Probabilistic normalization conditions of polytomous knowledge structures

Constructing a probability distribution of knowledge states reasonably is a fundamental and important issue in dichotomous probabilistic knowledge structure (PKS). Recently, the dichotomous knowledge structure has been extended to the polytomous knowledge structure, but the establishment of the probability distribution of the polytomous knowledge state has not yet been constructed. So it is necessary to find the relevant definition and normalization conditions for its establishment. To this end, this article builds two appropriate probability definitions of the polytomous knowledge state from different perspectives, and proves the equivalence of the two definitions. In addition, we obtain the probabilistic normalization conditions for the polytomous knowledge structure (Q,L,K), where (L,≤) is a linear order and supply some examples. The results generalize the according conclusions of the dichotomous knowledge states of learning space. Moreover, such findings provide the way for the construction of a number of probabilistic models with polytomous data for the applications of knowledge structure theory (KST).