Forward-graded and backward-graded structures of knowledge are two important classes of knowledge structures. Spoto and Stefanutti (2020) establish necessary and sufficient conditions for skill maps to delineate these structures. We introduce fuzzy skills to describe varying levels of proficiency in skills and extend the theoretical results of Spoto and Stefanutti (2020) for delineating forward- and backward-graded knowledge structures using fuzzy skill maps. The paper establishes necessary and sufficient conditions for fuzzy skill maps to delineate a backward-graded simple closure space, a forward-graded knowledge space, and a forward-graded simple closure space. Furthermore, the competence-based local independence model (CBLIM) with fuzzy skills is introduced and its unidentifiability is discussed.

知识结构的前向和后向是两类重要的知识结构。Spoto 和 Stefanutti (2020) 为技能图描绘这些结构建立了必要和充分的条件。我们引入模糊技能来描述不同的技能熟练程度，并扩展了 Spoto 和 Stefanutti (2020) 的理论结果，使用模糊技能图来描述前向和后向分级的知识结构。文章建立了模糊技能图的充要条件来划分后向分级的简单闭包空间、前向分级的知识空间和前向分级的简单闭包空间。此外，引入了基于能力的模糊技能局部独立模型（CBLIM），并讨论了其不可识别性。