Dimensions of knowledge structures

Abstract

A knowledge structure is inherently one-dimensional when its collection of states forms a chain. But how to define the dimension of a knowledge structure in general? We investigate four options: (i) the ordinal dimension, which is the dimension of the poset consisting of all states ordered by inclusion; (ii) for a knowledge space, the spatial dimension which is the least number of one-dimensional knowledge spaces which generate the space (a notion extending from learning spaces to knowledge spaces the dual of the convex dimension of an antimatroid); (iii) the bidimension, which is the bidimension of the membership relation from items to states, in either the intersection or the union version of the bidimension. Our results establish or disprove inequalities among the four dimension parameters for knowledge structures, for knowledge spaces, for terse knowledge structures, for terse knowledge spaces, and finally for learning spaces. We finally list some problems for future research.