On the polytomous generalization of knowledge space theory

Abstract One of the core assumptions of knowledge space theory (KST) is that the answer of a subject to an item can be dichotomously classified as correct or incorrect. Schrepp (1997) provided a very first attempt to generalize the main KST concepts to items with more than two response alternatives, but his work has not had a strong impact on the subsequent research on KST. The aim of the present article is to introduce a new formulation of the polytomous KST, starting from the work of Schrepp and broadening it to a wider extent. Schrepp’s generalization is revisited, and the fundamental closure conditions are reformulated and decomposed into a necessary and sufficient set of four independent properties of polytomous knowledge structures. Among them, two special properties emerge in the polytomous case that in the dichotomous one are neither testable nor immediately visible, since necessarily true. These properties allow for a straight generalization of Birkhoff’s Theorem with respect to quasi-ordinal knowledge spaces, and Doignon and Falmagne’s Theorem for knowledge spaces. Such findings open the field to a systematic generalization of many KST concepts to the polytomous case.

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