The inclusion degrees of fuzzy skill maps and knowledge structures.

Fuzzy skills can provide a representation of the latent cognitive abilities of individuals in terms of proficiency. A disjunctive model of the fuzzy skill map, in which individuals master at least one skill to reach the minimum requirement, always delineates a knowledge space. At the same time, a conjunctive model delineates a simple closure space, which needs individuals' proficiency of all skills to reach the minimum requirements. The restriction of the disjunctive model is too weak, and the conjunctive model is too strongly constrained. Between these two models, there is a lack of detailed classification on different numbers of skills that meet the minimum requirements. Thus, for a given fuzzy skill map (Q,S,τ), it is natural for us to propose the concept of a fuzzy skill inclusion degree, which is a ratio of the number of skills meeting the minimum requirements to the number of skills needed to solve the item q. This ratio can describe levels of different numbers of skills mastered, and it is the basis of the variable precision α-models in the following discussion. By the fuzzy skill inclusion degree, the variable precision α-models can divide knowledge structures into different levels between the disjunctive model and the conjunctive model. Meanwhile, both the disjunctive and conjunctive models become two special cases of the variable precision α-model. The knowledge structures delineated via disjunctive models are classified into different levels by α. The dual to the knowledge structure delineated via the variable precision α-model is also discussed. For a fuzzy skill multimap (Q,S,μ), the variable precision α-competency models of a fuzzy skill multimap are found, and different knowledge structures can be obtained through different values of α. When α=1, the variable precision α-competency model becomes a competency model of the fuzzy skill multimap.