Atomic representation and algorithms for polytomous knowledge spaces.

Classical knowledge space theory provides a rigorous framework for cognitive diagnosis, but its dichotomous response model fails to capture the graded nature of knowledge. While recent research has extended KST to polytomous responses through reductionist approaches, their practical adoption is hindered by computational complexity and the lack of construction methods. This paper introduces a novel framework based on polytomous closure spaces with three key contributions. First, we establish the theory of these spaces alongside an atomic decomposition that enables compact state representation. Second, we characterize granularity conditions that ensure complete atomic decompositions and establish the bijective correspondence between knowledge spaces and their atomic bases. Third, we develop algorithms for base extraction and knowledge space generation that leverage the atomic structure to reduce complex state operations to set computations. The theoretical framework maintains mathematical rigor through lattice-theoretic foundations while achieving computational tractability, providing a practical foundation for adaptive assessment of graded knowledge.