The dichotomous knowledge structure theory was introduced to polytomous items, which is proved meaningful in representing individuals' partial mastery of items. Given that different proficiencies in skills may be required to solve polytomous items, fuzzy skills can be applied to represent latent cognitive abilities of individuals. Then we consider how to construct polytomous knowledge structures by fuzzy skills. To characterize relationships between polytomous items and fuzzy skills related to solving these items, we introduce the concept of fuzzy skill function on polytomous items. Given a set $Q\times L$ of polytomous items and a set S of skills, a fuzzy skill function $(Q\times L,S,\sigma )$ assigns each response category $(q,l)\in Q\times L$ with a set ${\sigma }_{(q,l)}$ of fuzzy sets on S, where each element in ${\sigma }_{(q,l)}$ represents an approach for solving q to the level l. Notice that there are two special kinds of fuzzy skill functions: one is disjunctive, while the other is conjunctive. The former delineates polytomous knowledge spaces, while the latter delineates polytomous closure spaces. Based on these facts, we design algorithms for delineating the two kinds of polytomous knowledge structures. For any fuzzy skill function on polytomous items, all competencies and all weak competencies are enough to delineate a polytomous knowledge structure. Thus, we design an algorithm for delineating polytomous knowledge structures via fuzzy skill functions on polytomous items. This algorithm only depends on competencies and weak competencies of fuzzy skill functions on polytomous items. Besides, we take “Addition and subtraction of fractions” as knowledge domain to illustrate the process of constructing polytomous knowledge structures from fuzzy skills.

将二分知识结构理论引入到多分项中，证明对代表个体对项的部分掌握具有重要意义。鉴于解决多头项目可能需要不同的技能熟练程度，可以应用模糊技能来表示个人的潜在认知能力。然后我们考虑如何通过模糊技能来构建多分知识结构。为了表征多分项与解决这些项相关的模糊技能之间的关系，我们在多分项上引入了模糊技能函数的概念。给定一个集合$问\times \mathrm{大号}$多分项和一组S技能，一个模糊技能函数$(问\times \mathrm{大号},\mathrm{小号},\sigma )$分配每个响应类别$(q,l)\in 问\times \mathrm{大号}$用一套${\sigma }_{(q,l)}$S上的模糊集，其中每个元素${\sigma }_{(q,l)}$表示将q求解到水平l的方法. 请注意，有两种特殊的模糊技能函数：一种是析取的，另一种是合取的。前者描绘了多分支的知识空间，而后者描绘了多分支的闭包空间。基于这些事实，我们设计了描述这两种多分知识结构的算法。对于多分项上的任何模糊技能函数，所有胜任力和所有弱胜任力都足以描绘出一个多分知识结构。因此，我们设计了一种算法，通过对多项项目的模糊技能函数来描述多项知识结构。该算法只依赖于模糊技能函数对多分项的胜任力和弱胜任力。除了，