Morphological rules of similarity for hierarchical distributed representations

This paper presents and discusses four new criteria that are able to correctly identify hierarchical relationships (in top-down and bottom-up fashion) created upon binary distributed representations. Each criteria is mathematically formulated in terms of two separate rules of similarity, each one being able to identify one type of hierarchical relationship. The rules are also presented in terms of a given frame of reference in accordance with the hierarchical distributed representations. Several mathematical correspondences are shown among different criteria and the rules that compose them. It is proven that the new rules used to identify hierarchical relationships among two patterns represent the mathematical decomposition of the hamming distance among them. The combination of the rules able to identify one type of hierarchical relationship can be used to identify particular cluster of patterns with special meaning. This diminishes the need of defining state spaces with high-dimensionality.