知识空间理论使用数学语言对学习者进行知识评价与学习指导, 属于数学心理学的研究范畴. 技能与问题是构成知识空间的两个基本要素, 深入研究两者之间的关系是知识状态刻画与知识结构分析的内在要求. 在当前的知识空间理论研究中, 没有明确建立技能与问题之间的双向映射, 从而难以提出直观概念意义下的知识结构分析模型, 也没有明确建立知识状态之间的偏序关系, 不利于刻画知识状态之间的差异, 更不利于规划学习者未来的学习路径. 此外, 现有的成果主要集中在经典的知识空间, 没有考虑实际问题中数据的不确定性. 为此, 将形式概念分析与模糊集引入知识空间理论, 建立面向知识结构分析的模糊概念格模型. 具体地, 分别建立知识空间与闭包空间的模糊概念格模型. 首先, 建立知识空间模糊概念格, 并通过任意两个概念的上确界证明所有概念的外延构成知识空间. 引入粒描述的思想定义技能诱导的问题原子粒, 由问题原子粒的组合判定一个问题组合是否是知识空间中的一个状态, 进而提出由问题组合获取知识空间模糊概念的方法. 其次, 建立闭包空间模糊概念格, 并通过任意两个概念的下确界证明所有概念的外延构成闭包空间. 类似地, 定义问题诱导的技能原子粒, 由技能原子粒的组合判定一个技能组合是否是闭包空间中某一知识状态所需的技能, 进而提出由技能组合获取闭包空间模糊概念的方法. 最后, 通过实验分析问题数量、技能数量、填充因子以及分析尺度对知识空间与闭包空间规模的影响. 结论表明知识空间模糊概念不同于现有的任何概念, 也不能从其他概念派生而来. 闭包空间模糊概念本质上是一种面向属性单边模糊概念. 在二值技能形式背景中, 知识空间与闭包空间中的状态具有一一对应关系, 但这种关系在模糊技能形式背景中并不成立.

Knowledge space theory, which uses mathematical language for the knowledge evaluation and learning guide of learners, belongs to the research field of mathematical psychology. Skills and problems are the two basic elements of knowledge space, and an in-depth study of the relationship between them is the inherent requirement of knowledge state description and knowledge structure analysis. In the existing knowledge space theory, no explicit bi-directional mapping between skills and problems has been established, which makes it difficult to put forward a knowledge structure analysis model under intuitive conceptual meanings. Moreover, the partial order relationship between knowledge states has not been clearly obtained, which is not conducive to depicting the differences between knowledge states and planning the learning path of learners. In addition, the existing achievements mainly focus on the classical knowledge space, without considering the uncertainties of data in practical problems. To this end, this study introduces formal concept analysis and fuzzy sets into knowledge space theory and builds the fuzzy concept lattice models for knowledge structure analysis. Specifically, fuzzy concept lattice models of knowledge space and closure space are presented. Firstly, the fuzzy concept lattice of knowledge space is constructed, and it is proved that the extents of all concepts form a knowledge space by the upper bounds of any two concepts. The idea of granule description is introduced to define the skill-induced atomic granules of problems, whose combinations can help determine whether a combination of problems is a state in the knowledge space. On this basis, a method to obtain the fuzzy concepts in the knowledge space from the problem combinations is proposed. Secondly, the fuzzy concept lattice of closure space is established, and it is proved that the extents of all concepts form the closure space by the lower bounds of any two concepts. Similarly, the problem-induced atomic granules of skills are defined, and their combinations can help determine whether a skill combination is the skills required by a knowledge state in the closure space. In this way, a method to obtain the fuzzy concepts in the closure space from the skill combinations is presented. Finally, the effects of the number of problems, the number of skills, the filling factor, and the analysis scale on the sizes of knowledge space and closure space are analyzed by some experiments. The results show that the fuzzy concepts in the knowledge space are different from any existing concept and cannot be derived from other concepts. The fuzzy concepts in the closure space are attribute-oriented one-sided fuzzy concepts in essence. In the formal context of two-valued skills, there is one-to-one correspondence between the states in knowledge space and closure space, but this relationship does not hold in the formal context of fuzzy skills.

国家自然科学基金(11971211);河南省自然科学基金(222300420445);河南省高校基本科研业务费专项(NSFRF210318)