, software tools and numerous practical applications are available. The underlying assumption in knowledge spaces is that a knowledge state of an individual is represented by a set of items/questions which the individual has mastered/is capable of answering. In this paper, we propose an extension of knowledge spaces which accounts for gradedness of knowledge states. Namely, more often than not, to assess whether an individual has mastered an item is a matter of degree. We assume that a knowledge state is represented by a graded set with grades representing degrees to which an individual has mastered the items. The scale of grades becomes a parameter of our approach. If 0 and 1 are the only grades, our approach coincides with that of ordinary knowledge spaces. We develop basic concepts and results in the graded setting including bases of graded knowledge states and their computation and a logic of partial failure with its completeness theorem. We also present illustrative example. Our main aim is to demonstrate mathematical and computational feasibility of knowledge spaces with graded knowledge states.