Knowledge spaces with graded knowledge states

Knowledge spaces represent a framework for assessment of knowledge with solid theoretical foundations, methodology, software tools, and practical applications. The underlying assumption in knowledge spaces is that a knowledge state of an individual is represented by a set of items which the individual has mastered. In this paper, we propose an extension of the theory of knowledge spaces which accounts for gradedness of knowledge states. Namely, we assume that a knowledge state is represented by a fuzzy (graded) set with degrees representing levels to which an individual has mastered the items. If 0 and 1 are the only degrees, 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 an illustrative example. The main aim of this paper is to demonstrate mathematical and computational feasibility of knowledge spaces with graded knowledge states.