A unified framework for conjunctive and competence-based attribute functions

Abstract

Competence-based extensions of polytomous knowledge space theory have recently seen significant development, primarily through specialized conjunctive or disjunctive models. The conjunctive attribute map proposed by Stefanutti et al. (2023) assumes that a single, conjunctive set of attributes is required for each item-response, while other models, such as that of Wang et al. (2025), have explored purely disjunctive mechanisms. However, these frameworks do not readily account for scenarios where a given response can be justified by one of several distinct sets of attributes—a structure analogous to the competency model in dichotomous theory. This paper introduces a generalized attribute function which maps each item-response pair to a collection of attribute sets. This framework naturally subsumes the conjunctive model as the special case where this collection is a singleton. We establish a rigorous axiomatic foundation for this function, specified by several general conditions. We then derive the corresponding item-response function and the maximal polytomous assessment structure. We show that while our framework generalizes the conjunctive model, its disjunctive specialization is structurally distinct from other disjunctive approaches. The utility of this generalized model for capturing complex reasoning paths is demonstrated with a detailed example.